Hypergeometric Functions

Hypergeometric Functions


analytic functions defined for ǀzǀ< 1 using hypergeometric series. The name “hypergeometric function” was coined by J. Wallis in 1650. Hypergeometric functions are integrals of the hypergeometric equation

z(1 - z)w” + [γ - (1 + α + β)Z]w´ - αβw = 0

This equation has three regular singular points 0, 1, and », and is the canonical form of hypergeometric-type equations. The most important functions of mathematical analysis are integrals of equations of the hypergeometric type (for example, spherical functions) or of equations resulting from the hypergeometric-type equations by merging their singular points (for example, cylindrical functions). The theory of hypergeometric-type equations became the basis for the origin of an important mathematical discipline, the analytic theory of differential equations.

Between various hypergeometric functions

w = F(α, β γ; z)

there are numerous relationships, for example,

F(α, 1; γ, z) = (1 - z)-1F(1, γ - α;γ; z/(z - 1))


Whittaker, E. T., and G. N. Watson. Kurs sovremennogo analiza, 2nd ed., part 2. Moscow, 1963.
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1]; as the remaining integral in that equation could be evaluated analytically into a complicated expression involving hypergeometric functions, we preferred to calculate the integral numerically.
He begins by discussing generalized hypergeometric functions, of which Bessel functions are a special case, then applies the results to many useful representation of Bessel functions and their integrals by specializing the parameters.
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2](x, y) can be expressed in terms of hypergeometric functions of x/[(1 - 4y).
and these functions can be represented in terms of the hypergeometric functions and their explicit expressions are given in the paper [34].