Gamma function

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For the gamma function of ordinals, see Veblen function.
The gamma function along part of the real axis


\Gamma(n) = (n-1)!

The gamma function is defined for all complex numbers except the negative integers and zero. For complex numbers with a positive real part, it is defined via a convergent improper integral:

 \Gamma(t) = \int_0^\infty  x^{t-1} e^{-x}\,{\rm d}x.






"Find a smooth curve that connects the points (xy) given by y = (x − 1)! at the positive integer values for x."

A plot of the first few factorials makes clear that such a curve can be drawn, but it would be preferable to have a formula that precisely describes the curve, in which the number of operations does not depend on the size of x. The simple formula for the factorial, n! = 1 × 2 × … × n, cannot be used directly for fractional values of x since it is only valid when x is a natural number (i.e., a positive integer). There are, relatively speaking, no such simple solutions for factorials; any combination of sums, products, powers, exponential functions, or logarithms with a fixed number of terms will not suffice to express x!. Stirling's approximation is asymptotically equal to the factorial function for large values of x. It is possible to find a general formula for factorials using tools such as integrals and limits from calculus. A good solution to this is the gamma function.


A more restrictive property than satisfying the above interpolation is to satisfy the recurrence relation defining a slightly translated version of the factorial function,

f(1) & = 1 \ \text{, and} \\
f(x+1) &= x f(x),



Main definition[edit]

The extended version of the gamma function in the complex plane

The notation Γ(t) is due to Legendre. If the real part of the complex number t is positive (Re(t) > 0), then the integral


converges absolutely, and is known as the Euler integral of the second kind (the Euler integral of the first kind defines the Beta function). Using integration by parts, we see that the gamma function satisfies the functional equation:

\Gamma(t+1)=t \Gamma(t).

Combining this with Γ(1) = 1, we get:

\Gamma(n) = 1 \cdot 2 \cdot 3 \cdots (n-1) = (n-1)!\,

for all positive integers n.

The identity Γ(t) = Γ(t+1)/t can be used (or, yielding the same result, analytic continuation can be used) to extend the integral formulation for Γ(t) to a meromorphic function defined for all complex numbers t, except t = −n for integers n ≥ 0, where the function has simple poles with residue (−1)n/n!.


Alternative definitions[edit]

The following infinite product definitions for the gamma function, due to Euler and Weierstrass respectively, are valid for all complex numbers t, except the non-positive integers:

\Gamma(t) &= \lim_{n \to \infty} \frac{n! \; n^t}{t \; (t+1)\cdots(t+n)}
= \frac{1}{t} \prod_{n=1}^\infty \frac{\left(1+\frac{1}{n}\right)^t}{1+\frac{t}{n}} \\
\Gamma(t) &= \frac{e^{-\gamma t}}{t} \prod_{n=1}^\infty \left(1 + \frac{t}{n}\right)^{-1} e^{\frac{t}{n}}

where γ ≈ 0.577216... is the Euler–Mascheroni constant. It is straightforward to show that the Euler definition satisfies the functional equation (1) above.

A somewhat curious parametrization of the gamma function is given in terms of generalized Laguerre polynomials,

\Gamma(t)=x^t \sum_{n=0}^{\infty} \frac{L_n^{(t)}(x)}{t+n},

which converges for Re(t) < 1/2.

The gamma function in the complex plane[edit]

The absolute value of the gamma function on the complex plane.

The behavior of Γ(t) for an increasing positive variable is simple: it grows quickly — faster than an exponential function. Asymptotically as t → ∞, the magnitude of the gamma function is given by Stirling's formula

\Gamma(t+1)\sim\sqrt{2\pi t}\left(\frac{t}{e}\right)^{t},


The behavior for nonpositive t is more intricate. Euler's integral does not converge for t ≤ 0, but the function it defines in the positive complex half-plane has a unique analytic continuation to the negative half-plane. One way to find that analytic continuation is to use Euler's integral for positive arguments and extend the domain to negative numbers by repeated application of the recurrence formula,


choosing n such that t + n is positive. The product in the denominator is zero when t equals any of the integers 0, −1, −2,... . Thus, the gamma function must be undefined at those points; it is a meromorphic function with simple poles at the nonpositive integers. The residues of the function at those points are:


The gamma function is nonzero everywhere along the real line, although it comes arbitrarily close to zero as t → −∞. There is in fact no complex number t for which Γ(t) = 0, and hence the reciprocal gamma function 1/Γ is an entire function, with zeros at t = 0, −1, −2, ... The gamma function has a local minimum at x_{\mathrm{min}} \approx 1.46163 where it attains the value \Gamma(x_{\mathrm{min}}) \approx 0.885603 . The gamma function must alternate sign between the poles because the product in the forward recurrence contains an odd number of negative factors if the number of poles between t and t+n is odd, and an even number if the number of poles is even.



Other important functional equations for the gamma function are Euler's reflection formula

\Gamma(1-z) \Gamma(z) = {\pi \over \sin{(\pi z)}},

which implies

\Gamma(\varepsilon - n) = (-1)^{n-1} \; \frac{\Gamma(-\varepsilon) \Gamma(1+\varepsilon)}{\Gamma(n+1-\varepsilon)},

and the duplication formula

\Gamma(z) \Gamma\left(z + \tfrac{1}{2}\right) = 2^{1-2z} \; \sqrt{\pi} \; \Gamma(2z).

The duplication formula is a special case of the multiplication theorem

\prod_{k=0}^{m-1}\Gamma\left(z + \frac{k}{m}\right) = (2 \pi)^{\frac{m-1}{2}} \; m^{\frac{1}{2} - mz} \; \Gamma(mz).


\overline{\Gamma(z)} = \Gamma(\overline{z}) \; \Rightarrow \; \Gamma(z)\Gamma(\overline{z}) \in \mathbf{R} .



which can be found by setting z = 1/2 in the reflection or duplication formulas, by using the relation to the beta function given below with x = y = 1/2, or simply by making the substitution u = √t in the integral definition of the gamma function, resulting in a Gaussian integral. In general, for non-negative integer values of n we have:

\Gamma\left(\tfrac{1}{2}+n\right) &= {(2n)! \over 4^n n!} \sqrt{\pi} = \frac{(2n-1)!!}{2^n} \sqrt{\pi} = \sqrt{\pi} \left[ {n-\frac{1}{2}\choose n} n! \right] \\
\Gamma\left(\tfrac{1}{2}-n\right) &= {(-4)^n n! \over (2n)!} \sqrt{\pi} = \frac{(-2)^n}{(2n-1)!!} \sqrt{\pi} = \frac{\sqrt{\pi}}{{-\frac{1}{2} \choose n} n!} 



Another useful limit for asymptotic approximations is:

\lim_{n\to\infty} \frac{\Gamma(n+\alpha)}{\Gamma(n)n^{\alpha}} = 1, \qquad \alpha\in\mathbf{R}



For positive integer m the derivative of gamma function can be calculated as follows (here γ is the Euler–Mascheroni constant):

\Gamma'(m+1) = m! \left(  - \gamma + \sum_{k=1}^m\frac{1}{k} \right)\,.

For Re(x) > 0 The n-th derivative of the gamma function is:

\frac{{\rm d}^n}{{\rm d}x^n}\,\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} (\ln t)^{n} \,{\rm d}t.[3]

The gamma function has simple poles at z = −n = 0, −1, −2, −3, … The residue there is


Moreover, the gamma function has the following Laurent expansion in 1

\Gamma(z) = 1+\sum_{k=1}^\infty\frac{\Gamma^{(k)}(1)}{k!}(z-1)^{k},

valid for |z − 1| < 1. In particular

\Gamma(z) = \frac1z-\gamma+\frac12\left(\gamma^2+\frac {\pi^2}6\right)z-\frac16\left(\gamma^3+\frac {\gamma\pi^2}2+2 \zeta(3)\right)z^2+O(z^3).


In a certain sense, the log(Γ)-function is the more natural form; it makes some intrinsic attributes of the function clearer. A striking example is the Taylor series of log(Γ) in 1:

\ln \Gamma(z+1)= -\gamma z +\sum_{k=2}^\infty \frac{\zeta(k)}{k} \, (-z)^{k}\qquad \forall\; |z| < 1


Fourier series expansion[edit]

The logarithm of the gamma function ("GammaLn") has the following Fourier series expansion

\ln\Gamma(x) = \left(\frac{1}{2}-x\right)(\gamma+\ln2)+(1-x)\ln\pi 
- \frac{1}{2}\ln\sin\pi x  + \frac{1}{\pi}\sum_{n=1}^{\infty}\frac{\sin 2\pi n x \cdot\ln{n}}{n}\,, \qquad 0<x<1,

which has been long-time attributed to Ernst Kummer who derived it in 1847.[4][5] However, it was comparatively recently that it was discovered by Iaroslav Blagouchine that this series was first derived by Carl Johan Malmsten in 1842.[6]

An approximation for GammaLn is given by Numerical Recipes in C (First Edition):

GammaLn(x) = ln(2.506628274631 * s) - z
z = x + 4.5 - (x - 0.5) * ln(x + 4.5);
s = 1.00000000019001 + 76.1800917294715/z - 86.5053203294168/(z + 1) + 24.0140982408309/(z + 2) - 1.23173957245016/(z + 3) + 1.20865097386618E-03/(z + 4) - 5.395239384953E-06/(z + 5).

Raabe's formula[edit]

In 1840 Raabe proved that

\int\limits_a^{a+1}\log\Gamma(t)\,\mathrm dt = \tfrac12\log2\pi + a\log a - a,\quad a\ge0.

In particular, if a=0 then

\int\limits_0^1\log\Gamma(t)\,\mathrm dt = \tfrac12\log2\pi.

Pi function[edit]

An alternative notation which was originally introduced by Gauss and which was sometimes used is the Pi function, which in terms of the gamma function is

\Pi(z) = \Gamma(z+1) = z \Gamma(z)  = \int_0^\infty  e^{-t} t^z\,{\rm d}t,

so that

\Pi(n) = n!,

for every non-negative integer n.

Using the Pi function the reflection formula takes on the form



\Pi\left(\frac{z}{m}\right) \, \Pi\left(\frac{z-1}{m}\right) \cdots \Pi\left(\frac{z-m+1}{m}\right) = (2 \pi)^{\frac{m-1}{2}} m^{-z-\frac{1}{2}} \Pi(z).

We also sometimes find

\pi(z) = \frac{1}{\Pi(z)},


Somewhat interestingly, the Volume of an n-ellipsoid with radii r_1,\dotsc,r_n can be expressed as


Relation to other functions[edit]

  • The gamma function is related to the Beta function by the formula
\pi^{-\frac{z}{2}} \; \Gamma\left(\frac{z}{2}\right) \zeta(z) = \pi^{-\frac{1-z}{2}} \; \Gamma\left(\frac{1-z}{2}\right) \; \zeta(1-z).
And also in the following elegant formula:
\zeta(z) \; \Gamma(z) = \int_{0}^{\infty} \frac{u^{z-1}}{e^u - 1} \; \mathrm{d}u,
which is valid only for Re(z) > 1.
\log\Gamma(x) = \zeta_{H}'(0,x) - \zeta'(0),
where ζH is the Hurwitz zeta function, ζ is the Riemann zeta function and the prime (') denotes differentiation in the first variable.
\langle\tau^n\rangle \equiv \int_0^\infty {\rm d}t\, t^{n-1}\, e^{ - \left( \frac{t}{\tau} \right)^\beta} = \frac{\tau^n}{\beta}\Gamma \left({n \over \beta }\right).

Particular values[edit]

Some particular values of the gamma function are:

\Gamma(-1) & = (-2)! && = \infty \\
\Gamma(0) & = (-1)! && = \infty \\
\Gamma(1) & = 0! && = 1 \\
\Gamma(2) & = 1! &&= 1 \\
\Gamma(3) & = 2! &&= 2 \\
\Gamma(4) & = 3! &&= 6 \\
\Gamma(-\tfrac{3}{2}) & = \tfrac{4}{3} \sqrt{\pi} &&\approx 2.363271801207 \\
\Gamma(-\tfrac{1}{2}) & = -2\sqrt{\pi} &&\approx -3.544907701811 \\
\Gamma(\tfrac{1}{2}) & = \sqrt{\pi} &&\approx 1.772453850905 \\
\Gamma(\tfrac{3}{2}) & = \tfrac{1}{2}\sqrt{\pi} &&\approx 0.88622692545 \\
\Gamma(\tfrac{5}{2}) & = \tfrac{3}{4}\sqrt{\pi} &&\approx 1.32934038818 \\
\Gamma(\tfrac{7}{2}) & = \tfrac{15}{8}\sqrt{\pi} &&\approx  3.32335097045



The gamma function can be computed to fixed precision for Re(z) ∈ [1, 2] by applying integration by parts to Euler's integral. For any positive number x the gamma function can be written

\Gamma(z) &= \int_0^x e^{-t} t^{z-1}\, \frac{\mathrm{d}t}{t} + \int_x^\infty e^{-t} t^{z+1}\, \frac{\mathrm{d}t}{t} \\
&= x^z e^{-x} \sum_{n=0}^\infty \frac{x^n}{z(z+1) \cdots (z+n)} + \int_x^\infty e^{-t} t^{z}\, \frac{\mathrm{d}t}{t}.

When Re(z) ∈ [1, 2] and x ≥ 1, the absolute value of the last integral is smaller than (x + 1) e−x. By choosing a large enough x, this last expression can be made smaller than 2N for any desired value N. Thus, the gamma function can be evaluated to N bits of precision with the above series.

The only fast algorithm for calculation of the Euler gamma function for any algebraic argument (including rational) was constructed by E.A. Karatsuba,[7][8][9]

For arguments that are integer multiples of 1/24 the gamma function can also be evaluated quickly using arithmetic-geometric mean iterations (see particular values of the gamma function).


 \ln (\Gamma(z)) = \ln (\Gamma(z+1)) - \ln(z)


 \ln (\Gamma(z)) \approx (z - \tfrac{1}{2}) \ln(z) - z + \tfrac{1}{2}\ln(2\pi).


 \ln(\Gamma(z-m)) = \ln(\Gamma(z)) - \sum_{k=1}^{m} \ln(z-k).


\Gamma(z)\sim z^{z - \frac{1}{2}} e^{-z} \sqrt{2\pi} \left( 1 + \frac{1}{12z} + \frac{1}{288z^2} - \frac{139}{51840 z^3} - \frac{571}{2488320 z^4}
        \right) \qquad \qquad \text{as }|z|\to\infty\text{ at constant}\quad |\arg(z)| < \pi

In a more "natural" presentation:

\ln \Gamma(z) \sim z \ln (z) - z - \tfrac{1}{2} \ln \left (\frac{z}{2\pi} \right ) + \frac{1}{12z} - \frac{1}{360z^3} +\frac{1}{1260 z^5}\qquad \qquad \text{as }|z|\to\infty\text{ at constant}\quad |\arg(z)| < \pi



where the Bk are the Bernoulli numbers.



Integration problems[edit]

The gamma function finds application in such diverse areas as quantum physics, astrophysics and fluid dynamics.[11] The gamma distribution, which is formulated in terms of the gamma function, is used in statistics to model a wide range of processes; for example, the time between occurrences of earthquakes.[12]

The primary reason for the gamma function's usefulness in such contexts is the prevalence of expressions of the type f(t)\,e^{-g(t)} which describe processes that decay exponentially in time or space. Integrals of such expressions can occasionally be solved in terms of the gamma function when no elementary solution exists. For example, if f is a power function and g is a linear function, a simple change of variables gives the evaluation

\int_0^\infty t^b e^{-at} \,dt = \frac{\Gamma(b+1)}{a^{b+1}}.


It is of course frequently useful to take limits of integration other than 0 and ∞ to describe the cumulation of a finite process, in which case the ordinary gamma function is no longer a solution; the solution is then called an incomplete gamma function. (The ordinary gamma function, obtained by integrating across the entire positive real line, is sometimes called the complete gamma function for contrast).

An important category of exponentially decaying functions is that of Gaussian functions


and integrals thereof, such as the error function. There are many interrelations between these functions and the gamma function; notably, the square root of π we obtained by evaluating Γ(1/2) is the "same" as that found in the normalizing factor of the error function and the normal distribution.


Another important special case is that of the beta function

\mathrm{\Beta}(x,y) = \int_0^1 t^{x-1}(1-t)^{y-1}\,dt = \frac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}.

Calculating products[edit]





\prod_{i=a}^b \frac{P(i)}{Q(i)} = \left( \prod_{j=1}^m \frac{\Gamma(b-p_j+1)}{\Gamma(a-p_j)} \right) \left( \prod_{k=1}^n \frac{\Gamma(a-q_k)}{\Gamma(b-q_k+1)} \right).




Analytic number theory[edit]


\Gamma\left(\frac{s}{2}\right)\zeta(s)\pi^{-\frac{s}{2}} = \Gamma\left(\frac{1-s}{2}\right)\zeta(1-s)\pi^{-\frac{1-s}{2}}.


\zeta(z) \; \Gamma(z) = \int_0^\infty \frac{t^{z}}{e^t-1} \; \frac{dt}{t}.




18th century: Euler and Stirling[edit]

Daniel Bernoulli's letter to Goldbach, 1729-10-06


n! = \prod_{k=1}^\infty \frac{\left(1+\frac{1}{k}\right)^n}{1+\frac{n}{k}}\,,





19th century: Gauss, Weierstrass and Legendre[edit]

The first page of Euler's paper

Carl Friedrich Gauss rewrote Euler's product as

\Gamma(z) = \lim_{m\to\infty}\frac{m^{z}m!}{z(z+1)(z+2)\cdots(z+m)}

and used this formula to discover new properties of the gamma function. Although Euler was a pioneer in the theory of complex variables, he does not appear to have considered the factorial of a complex number, as instead Gauss first did.[16] Gauss also proved the multiplication theorem of the gamma function and investigated the connection between the gamma function and elliptic integrals.


\Gamma(z) = \frac{e^{-\gamma z}}{z} \prod_{k=1}^\infty \left(1 + \frac{z}{k}\right)^{-1} e^{\frac{z}{k}},



It is justified to ask why we distinguish between the "ordinary factorial" and the gamma function by using distinct symbols, and particularly why the gamma function should be normalized to Γ(n+1) = n! instead of simply using "Γ(n) = n!". Consider that the notation for exponents, xn, has been generalized from integers to complex numbers xz without any change. Legendre's motivation for the normalization does not appear to be known, and has been criticized as cumbersome by some (the 20th-century mathematician Cornelius Lanczos, for example, called it "void of any rationality" and would instead use z!).[17] Legendre's normalization does simplify a few formulas, but complicates most others. From a modern point of view, the Legendre normalization of the Gamma function is the integral of the additive character e−x against the multiplicative character xz with respect to the Haar measure dx/x on the Lie group R+. Thus this normalization makes it clearer that the Gamma function is a continuous analogue of a Gauss sum.

19th-20th centuries: characterizing the gamma function[edit]





Reference tables and software[edit]







See also[edit]


  1. ^ Kingman, J.F.C. 1961. A convexity property of positive matrices. Quart. J. Math. Oxford (2) 12,283-284.
  4. ^ Harry Bateman and Arthur Erdélyi Higher Transcendental Functions [in 3 volumes]. Mc Graw-Hill Book Company, 1955.
  5. ^ H.M. Srivastava and J. Choi Series Associated with the Zeta and Related Functions. Kluwer Academic Publishers. The Netherlands, 2001
  6. ^ Iaroslav V. Blagouchine Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results. The Ramanujan Journal, 2013.
  9. ^ E.A. Karatsuba "Fast Algorithms and the FEE Method".
  16. ^ Remmert, R., Kay, L. D. (translator) (2006). Classical Topics in Complex Function Theory. Springer. ISBN 0-387-98221-3. 
  19. ^ Berry, M. "Why are special functions special?". Physics Today, April 2001


External links[edit]