It is a type of smooth curve, lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A type of smooth curve, lying in a plane, defined by its geometric properties or by equations for which it is the solution set.
In mathematics, a hyperbola (plural hyperbolas or hyperbolae) is a type of smooth curve, lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola may be defined equivalently as the locus of points where the absolute value of the difference of the distances to the two foci is a constant equal to 2 a, the distance between its two vertices. Hyperbolic Functions. The hyperbolic functions, , , , , (hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cosecant, hyperbolic secant, and hyperbolic cotangent) are analogs of the circular functions, defined by removing s appearing in the complex exponentials. For example, (1). so. hyperbolic function. n. 1. (Mathematics) any of a group of functions of an angle expressed as a relationship between the distances of a point on a hyperbola to the origin and to the coordinate axes. The hyperbolic cosine, defined by the equation cosh x = 1 / 2 (ex + e-x). c. The hyperbolic tangent, defined by the equation tanh x = sinh x /cosh x. d. The hyperbolic cotangent, defined by the equation coth x = cosh x /sinh x. Definition of the hyperbolic sine function for a complex argument. In the complex ‐plane, the function is defined by the same formula used for real values: Here are two graphics showing the real and imaginary parts of the hyperbolic sine function over the complex plane. The values of the hyperbolic sine function for special values of its argument can be easily derived from the corresponding values of the circular sine in special points of the circle: The values at infinity can be expressed by the following formulas: n. Any of a set of six functions related, for a real or complex variable x, to the hyperbola in a manner analogous to the relationship of the trigonometric functions to a circle, including: a. The hyperbolic sine, defined by the equation sinh x = 1 / 2 (ex-e - x). b. The hyperbolic cosine, defined by the equation cosh x = 1 / 2 (ex + e-x). c. The hyperbolic tangent, defined by the equation tanh x = sinh x /cosh x. d. The hyperbolic cotangent, defined by the equation coth x = cosh x /sinh x. Values in points. The values of the hyperbolic sine function for special values of its argument can be easily derived from the corresponding values of the circular sine in special points of the circle: The values at infinity can be expressed by the following formulas: General characteristic where is the hyperbolic cosine and is the hyperbolic sine. The hyperbolic functions arise in many problems of mathematics and mathematical physics in which integrals involving arise (whereas the circular functions involve). The hyperbolic functions, , , , , (hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cosecant, hyperbolic secant, and hyperbolic cotangent) are analogs of the circular functions, defined by removing s appearing in the complex exponentials. For example, (1). so. $y=\log_{1/e} x=-\ln x$. Here is a logarithmic function with a base between 0 and 1. $y=\log_{-e} x$. If one attempts to use a negative base in a logarithmic function, points will be scattered along two curves (both asymptotic to the $y$-axis), and a great many more values of the function will be undefined. $y=\coth x=\dfrac{e^x+e^{-x}}{e^x-e^{-x}}$. This is the graph of the hyperbolic cotangent function. $y=\operatorname{sech} x=\dfrac{2}{e^x+e^{-x}}$. This is the graph of the hyperbolic secant function. $y=\operatorname{csch} x=\dfrac{2}{e^x-e^{-x}}$. This is the graph of the hyperbolic cosecant function. $y=\sinh x=\dfrac{e^x-e^{-x}}{2}$. The hyperbolic sine function is asymptotic to a pair of exponential functions. $y=\cosh x=\dfrac{e^x+e^{-x}}{2}$. The hyperbolic cosine function is also asymptotic to a pair of exponential functions. This function is also known as a catenary, which is the shape taken by a chain suspended between two points. $y=\tanh x=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}$. This is the graph of the hyperbolic tangent function. $y=\coth x=\dfrac{e^x+e^{-x}}{e^x-e^{-x}}$. This is the graph of the hyperbolic cotangent function. $y=\operatorname{sech} x=\dfrac{2}{e^x+e^{-x}}$. This is the graph of the hyperbolic secant function. $y=\operatorname{csch} x=\dfrac{2}{e^x-e^{-x}}$. This is the graph of the hyperbolic cosecant function.
https://en.wikipedia.org/wiki/Hyperbola http://mathworld.wolfram.com/HyperbolicFunctions.html http://www.thefreedictionary.com/Hyperbolic+curve http://functions.wolfram.com/ElementaryFunctions/Sinh/introductions/Sinh/ShowAll.html http://www.thefreedictionary.com/Hyperbolic+curve http://functions.wolfram.com/ElementaryFunctions/Sinh/introductions/Sinh/ShowAll.html http://mathworld.wolfram.com/HyperbolicFunctions.html http://www.milefoot.com/math/planecurves/expfcns.htm http://www.milefoot.com/math/planecurves/expfcns.htm
what is a hyperbolic curve