Good morning. In the previous lecture, we discussed the differential calculus part of complex analysis. Today, in this lecture, we go into the integral calculus part of it, and we study integrals in the complex plane. Our two main topics of discussion will be Cauchy's integral theorem and Cauchy's integral formula. Before going into those two topics, we spend a little time to build the ground for it. First, we discuss the concept of line integral in the complex plane.  Then, we can define its line integral over a smooth curve c in this manner.  And then, the product of this, these two will give us the real part as u d x and minus v d y, which is here. And the imaginary part will be v d x plus u d y, which is here. Now, this real integral plus i into this real integral will give us the line integral of f z. So, this will be a complex function, obviously, because this is the real part and this is the imaginary part. Now, this we have defined for the time being over a smooth curve c. Now, extension of this same definition over piecewise smooth curve is obvious, because if the smoothness is interrupted at, say, a finite number of points, then considering those finite number of points as the end point of one segment of the curve and starting point of the next segment of the curve, we can build the same line integral over several segments. So, now, if particularly if we have a parameterization of the curve z in the form of z of t with respect to a real parameter t, which varies from a to b. That is, as this parameter t varies from real number a to real number b over an ordinary interval, then if the corresponding z of t traces that curve with this derivative never vanishing, then this line integral can be reduced to a simple definite integral, because in that case, we can put f z as f of z of t, which means this turns out to be in terms of t and then d z can be put in terms of z dot into d t. And in that case, this entire thing f and z dot, the product of f and z dot becomes the integrand and integral is with respect to this real parameter t and then it is a definite integral. So, with the parameterization of the curve c available, the line integral can be reduced to a definite integral in real variable quite easily. Now, if the closed curve, if the simple, if the smooth curve c is taken as a closed curve, simple closed curve without self-interceptions, then you can talk of contour integrals, which is integral over a closed curve and then the symbol is used like this. Let us take an example, which will serve two purposes, one to demonstrate the evaluation of integrals using this kind of formulae and second to have a little result in hand, which we will be using later for a very important step in later derivations. We try to find out this particular integral for integer n, that is z to the power n, simple function z to the power n, we try to find the integral of this around a circle, which is centered at 0 at origin and with radius rho. Then here, you see that the variable theta is the parameter, it takes the place of t in this formulation. In that case, this for evaluating this contour integral over the circular contour, we consider z as this and then d z will be i into this whole thing, because the derivative of this will be rho into e to the power i theta into i. So, from this step to this step, n powers of rho come from this and the last, another additional power comes from here, from the d z. So, rho to the power n plus 1 and since rho is constant, it is a circular contour. So, that we can take outside the integral sign and then similarly, e to the power i theta, n powers of that come from here and one power comes from here. So, e to the power i theta also is raised to n plus 1 power and in the derivative of this factor i comes, which is here. So, this entire thing has been now reduced to this and now since here, the variable of integration is theta. So, the integral is evaluated over theta with the limits 0 to 2 pi, one circle will have 0 to 2 pi interval of the parameter angle theta. Now, you see that the contour integral has been reduced to an ordinary definite integral in this manner. Now, if we try to evaluate this integral, we notice that if n is equal to minus 1, then this entire power vanishes and this entire integral integrand becomes 1 and in that case, it is a simple integral of a constant function 1 and that will give us 2 pi and therefore, for n is equal to minus 1, you get 2 pi from here and you get 2 pi i as the integral. On the other hand, if you have n as anything other than minus 1, say 0, 1, 2, 3 and so on, then this turns out to be an integer, which is going to give us e to the power i phi, where phi is n plus 1 theta. Now, we know that e to the power i phi is cos phi plus i sin phi and as we try to integrate that, we will find that both cos phi and sin phi integrated over 0 to 2 pi will produce 0. So, real part as well as imaginary part will give us 0 and therefore, n not equal to minus 1, we get this integral as 0. So, this small result we will keep in hand, which will be useful later. Now, another small result, which will be quite useful in later discussion is conveniently named as the M L inequality. Now, here what we say is that if c is a curve of finite length L and f z is a function, which is bounded and the bound is M, bounded means its absolute value is bounded. So, nowhere is the value of f z of absolute value greater than M. So, if the value of f z, its absolute value is bounded by M over that curve, then we say that the absolute value of this integral will be always less than equal to what we would get if we replace this f z and d z by their absolute values. So, then this is 1 less than equal to that is this is 1 inequality and then rather than f z absolute value, if we put its supremum M, then this side will be further increased. So, therefore, this will be less than equal to actually less than because of this strict inequality, this thing will be less than the integral that we would get if in place of this absolute value of f z, if we put the supremum value M and that value being constant can be taken out of the integral. And then the absolute value of d z integrated over c actually gives us the length of the curve and that is M L. So, what we get as a result? The result is that if the value of the function, absolute value of the function is bounded by this M and the curve is of finite length L, then this absolute value of the line integral is bounded by the product of M L and L. So, this is a convenient result which will be of use later also. Now, we come to one of the important topics of our discussion and that is Cauchy's integral theorem. The premises are that c is a simple closed curve, closed curve and non intersecting. In a simply connected domain d, we have already discussed what is a simply connected domain in the context of vector calculus. Here in the complex plane, it is the similar domain in which there is no interior boundaries, no interior whole, a single boundary which is also non self-intersecting that is simply closed curve, simple closed curve. Then if we take this function which we consider as analytic in this domain. Now, what we have? We have a simply connected domain d in that we have an analytic function which is every analytic everywhere in this domain and we have a simple closed curve c in that domain and then we want to evaluate what is the contour integral of that function over that particular closed curve. So, if the derivative, now one point is already clear that since the function is analytic, so it will possess derivative. So, if that derivative is continuous, then we can apply Green's theorem to develop this integral, this is the integral. So, here we talk of this integral and we remember what is Green's theorem in the plane and that is the line integral of a function with components f 1, f 2 which is given like this f 1 d x plus f 2 d y over a simple closed curve c is given by the double integral over the region enclosed by c of del f 2 by del x minus del f 1 by del y. Simply stated, line integral of function f over the closed curve c turns out to be equal to the surface integral of its curl over the surface bounded by c. Now, that is the Stokes's theorem and the special case is Green's theorem in the plane.     So, minus del v by del x minus del f 1 by del y del f 1. So, f 1 is u. So, we will have minus del u by del y.  So, we will have here v is taking the place of f 1. So, and v and u is taking the place of f 2. So, u here del u by del x minus del v by del y this will be the imaginary part. And then you will find that since the function f is analytic that means u and v it is real and imaginary components will satisfy the Cauchy Riemann conditions which means del u by del x is equal to del v by del y and del u by del y will be equal to minus del v by del x. That means both of these integrants will vanish and therefore the entire integral will vanish. So, by applying Green's theorem in the plane in this case we get this as we have developed here. And then Cauchy Riemann conditions will imply that both of these integrants will vanish and we will get the integral as 0. Now, one important premise that we have used is that the derivative is continuous. Now, this makes the proof simple. However, there is a more rigorous proof by Gorsuch which does not use the hypothesis of continuity of f from z. That gives a little formal advantage to the proof by Gorsuch in the sense that in that case if the continuity of the derivative is not used even without using it if it is proved then the continuity of f prime appears as a consequence. Therefore, many authors will many authors consider the name this Cauchy's theorem as Cauchy Gorsuch theorem. And what it means? It means that if f z is analytic in a simply connected domain D then the contour integral over a simple closed curve is 0 for every simple closed curve in C in D. That means that if in the domain the function is analytic everywhere then whatever closed curve simply closed simply a simple closed curve we can frame in that. So, around every contour that analytic that integral contour integral will vanish. That means there is no information content of the function in a domain in which it is analytic everywhere. This integral will always vanish over every contour. Now, as we continue in that path so we find that on a simple closed curve if we take two points say this is a simple closed curve. So, we take this as one point and this as another point. Now, around this contour if the contour integral turns out to be 0 then this is one line integral from z 1 to z 2 through this path and z 1 to z 2 through this path is another contour integral. So, if we take another line integral so now this contour integral from z 1 to z 2 and back to z 1 is actually the sum of two line integrals z 1 to z 2 along this curve and then z 2 to z 1 along this curve which is the second one is equivalent to the negative of the line integral from z 1 to z 2 along this curve. So, this minus this this line integral minus this line integral will mean this line integral plus this line integral which is a contour integral which should vanish which is 0. So, that means if you take two points z 1 and z 2 on the closed curve c then these two points will open the contour open the closed curve into two open curves open path c 1 and c 2 c 1 as this and c 2 as this both from z 1 to z 2 and then as we apply Cauchy's theorem on c with the two parts c 1 in the forward direction and c 2 in the reverse direction then we will find that the contour integral turns out to be integral over line integral over c 1 minus line integral over c 2 which should vanish by Cauchy's theorem. Then that will mean that this line integral and this line integral are equal and then we can say that this line integral depends only on the end points and not on the curve along which we are going because as long as we keep these two end points in hand for any other curve we can argue in the same manner. So, this is the important result that we get that is for an analytic function in a simply connected domain this line integral from one point to another is independent of the path and depends only on the end points as long as the path is completely contained in D. As outside D there is no guarantee of analyticity in D we have taken the premise of analyticity. So, analyticity is the fundamental requirement here. Now, as a consequence we can talk of this that is keeping one fixed point z 0 as say z 1 in place of z 1 we keep one point fixed z 0 and the other point if we keep a variable point z then we can define a new function of z and call it say capital F. Now, what does this formulation suggest? This formulation is looking similar to the definition of an indefinite integral because we are keeping this z as the complex variable itself and as a result we are not getting a value of the integral, but we are getting a function of z. Now, this will suggest the idea of indefinite integral, but then the implication of a definite integral the direct meaning of a definite integral is the sum of infinite number of small infinitesimal contributions that we have already considered. However, the idea of indefinite integral is that of a function capital F whose derivative turns out to be this integral. Then before formally accepting this function define in this manner as the indefinite integral of small f we should ask if we define a function like this and call it capital F then is capital F differentiable because if capital F is differentiable and if its derivative happens to be this integral then we will be able to accept this formulation as the indefinite integral of small f because the idea of indefinite integral is comes through anti differentiation that is the reverse process of differentiation. So, we ask this question is f z differentiable or if f z analytic and is the derivative equal to small f to get an answer of that we consider this from z we make a little variation delta z and then f of z plus delta z minus f z by delta z we try to evaluate the difference of this quantity with the value of small f. Now, if f z is actually differentiable and if its derivative happens to be f then the limit of this as delta z tends to 0 should be same as f small f and in that case this difference should tend to 0 as delta z tends to 0. So, we try to first evaluate this difference that is simplify it and then see whether it has a limit and what is that limit that limit should be 0 for our purposes. Now, for this 1 by delta z is common so we keep it common capital F at this case point and the original point are evaluated based on this formula. So, in the 1 in 1 place in place of z we use z as it is in the other place we use z plus delta z. So, capital F of z plus delta z is this integral evaluated from z 0 to this point and capital F of z is here that is the argument of the function appears only in the upper limit of the definite integral. And everything else is same now here since f is analytic so the integral from z 0 to this minus z 0 to this will turn out to be the integral of f from z to z plus delta z from the difference of the 2. And if we say that this f z for the purposes of this integral business is constant because it is outside the integral sign, but we can always say that this is f z into d psi integrated from this to this. Now, that is basically verified from the evaluation of this second integral. So, consider this integral of f z this part integral of f z from here to here f z is independent of psi. So, f z into psi and the difference of psi will come out to be delta z. So, integral of this will turn out to be f z into delta z and delta z cancelled it is f z only. So, that means this f z can be inserted into the integral sign in this manner. Now, as we get this then we say that this difference is equal to this. And then if the function is continuous which it must be because the function is analytic. So, analytic function must be continuous. So, since f is continuous then for whatever small epsilon neighborhood you demand between the function values that small epsilon neighborhood you can achieve by taking a suitable small delta neighborhood of the variable itself that is the direct meaning of continuity. So, by taking a suitable delta neighborhood you can achieve an epsilon neighborhood of the function value whatever small epsilon is demanded. And in that case we choose delta z to be smaller than this delta which will ensure that this is always less than epsilon. So, in that case that epsilon constant is taken out. So, you get epsilon by delta z and in the integrand nothing remains just one. So, that means that the integral will then be simply delta z and that delta z will cancel this delta z and will give you epsilon. So, that means this whole thing is less than epsilon. And what we have discussed already for whatever small epsilon neighborhood is demanded. That means epsilon can be indefinitely reduced in size. And that means this will tend to 0 as delta z tends to 0. And that means that indeed capital F is analytic. And the limit of this as delta z tends to 0 turns out to be equal to f z because the difference tends to 0. So, therefore, we say that if f z is analytic then there exists another analytic function capital F whose derivative happens to be f. That means if the function f small f is analytic then we knew earlier that its derivative exists. Now, we come to know that its integral also exists which is capital F whose derivative happens to be this. And the earlier definite integral that we were talking about earlier definite integral which we defined based on path independence that turns out to be available from this formulation. Now, this gives us the same path independence gives us another important tool and that is called principle of deformation of paths. Say we take an analytic function f z which is analytic everywhere other than perhaps a few isolated points like this point s 1, this point s 2, this point s 3 at which the function is not analytic everywhere else the function is analytic. Apart from those few finite number of isolated points then what we can do if we can construct a domain as shown by this dashed line which does not enclose these isolated points where the function is not analytic. Then we can say that any path connecting two points z 1 and z 2 in that can be continuously deformed to any other path without crossing as long as the path remains within the domain d d bar then the path say c 1 can be continuously deformed to c 2 or c 3 without sweeping over any of these non-analytic points. And in that case the integral of the function from z 1 to z 2 along all these paths will turn out to be same as long as you do not sweep over these non-analytic points. If you sweep over the non-analytic points then this guarantee will not be there that is the path the line integral along c 1 along c 2 along c 3 will be all same because they fall within the domain that does not enclose the non-analytic points. On the other hand the same will not be true for a path which is c star because from c 1 c 2 if you want to change to c star through continuous deformation then the deformation will sweep through this point. So, this we can say so that means the line integral remains unaltered through a continuous deformation of the path of integration with fixed end points because we have already seen the path independence of definite integral as long as the function remains analytic. And as you sweep across sweep through s 1 as the curve sweep through s 1 or s 2 or s 3 that analyticity is not guaranteed. So, you cannot claim the equality of the line integral over c star. So, the line integral will remain unaltered through the continuous deformation of the path of integration with fixed end points as long as the sweep of this deformation does not enclose any of the non-analytic points. Now, we try to extend the theorem which is integral theorem to multiply connected domain now here consider this multiply connected domain the outer boundary is this and there are three inner boundaries that means the domain consists of this region and not the interior of this not the interior of this and not the interior of this. Now, to evaluate the integral over this we say that it is guaranteed that the function f z is analytic over this domain, but about the interior of this internal boundaries nothing has been said that means the function could have a singularity could be non-analytic somewhere inside inside these interior boundaries. So, what we do is that we consider a little changed contour that is we take the contour as this curve says we start from here we will come back to here at the end start from here from this arrow we go like this and here we make a cut and come along this arrow and then take a clockwise turn along C 1 clockwise means actually negative and then along the same cut line we go back here and then come here make another cut come here another clockwise turn go back and then come here along the cut come along the arrow another third clockwise turn and then here and come back. So, this is a closed contour and this does not enclose any of the singularities this encloses the domain correctly as long as we keep this forward movement and backward movement along this cut lines very very close as close as we want. Now, the domain which this contour has enclosed is continuous is a domain in which the function is analytic everywhere and therefore, we say that over this contour the line integral vanishes the contour integral vanishes and this contour has enclosed as traverse through curve from here to here and then here to here here to here which is the curve C that is this and then it has enclosed this line segment l 1 forward and backward cancelled this line segment l 2 forward and backward cancelled this line segment l 3 forward and backward cancelled and the three interior curves C 1 C 2 C 3 in the negative manner clockwise. So, that is why minus the contour integral over C 1 C 2 and C 3 this whole thing is 0 because f z is analytic everywhere in the contour that has been actually traversed. Now, this gives us the result that if f z is analytic in a region bounded by the contour C as the outer boundary and non overlapping contour C 1 C 2 C 3 etcetera as inner boundaries in this manner then the contour integral over that outer boundary turns out to be equal to the sum of this plus this plus this that means whatever are the inner boundaries around those inner boundaries if we develop the contour integrals then the sum of these contour integrals is going to be the contour integral over the outer boundary that means that whatever contribution to the contour integral over this curve C is to be made that must be made by the contours enclosing the non analytic regions inside the interior boundary only those places where the function is analytic will make no contribution to this contour integral. So, that means that if we find that in the domain there are a few spots of non analyticity then around that non analyticity non analytic small interior parts if we evaluate the small contour integrals and then sum of these then that will account for the entire contour integral over the outer boundary and in the region in which the function is analytic from that region no fresh contribution will be made. Next we proceed to develop the Cauchy's integral formula suppose f z is a function which is analytic in a simply connected domain D and then at a point z 0 in D and a simple closed curve C enclosing that point z 0 we want to evaluate this integral. Contour integral of f z by z minus z 0 and the Cauchy's integral formula tells us that this happens to be the expression for this integral that is integral around the curve C closed curve C is given by 2 pi i times the function value at the interior point. Now we see that this function f z is analytic but f z by z minus z 0 is certainly not analytic at z 0 because at z 0 it is singular. So, which closed curve we consider as C we can consider any closed curve as long as it can be obtained through continuous deformation of any other curve. So, without loss of generality we consider C as a circle with center z 0 and radius rho because any other curve could any other such simple closed curve enclosing z 0 could be reduced to this circle through continuous deformation over the analytic domain. So, if we consider this particular curve then this integral we can evaluate as splitting it first in two parts that is we consider this f z as f of z 0 plus f z minus f of z 0 that f of z 0 first part we take here and that is that being independent of z comes out and we get this and the rest of it is here. Now this integral we have already evaluated earlier when we were considering the integral of z to the power n. So, now through a coordinate shift this will give you the same thing and this is the for the power n equal to minus 1 in which case this was giving us 2 pi i. So, this part we know and from continuity of f z we can find out a suitable delta in order to make this less than epsilon for any small epsilon and then z minus z 0 we can take less than delta that is a small delta neighborhood which will give this as within a an epsilon neighborhood and then if we take the radius of the circle that rho the radius of the circle as even smaller than the corresponding delta then that will ensure that this whole thing this is less than epsilon and whatever delta is necessary for this if we take even smaller rho as the z minus z 0 as the radius of the circle then this thing will be less than epsilon by rho the lower the denominator is taken as even smaller. So, this side becomes larger so with rho less than delta. So, then this fellow will be less than epsilon by rho and then as we consider the upper bound of this then this will give us less than epsilon by rho into the size of the into the length of the path and the length of the path is 2 pi into radius 2 pi rho. So, the integrand is bounded by epsilon by rho and the path length is equal to 2 pi rho. So, this integral will be limited the integral absolute value will be limited by the product of these 2 from m l inequality that means epsilon by rho into 2 pi rho that means 2 pi epsilon. So, this fellow will be bounded by 2 pi epsilon and that will happen for any epsilon whatever small that means epsilon can be demanded smaller and smaller and smaller and accordingly we will get delta and therefore, rho which is even smaller smaller smaller. So, as we make it small enough we will be able to make this in the limit vanish and then this integral will be evaluated as this and the integral of this we already know. So, the integral of this will give us 2 pi i. So, 2 pi i f of z 0 that is the result. So, this establishes the Cauchy's integral formula and what are the applications of this formula? There are two significant applications one is evaluation of contour integral as it is proposed that is if we want to evaluate a contour integral like this. Then by evaluating the function at the interior point at z 0 around which the contour is there. So, at that point z 0 if we evaluate the function f then we get the integral of this function which happens to be singular. Note that f is analytic, but f z by z minus z 0 is the integrand which is not analytic. So, when we want to find out the contour integral of this around a contour inside which there is a point of singularity then singularity of this integrand f z by z minus z 0 f z is still analytic. So, at that point if we evaluate f z that is f of z 0 multiply that with 2 pi i then we get the integral of this singular function. Singular at z 0 analytic everywhere else because 1 by z minus z 0 has a single singularity at z 0 and f z itself has no singularity anyway. So, this rational function f z by z minus z 0 has a single singularity at z 0 around which we have the contour. So, evaluating the function at z 0 function f at z 0 will give us the integral of this that is the direct application that is if a function g z is analytic on the contour and in the enclosed region then Cushy's theorem implies its integral to be 0. On the other hand if the contour encloses a singularity then Cushy's theorem does not give us the result, but then if g z a has a singularity at z 0, but it is removed by multiplying with z minus z 0 that is if f z like this is analytic then this formula Cushy's formula supplies a non-zero contribution to that integral around that particular point. So, this is the purpose of evaluating the contour integral. Now if it happens that we do not know the function value at the integral points, but we know the function value at the boundary points. If that is the situation then the same formula can be used for the reverse purpose and that is the solution of boundary problems that is evaluation of function at an interior point. If finding the integral on the left hand side is relatively simple then we can use the same formula to evaluate f z f z 0 that is if the function is easy to integrate over a contour and because its value over the boundary points is known and in the integer point we want to evaluate the function then this same integral formula can be used to develop this integral divided by 2 pi i and that gives the function value at the integer point z 0. This is the application of this integral formula for solving boundary value problems. So, this is for example in the previous lecture we discussed this integral formula Poisson's integral formula which is this. Now we can develop this formula now say we take the boundary point as capital R phi in this manner z and the integer point z 0 as this small r theta. We want the values at z 0 integer point and we can easily find the value of the function on the boundary point. So, capital R is larger than smaller, smaller is interior point. So, if we put it in the Poisson's formula then we get 2 pi i f z 0 is equal to integral over the circle and that means 0 to 2 pi of phi that is the boundary integral and this is f z by z minus z 0 and this whole thing is actually d z. So, if z is this then we will get d z as this and through this substitution through this change of variable this phi will vary from 0 to 2 pi and now this if we try to substitute everything directly and simplify a lot of imaginary terms will remain. So, to get rid of that what we can do is that we consider a complementary term. So, in place of smaller if we put capital R square by R then you see here and here we put that small r in place of small r we put capital R square by R and in that case if small r is inside smaller than capital R then this will be larger than capital R and that will mean that the corresponding point this will be outside this contour where the function is anyway analytic and therefore, the line integral will be the contour integral will be 0. So, for that 0 is equal to 0 to 2 pi the same thing in place of smaller if we write capital R square by R then we will be talking about the contour integral around the point which is actually outside. So, that integral is 0 and now through a little simplification what has been done in this simplification small r has been multiplied. So, this small r has gone it has appeared here and the small r will be appearing here this capital R has been cancelled with this R here. So, one R will vanish from here the other one R will go from here the capital R will remain e to the power i phi and e to the power i theta with these two factors we are dividing it. So, e to the power i phi will go off e to the power minus i theta will remain. So, in the denominator in a numerator that will happen in the denominator e to the power i phi will vanish here e to the power minus i theta will come and the reverse will happen here. So, we get this now as we subtract this from here in a left side 0 is subtracted so the same thing remains on this side. However, a lot of things happen first of all this is common. So, this remains and this i is common here as well as here. So, this i goes outside that also remains and other than that what will remain this R e to the power i phi by this and this small r e to the power minus i theta divided by this. So, this is positive powers this is negative powers everywhere. Now, as we simplify this then we will find that those terms which would give imaginary parts they will cancel out and we get this result through simplification. And in the denominator as we multiply out we get this and this is exactly what we studied in the what we saw in the Poisson's integral formula earlier. So, this is f of z that is u plus i v is equal to this same u plus i v over the boundary points are here. So, at the interior point the function gets its value through the integral over boundary points integrated from 0 to 2 pi this gives us the Poisson's integral formula for a disc. And if we take the real and imaginary parts separately say writing only for real part then we will get the corresponding u is equal to 1 by 2 pi this integral everywhere same thing here that function u will come over the boundary points. So, this establishes the Poisson's integral formula this is just an example for the unit disc for the disc as the domain, but for other such domains also this kind of formulas for solution of boundary problems can be established. Now, another important aspect in the evaluation of integrals is this you know we have discussed that Poisson's integral formula evaluates contour integral of g z if the contour encloses a point z 0 where the function g z is non analytic, but g z into z minus z 0 is analytic that is one multiplication with z minus z 0 makes it analytic in that case we get the integral evaluation through Poisson's integral formula that is if g z into z minus z 0 is analytic then g z turns out to be an analytic function divided by z minus z 0 and integral of such a function is given by the Poisson's integral formula. But then if it happens that even this is not analytic, but to make it analytic another multiplication of z minus z 0 is required then what to do through further processing of our Poisson's integral formula we can find a way for that also. So, this is the Poisson's integral formula that we have discussed earlier. Now, if we try to differentiate this expression with respect to z 0 then we get f prime z 0 and these for this formal expression can be found by differentiating under the integral sign noting that z 0 and z are two different variables z is a boundary variable and z 0 is an integral variable. So, if we differentiate with respect to z 0 and that is independent of z then differentiation under the integral sign will proceed directly and that differentiation and integration order can be changed. So, the derivative of this will be the integral of the derivative of the integrand with respect to z 0 and with respect to z 0, differentiating this is easy because we get f z into 1 by z minus z 0 square negative and another negative sign will come because of this. So, two negatives will cancel and we will get this. The derivative for the derivative also will appear as minus 2 f z by z minus z 0 to the power 3 into minus 1. So, minus minus again will cancel at 2 will appear and this power will get raised to 3 and so on. We can go on differentiating these. The expressions of these derivatives can be established simply by differentiation under the integral sign. Now, the actual rigorous establishment that this really happens will require an exercise similar to what we earlier conducted to establish the analyticity of the integral of f z. In a similar manner, we could consider this to establish the this derivative. To establish this derivative, we need to consider an expression like this and through several simplifying steps in which we will try to separate out this part and whatever is the difference, we will try to expand that and then use appropriate considerations to show that this tends to 0. As we do that, finally through the application of M L inequality, we will be able to show that this part vanishes in the limit and that will show that this really tends to this and this is this tends to this value as delta z tends to 0 and that will mean that this turns out to be the derivative of f z 0 that is f prime z 0. And if this we can do the detailed derivation, you can consult from the textbook and if this part is established then by a similar exercise over this function, we will be always able to evaluate this this this this and so on. And this shows that an analytic function possesses derivatives of all orders at every point in its domain and this is the great property of analyticity that we discussed in the previous example. Analyticity, analyticity implies much more than mere differentiability, it implies differentiability up to any order desired. Now, with this established if g z is non analytic at a point z 0, but then g z into z minus z 0 to the power anything 3 4 5 6 say n turns out to be analytic. Then for that n what we do is or say for n plus 1 say for n plus 1 it is analytic then what we can do we use this. And then the g z is expressed as that analytic function g z into z minus z 0 to the power n plus 1 which is f z and then this integrant becomes g z. So, integral of g z d z will turn out to be 2 pi i the n s derivative evaluated at z 0 divided by factorial n. So, that way if as long as z minus z 0 multiplied a finite number of times makes the function analytic then to the use of some suitable derivative formula in this series we can evaluate the contour integral of g z. And this is the great property which will be used in our next lecture also which will be a complete analysis which will completely stress on the singularities of complex functions. So, we use the results that we have found in this lecture into the study of singularities of complex functions in the next lecture. Thank you.


Good afternoon everyone, thank you for joining this open-air provide community call, it's a great pleasure to to run this call number 15, and it's always a great pleasure to have you with us in this call to present some of the novelties that we have in the provide the provide service or in the open-air services or functionalities that are interesting, useful for content providers, for repository managers, pre-system managers, not the repository managers, so it's a it's a it's really a great pleasure. So our focus today will be on on a novelty in fact that we have available currently in beta in the validator for you to test, I will we will also demonstrate that for you to be aware of this adaptation of the fair data maturity model indicate in our in our provide in our open-air validator, I think this will be an interesting facility and there is my calling from Bielefeld University of Bielefeld will detail this but the agenda for today is in fact so to dedicate some time to this work that we have done, mainly our team from Bielefeld University together with some other colleagues, we have done an adaptation of the fair data maturity model indicator that are an output from the research data alliance in our validator and we can have a discussion around around this, so I hope that this will be interesting. Later on I will also present and share with you where it is in the validator in the beta version of the provide service but let's now proceed with this with this presentation just one highlight in fact that is the novelty it's about the fair assessment in beta so now it's available in in validator we were discussing a bit in the provide team if we wanted to have in production we the kind of beta stamp but we decided as we have this workflow now running quite well in open-air infrastructure where we put everything in the beta environment of our services we have decided to do the same for this new functionality of the validator so you can test this in beta I will tell you how and then you can contribute also for with some feedback and test with your own repository etc. I would like also to highlight because every time we have these calls there are some newcomers but we'd like also to highlight that we have them then the last call was dedicated to our open-air services are contributing to the OSC and I know that this is quite relevant for some of you the presentation was done by Paul Umangi the technical director of open-air recordings and slides are available so we can share the links here in the chat and you can check it if you were not able to to join this call I think this is important for you also to clearly understand that your participation in provide will also allow you to be part in a way of fiosk okay two reminders that you are already aware and that we but as this is something sometimes also important for some newcomers one is about the need to subscribe to receive the metadata enrichments you are aware of this but sometimes people don't don't don't catch it so well so we're now we did some in the in the recent months as we did some changes in the in the way that we are exposing metadata enrichments and the broker events are generated we just show present a sample of of 100 events in the in the dashboard in order for you to receive all you need to subscribe you can filter or for what what you want a specific event or with a specific place and then you subscribe and you will receive we are standing quite in a in a in a very good periodicity the the notification so we are running the and the plating the broker events and for sure you we are aware that you have already received every month at least one one notification with the with new with new events of course depending on what you have subscribed and and if there is already if there is if there are new events or not the other thing is that we don't have novelties about the registration process for Cree systems but we we are working on that and for sure we'll allocate time to this activity in the coming months but for you to be aware if you are coming from this area or if you are related with any Cree system be aware that the Cree systems we can register currently in the open air infrastructure but in the near future we will have a process of registration in open air and we will use the the the Dries the directory from Eurocris to to become the alternative source for the registration process so be aware of that so in order to have your Cree system in open air you need to register in in Dries from from from Eurocris so I think this is important for you to to know this is not available now in provide in the registration process but will be available soon as we did a project with Eurocris in order to prepare the directory to communicate let's say to simplify with with open air so we will present this later but related with the first novelty in beta you you will see if you access beta dot provide dot open air dot u you will see this this new box in the in the validator functionality for the fair assessment and you can play with this so let's present this later and now I I will also just because I didn't put it here but I want to say something we are a bit late on that but we don't have I know that some of you are expecting for the multi-user access some of you have already requested that we have a functionality to for you to be able to give permissions to only one login as a manager of the C3 this is not ready yet unfortunately we have some delays but so it's in our in our to-do list every every two weeks we remind ourselves about the need but this is related with other authentication services internally in open air and it's not so easy to to to to solve okay Andreas so now my colleague Andreas so Andreas you can also present yourself but we'll run this presentation about this fair this verification of of your content provided in open air so of what is yours Andreas and many things for your availability to run this presentation thank you Pedro good afternoon also from my side so um yes today we have a topic about the certifications of our guidelines and focus in this community call for the literature guidelines or I will come a little bit later to you with this topic my name is Andreas Czerniak from Bielefeld University Library in Germany and in the face of open air at once we are leading the work package and task for the guidelines not only the guidance that I present now but also the others and under in during this phase we are working on the verification that's our what we call it because our guidelines are come from 10 years ago and we will check the elements that we have in the guidelines and recommendations there this in a moment so my topic topics for the presentation today is to give a short overview about the guidelines and especially for the institutional and thematic guidelines name mostly name called as literature guidelines and some fundamentals here um and what is new in the guidelines in the process in the last phase we are working on some up to updating these guidelines and not only on the fair process but only on something related to the guidelines and I will present this also here and after this I'm presenting the verification process and last but not least the live demo from beta provide an overview so as you know mostly the landscape of the repositories are quite diverse so at the match today we have institutional thematic repositories and during the last years and then there are something more repositories data repositories journal repositories from publishers and also software repositories and other research products and we see a lot of potential in the current research information systems and for all these kinds of repositories we have guidelines in the from open air but these repositories represents also different type of resource types and what we see is resource types in the repositories for literature articles preprints for data sets um also clinical pilots here um software software research products and newly for projects funders organization units instruments and equipment so if we map these repositories to the research output types that we see here or we've found that the institutional and thematic repositories covers mostly all of things in the meantime so not only literature which we started from but also the data sets software and other research products only the chris systems have all entity types in there and additionally to the four research type output types there also information about projects funders persons organization units instruments and all with persistent identifier linked in the if we take a look into the other repositories and the research output there for data repositories they are mostly data sets and literature at the moment for journal publishers mostly literature and so on if we come to our uh actual guidelines that we are focused on today is on the institutional and thematic repository guidelines and this covers all these four sets of research outputs these the guidelines have as I said before an evolution and starts in 2010 in 2010 there was the first version of the literature guidelines published and the work behind is starting in 28-29 and coming from the driver project before Okma starts after two years the first revision of the guidelines was made and um there's also seen that our guidelines for data are important so in 2012 there also published literature guidelines and guidelines for data archive repositories if we see um the years after um there are another version of literature guidelines version three there's also a version for data guidelines and from this evolution we know okay literature is not the only ones in the repositories yet and we should revise our guidelines to an institutional and thematic repository guidelines for institutional and thematic repositories it's called version four and mostly named as literature guidelines version four but that's not really true anymore it's a mix of from these resources that we've seen before and uh in 2018 we open air uh revise the content acquisition policy until 2018 open air covers only open access um articles and uh with the new content acquisition policy um open air collect and harvesting open access content but also non-open access content so it's open the word open here to the wider world in the same year 2018 um it's uh was in the revision of the chris a reef guidelines published and both are uh now the actual ones of guidelines that we have and we end the phase to revise these in the last year and also this year and updating the guidelines for institutional and thematic repositories but also the data archive guidelines and the guidelines for chris managers but we focus today on uh the institutional and thematic repository guidelines these guidelines mostly called as literature guidelines um based on Dublin core and data sites metasema format it's uh just uh describe uh publications in text for text will and for data publications have and have controlled the capularies um and you can see here a screenshot of the application profile which i um don't go into depth now but you can see is in on cenoto or at guidelines dot open air dot who and we are in the face to updating or the latest guideline version for institutional and repository um and we end the states of the release candidate one for the version for one we have made some changes we have some minor changes and the first one is to to namely to include fair and the reference to the 88 fair data majority working group output here to our guidelines so as a scene we are started in 2010 with the guidelines was mostly to um focus on open access on finding articles for accessing articles on open access and so on so uh this is not mentioned yet in the guidelines um but uh we will see we have this in the guidelines um since 2010 roundabout so this is also an adoption uh an adoption of the guidelines to the 88 fair data principles here um i will come back in a minute to this topic this important topic the other one is uh we are hands um something for aggregators in the last years we see um on the or epmh interface some gap um in the identification of repositories and together with our partner um and infrastructure la referencia or we um and develop a proof of concept to identify to identify repositories in the provenance element so if you are familiar with it this or um this is uh the about element the about note from the epmh interface and the provenance element in this go not really in the depth here but we um add some uh two simple elements for repository id and repository name repository id is an important one here um to identify which is the repository that comes that's behind this record and um we choose and discuss here the um identifier for open door or also for re3data um and the uh identifier from these registry also can it enhance this from from other repositories so the first element is a repository id was a registry and uh followed by the identifier of this registry the repository name is nice to have good to have in some cases to identify for human which are the repositories behind this identifier this was one enhancement and proof of concept for aggregators and the other enhancements that we are have done in the last month are for contributor routes we are um rely on the credit uh attack ceremony um to add more rules of contributors here um this is in line also in line with other um initiative like orchard orchard is also integrate the credit taxonomy here for the rules we also extend and refine the descriptions uh from of the elements and attributes and also um have more and hopefully better examples for the elements here and we see also that we should update links um that we have in the guidelines um like for crosser funders the domains change and links change and so on this was updated also in uh this revision so um I will share afterwards the linked for the guidelines there will in the presentation too I see the chat here with the question so first of all I've come to the verification process um verification our guidelines um that we have it's based on the ADA fair data majority model specification in guidelines it's published in June 2020 and um it covers the findable the accessible interverbal and the reusable and the specification and guidelines um share an excellent feed or the identification how fair is the elements that we have and here in the overview on the right side you see we are the overview of our analysis that we are done and that I present in the moment in detail um findable uh it's very grateful accessible that we have something to do or um that we are on a discussion at the moment interverbal and reusable reusable so in detail findable um the ADA share an excel sheet with um identical identify fair identity identity indicator and um we go through these sheet and um map this uh to the elements that we have are in the guidelines version four um or um you see here this is important also for the validator um here the identity identifier on the left column and the description and priorities met in the guidelines or not and um what we have done here we do this for all of these identifiers in the streets and we see oh very fine we can we have this all one in we are fit the findable with our guidelines um with resource identifiers or implicit via or e p m h interface identifiers and so on so if you have if we harvesting your data source your repository via or e p m h and um you have these identifiers regarding our guidelines version four you are um hundred percent findable perfect so these um analysis and evaluation uh if we done for all the fair elements and it's the same procedure um for accessible um you see here the identifier of the description of the identifier priorities and fits in the guidelines or not and um it is the scope of our guidelines or not and the result is presented on the right side so there are some gaps that we have in the guidelines and we discuss these gaps how can we um mention this in the guidelines something is not directly under control of the guidelines um something is in the control of open air but not really in the guidelines so we are in the discussion phase at the moment um for uh on the one end for better understanding these fair principles for the guidelines as I see here the the challenge is to have these um to to adapt these guidelines uh from the data for a fair data majority model from the area to the literature part or not only literature not only um so institutional and thematic repositories this is a challenge then we have not only um data in it we have also articles we have also software as we seen before so this is um that we um map would like to map in our analysis and see some gaps here um it's something like mostly isn't are the gaps in the guidelines not mentioned um regarding the the data um here's an example um day for the data that we see here the gap is not available for the guidelines data is accessible through an access protocol that supports authentication and authorization so this is not really in um under control of the guidelines we cannot check these ones through the guidelines and um at the moment we think that this uh not should not part of the guidelines and uh could not be mentioned here but we have this in the principles and um this we have done this analysis and evaluation for yeah something that we are discussed at the moment for for the elements uh for i um you see uh you see this analysis here and um the last one is reusable same thing here um that we have said we fit a lot of for of the reusable for like license and so on uh metadata is machine readable we are now standable but not all not all is mentioned here um and we are in under discussion to make it more fair and make it and describe this more on the element um level the conclusion that we have from this verification process is um that um it is that is possible to make an adoption adaptation of the guidelines to the ADA data fair data principles um and we see that the guidelines that we have at the moment are primarily fair so um it's not overall because on the uh the the sections of data and um accessible access of data sets and um but in most cases we fit to the fair principles for um from the ADA and um as i seen in the slides as you seen in the slides before the evolution of the guidelines was key so uh we are under at the moment under the discussion um to go forward on the guidelines regarding the fair and other um sections that we have and um for this this is important to um modify the guidelines over the years and fit it to the reality of the repository of all the information system brought here the references um the new guidelines oh the latest version that we are working on is um at the read the docs you see the link and i see the link is also shared by paolo thank you and examples and schemas are available at github at github there's also an issue checker that we use for um feedback channels you can make a feedback on our guidelines that we have directly on read the docs it's um there isn't this is possible but uh this requires also um look in in the annotation service of read the of read the docs and you can make the annotations and comments directly on the latest guideline pages here the other thing is to create an issue at github under the guidelines repository for this you need also a github login the last one is to um comment our google document without any credentials um and um there you in the google document you see also some discussions and links to um really related issues in github and so on so um you have here an overview about a discussion about the new version of the guidelines here so um peter would you like to take over and give you the floor to you to better provide to present or like go forward to and you present better provide and the fair assessments um afterwards yes yes thank you andreas many thanks so i can proceed just in order to allow people then to to test and to run some tests and to for sure they will have doubts i think this is a a lot of information i understand that some of the the participants of this call are have already very good knowledge about the the the fair maturity indicators some others don't have so i think it's important to have some time all this presentation is available for you you can check the details of them of the fair indicators but let's um um i was giving some time some wording in order to follow make me presenter uh but um so i will just uh check with you so um if you don't have access to beta so just go to to better uh dot provide dot open air dot you and and you may have access to to better if you if you don't have you can you can put it in the shot and we will make sure that you can have access to this to this version okay um you will see in the in the validator this new box here to perform a fair assessment on your metadata um and you will be allowed to do to do that i was running some tests just before this this is something new so i was running some tests so here is where you have this functionality functionality available you can run some tests uh you put here the the the OIP imagine um URL for for your the interface of your repository uh and run the the the test so i'm just performing some tests i have already here some in the validation history so but you can you can run this test what you will be able to see here in this fair assessment is the mainly the rules that uh we that um Andreas have presented uh that we are able to test as as as you saw some of the um of some of the fair indicators as we are not possible for us to to to check so you can you can run this um you can select or just have them all selected as this is the same for the other for the other guidelines and then here you can perform in all records or just select the number of records just to have a kind of example uh and you or you can also check if you want to to check some of the specific sets that you have in your repository and perform the the validation so like for like for all the other validations so you receive an email i was doing now tests during the presentation of Andreas just for you to see you receive an email saying that one email saying that the validation started and another email that the validation uh finish and you can then browse in the history the the the validation so you will see it in the validation history if it's already there so as you can see we have several um validations here so all jobs that i have submitted some of them are unsuccessful others are successful some failed so um i have so you can then you can check the status here in the in the steps so the ongoing those that have failed and those that are successful and then from the successful is like the other the for the other situations you can you can test you can test so you can check the where are the the errors so we have here first kind of overview of the results for the for the 100 i i did a test of for a sample of 100 records and then you can identify clearly the where we have where you you may have errors okay and then the errors are identified in specific records and you can check the records of course here is where it comes the the dots that we can clarify and that you you we may need to to provide you some support or you may need to understand better what we have in the indicators here we have some errors about um licenses if licenses is explicitly in the in the metadata record or not um if if there is a specific representation of a standardized format this there are some errors that you can see and that you can go to the to the record like for the other situations i understand that then you may have some some dots but this is the way that it works and it's the the way that the other validator also works i think it's now what is interesting so we are in a better environment we are in this phase of receiving your feedback as andrea stated i think what is relevant here for us is that we are um so we just want to stop sharing and what we really want here is to to provide a service i think it's an edit value service uh to assess this um the fairness uh of your your data source of your repository against well established uh indicators from from rda um and then you we can also collect your feedback and make it useful make it really useful receive your feedback so i think all the links to to contribute uh directly in in so we have andrea is just for clarification we have one google document where people can comment and but they can also directly in the guidelines comment make comment yes there are three feedback tunnels so um you can give us feedback through the read of docs functionality of annotations um but we see this um this requires look in and um then we set up feedback document on google directly so with our credentials you can give us uh your suggestions and um for topics in the guidelines so and um we share this information on the fair assessment and the functionalities of the validator that we have uh first with you um today so to see also how fits the validator and the roots that we that we have in the last years um our needs regarding fair and um how can we um use this functional functionalities for the continuous validation of repositories also so if we um have this functionality established in beta provide and um you see here uh only five or six rules that we implement at the moment and we implement more um then we can also um have these functional functionalities in the continuous validation of data sources and use all the features that open air and provide have so um yes so we just wanted to share this with you in this community call in order for you to be aware of this work we'll be aware of this added value functionality from from from open air provides if you have questions you can you can ask i'm sorry peter that's a question in the in the minutes so i'll just i'll just put it for andrea so if you want i can read it so john asked regarding the european commission because for horizon europe deposit in a trusted repository um so um that's a very good question so um i think also peter for um yes yeah so we can consider so when we we talk about trusted repositories um so there is um there is we can have a discussion about that for sure repositories that are part of for the time being i think this which should should be the approach i think we all aim to have in the future um seals that clearly state the how trustful is that the specific repository but i think we can understand now that for sure the commission will follow an orientation that is all repositories that are part of the directories accepted by the communities and part of open air are considered uh trustful repositories this is uh my understanding this is the understanding from from open air things are evolving um currently it's not mandatory to have specific seal for example um but maybe in the future that currently we don't have it and for sure to jordan i think and you are part of the community you may have also an opinion repositories that are part of mainly of the main directories that we have we can say a way through that the open door for sharing for sure they are repositories that are trustful repositories repositories that are part of open air are validated by open air and part of the open air infrastructure are repositories that the commission can consider trustful to to to have the outputs from the funded projects this is uh what we can say things are evolving and for sure we will learn more details even the mandate from the commission need some more guidance for some specifications i'm not sure if they will provide more guidance on this level of trustfulness but for the time being i think this is quite difficult what we need to have so and to have some more information to jordan if we don't have any other question so this is a journey the verification this this compliance with fair indicators fair metrics uh improving the probability of of of of repositories it's a journey and for some it's a long journey for others it's a short journey it's a journey that needs to be inclusive so if we start to have kind of of a clear orientation what is considered a trusted repository we will not be part we are we are not contributing to a good journey and we are not being inclusive so the principles that at least i was part of the a osc fair working group that have contributed with the recommendations about the certification of of of yosc service providers and the osc repositories etc and one of the recommendations was that because this is a journey we need to be inclusive so there is a journey to to ensure that everyone improves the interoperability everyone improve the the fairness of of the repository the fairness of the digital objects etc so let's stay calm this is my my clear opinion let's stay calm and then trust in some of the tools that are part of the community like the directories and then do the do our best i hope that this is a good answer for you jordan and partially my opinion in partially also the the view of open you have another question in the shot yeah julian thank you for your best and then yes it was a good suggestion from from alessandra so if the repository has a good fair assessment in all mail it would be received for batch yeah very good very good very good suggestion i think this is but but for that we need to be confident i think this is a very good suggestion at center but we need to be confident of what we are offering and to be confident it must be from from our team and must be also from the the feedback that we receive from you but but many things and i think it's something that we for sure we can follow and it's julian as i didn't saw the comment from alexander previous but it's julian i have a question but it's not related to the topic okay but we can reply so every every so do we have news about the new user interface for provider interface i'm thinking about the ability to decline the septic nor the enrichment suggested will it come for this place okay okay thank you the for this place for for this is for this place for this place crease it's available for this place seven will be made available the same approach can be done for this space version six but someone from the community need to to integrate this i can say that you know in portugal we are thinking to contribute as we are going to to to to give some some as we are going to update versions of this space that we asked in our national network of repositories we may be available available to work on this at least we are pushing for that let's see if we can use it on okay so and there's a question for milan um this is good question thank you um yes we uh the question was to um will you combine guidelines for literature and data repositories into one guideline uh we export our metadata currently to both yes that is what we see in the last uh years um that uh we have we collect metadata from the literature where you end point and also from data site or data where you end point and we're thinking about to have umbrella for our guidelines not especially for literature or data so also for uh software and other research products um to fit the needs and to to generalize these guidelines into one and have special parts for if you like literature if you like data repositories and so on so yes we are thinking about this and to have your view to on the guidelines so that you don't choose uh one of the guidelines in the validator also um or in the registration process so as I present before in the in the in one slide in institutional arithmetic repository there are a lot of research output at the moment and I think we can combine and have an umbrella for this okay anything so we need to we need to close this this community call so I think we can we can give more some more minutes but Andreas like I leave the floor to you and to Paula to close the meeting I need to move to another webinar that I'm in fact managing and they have already opened which is not easy so we can close the if you have any comments I think what what we mentioned here is quite relevant so this is a process Paula I leave the floor to you to close the the community calling for Andreas to reply to any other questions but usually we close it on time so feel free to close it whenever you want but I just need to leave many things for joining this call we have some more time to to to close thank you thank you Peter also the last was to request for comments on our latest guideline version um and also um we want to hear from your side your ideas or your talks about this and please until 20 of June so afterwards we would like to go forward to make a final version on these guidelines thank you thank you thank you very much for joining today and I would like also thank you my colleague Jörgen Schierwagen is done a lot of work on these fair assessments thank you very much


Today I'm going to be responding to a therapist on TikTok who basically claims that Christianity Christian teaching is traumatic for people that are engaged in it. So it's not just people that call themselves Christians that do terrible things and cause trauma and hurt people. No, it's actually the teaching itself. This is a really interesting take that I haven't heard before. So let's watch this video and I'll comment on it.  It was the way it was taught and that is simply not true. Simply understanding Christianity in and of itself is traumatizing.  Internalization of that belief in and of itself one will cause anxiety because it creates a problem in you that can only be solved by something external from you. In fact it starts off by telling you that you're born with an issue. So the trauma in religious trauma is simply understanding the religion. That's where the trauma comes from. You're in a constant state of worries. Jesus is coming soon. Jesus is coming soon. That sounds a lot like anxiety. This is really interesting to me. I want you to step back from the Christian worldview for a second and just imagine okay take this man's you know point to heart and say okay you know Christianity is traumatic for people and the belief that the teaching itself it's not good for people's hearts it's not good for people's souls. So what are they to believe alternatively like okay outside of a God that created us are you saying that no God created us are you saying that we're just kind of evolves protoplasm over millions and millions of years that we have no intrinsic worth or dignity or value. You think that belief is better for people's mental health really that's really really concerning to me. Furthermore in that worldview how do you get any kind of sense of identity or love or or truth like it's all just kind of subjective absurdity in that world we're just animals we're just chemical reactions there's no kind of emotional security that can be found because look it's all just meaningless. I don't know about you but that would send me into a mental death spiral very quickly and so when I look at Christianity when I look at the teachings of Christianity here's what I find right we find ourselves created in the image of God right and we're created and he calls us very good but we've rebelled against him so but ultimately we have that security okay first security in knowing that we're created in the image of God that's good but then there's a reality here that we have rebelled and he says this is going to cause anxiety the fact that God if we don't obey him or listen to his commands we're going to be sentenced to hell the question isn't does that belief cause anxiety the question should be does that belief is that belief true right because if that belief is true right then it should be something that we should take account for we should we should okay well I need to do something about this think about this the idea that oh if I walk across the street and I'm not looking both ways that I could get hit by a car that could cause anxiety absolutely but the the but the the solution to that is not believing that I won't get hit but rather understanding come to terms with okay this is the reality of the situation this is the truth of the situation and what can I do about it what okay I can look both ways and so here this idea that we need to just expel any kind of belief that might cause anxiety or fear it's like no no not if this is the truth and ultimately we should have a fear of God and an understanding of of his judgment and his wrath a lot of people don't understand how could God judge people and this is a little bit of a sidetrack but how could God you know send sentence people the hell it just seems unreasonable or maybe just extreme but when we understand God as a judge like think of God as a judge and you're standing before God and you're saying okay God you know I realize I've done some bad stuff right I know I'm not the perfect person but ultimately like I can walk old ladies across the street I can do all these sorts of good things just kind of forgive me and God looks at you and he says well no I'm a good judge I you have to kind of serve the punishment for the crime if I let you go I'm not being good and because God is infinitely holy that means that the punishment that we deserve should be eternal death as well but that what's makes the beauty of the gospel so relevant to us is that God in his mercy sent Jesus fully God and fully man to die on the cross the death we deserve to die for our sins against God that we would be raised to newness of life that we would find identity and security in him being adopted into his family I think about the ramifications and the implications of actually understanding the gospel in its true light there's so much security and identity to be found the idea that I can be forgiven and the idea that I am a child of God created in his image that I can't earn my own salvation that I can't fix what is wrong with me but rather somebody God has come and he's fixed it for me he's done all the work for me he says that it's it's a troubling thing to to believe that you can't fix yourself right to need to look beyond yourself in order to fix what's wrong with you I actually think that's a really comforting feeling because ultimately we know we don't have what we need in and of ourselves right we can exist in denial or create excuses for our sin and we can try to convince ourselves that we're okay that we're at peace but we don't have what we need on the inside to really find that peace to find that acceptance we need somebody external somebody objective that can look at us and say you are my child you are forgiven you don't need to strive anymore this is your identity when we try to manufacture that out of just nothingness it's unstable there's no really no real concrete foundation now we need to talk about the fact that Christian teaching can be distorted and manipulated in such a way where it actually does cause emotional harm to us and I've experienced that in terms of shame so for so long I kind of operated on this belief that I need to earn God's love for me that I needed to do something to prove that I was enough to God that I needed to measure up and ultimately what I was finding was I was falling short on every account that I wasn't doing enough to to merit God's favor I wasn't enough as a person I dwelled in insecurity and lack of self-confidence because I just had no concept that that I could be a be and do enough right and so then I just kind of wallowed in the shame even as as a Christian but then understanding the truth of scripture was what freed me from that bondage of shame it wasn't that oh you're perfect just the way you are but rather hey even in the midst of your weaknesses you can be secured in Christ because his righteousness is enough for you he calls you his child he delights over you like can you believe that he delights over you and he bears your burdens he's bore your shame on the cross that you wouldn't have to bear that anymore that you are invited into rest these are the things that that I've tried to internalize over the last few years that have really been a freeing process where somebody that's in an atheistic worldview they're navigating shame or guilt the solution to that is just to dispel those feelings entirely to say well shame is just kind of a societal concept and and guilt is just like morality is is very subjective and so you really shouldn't feel guilt for anything in particular you should just have you know radical acceptance of every behavior that you've ever had and disregard the shame is just meaningless it's like no like actual when when we look at the truth of what God has done in our world he set it up in such a way when we do sin we experience a level of shame and guilt and that causes us to look to God but then when we are in Christ when we do sin that shame and guilt is designed to draw us back to God to reaffirm the fact that hey actually this isn't me anymore right this isn't who I am where before when we were outside of Christ and we were sinning it was a testimony to who we were now that we're in Christ right that shame and guilt is a testimony to the fact that this is not who we are I am not this person anymore I am in Christ I am loved I am forgiven like these are the truths that actually bring emotional healing and when I think about it I think when we talk about anxiety or any kind of like mental health or trauma one of the most healing things is those things can only heal in relationship they can only heal in in safe community and I think about the ultimate safe relationship is with God right he is the ultimate comforter he is the ultimate father who can listen to the the cries of our heart and he affirms us in in our identity and that is a securing that's a healing thing if you're navigating shame if you're navigating anxiety if you're navigating depression to have that perfect relationship and then also within the the body of Christ in general having those relationships where we can say hey look this isn't just my own opinion this is what God says of you that is powerful we're in the atheistic worldview it's just everybody's opinion hey man yeah you're you're good don't don't feel so bad you know I I know you're going through some stuff but it's okay like it's not a big deal at the end of the day um we're just all stardust like really is that emotionally more healing than realizing that God has created us and and that he is with us and that his his nothing can separate us from the love of God don't believe the the the narrative that Christianity is just some cold dusty old religion that has no really connection to our emotional life no it does God has things to say about it the all these beliefs all these teachings the truth of scripture has implications to how we see ourselves how we operate in this world emotionally how we navigate mental health it's all connected I hope you guys got something from this video I hope you were encouraged in some way I'd love to hear your thoughts in the comments down below about how you kind of navigate some of these more challenging emotional issues in light of scripture in light of what Jesus has said about you I'd love to hear from you a huge thank you to everyone on patreon you guys know this is my full-time gig this is what I love to do this what I'm passionate about you guys enable me to continue to make this content I'm not big on accepting sponsors or doing promo deals I want to focus on the heart of the content and by you guys signing up on Patreon even today on the description down below you can enable me to continue to do that so thank you so much and I will see you guys next time God bless