Calculate Appell's First Hypergeometric Function of Two Variables

This function computes the numeric value of Appell's first hypergeometric function F1(a; b1, b2; c; x, y) using integral representation. The Appell hypergeometric function is a generalization of the hypergeometric function to two variables.

Usage

AppellsF1(a, b1, b2, c, x, y)

Arguments

a

A numeric value representing the first parameter.

b1

A numeric value representing the second parameter.

b2

A numeric value representing the third parameter.

c

A numeric value representing the fourth parameter.

x

A numeric value representing the first variable.

y

A numeric value representing the second variable.

Value

A numeric value of Appell's first hypergeometric function F1(a; b1, b2; c; x, y).

Details

The function uses the integral representation: $$F_1(a; b_1, b_2; c; x, y) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)} \int_0^1 t^{a-1}(1-t)^{c-a-1}(1-xt)^{-b_1}(1-yt)^{-b_2} dt$$

Examples

# Calculate F1(0.5; 0.5, 0; 1; 0.96, 1.2)
AppellsF1(0.5, 0.5, 0, 1, 0.96, 1.2)
#> [1] 1.920117

# Calculate F1(1; 1, 1; 2; 0.3, 0.4)
AppellsF1(1, 1, 1, 2, 0.3, 0.4)
#> [1] 1.541507