# AppellF1

AppellF1[a,b1,b2,c,x,y]

is the Appell hypergeometric function of two variables .

# Details • AppellF1 belongs to the family of Appell functions that generalizes the hypergeometric series and solves the system of Horn PDEs with polynomial coefficients.
• Mathematical function, suitable for both symbolic and numerical manipulation.
• has a primary definition through the hypergeometric series , which is convergent inside the region .
• The region of convergence of the Appell F1 series for real values of its arguments is the following:
• • In general satisfies the following Horn PDE system »: .
• reduces to when or .
• For certain special arguments, AppellF1 automatically evaluates to exact values.
• AppellF1 can be evaluated to arbitrary numerical precision.
• AppellF1[a,b1,b2,c,x,y] has singular lines in twovariable complex space at and , and has branch cut discontinuities along the rays from to in and .
• FullSimplify and FunctionExpand include transformation rules for AppellF1.

# Examples

open allclose all

## Basic Examples(8)

Evaluate numerically:

Evaluate symbolically:

The defining sum:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Series expansion at a singular point:

## Scope(26)

### Numerical Evaluation(4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate AppellF1 efficiently at high precision:

### Specific Values(4)

Values at fixed points:

Evaluate symbolically:

Value at zero:

For simple parameters, AppellF1 evaluates to simpler functions:

### Visualization(4)

Plot the AppellF1 function for various parameters:

Plot AppellF1 as a function of its second parameter :

Plot the real part of :

Plot the imaginary part of :

Plot the real part of in three dimensions:

Plot the imaginary part of in three dimensions:

### Function Properties(9)

Real domain of AppellF1:

Complex domain of AppellF1:

AppellF1 is not an analytic function:

Has both singularities and discontinuities: is neither nondecreasing nor nonincreasing: is injective: is not surjective: is neither non-negative nor non-positive: is neither convex nor concave:

### Differentiation(3)

First derivative with respect to y:

Higher derivatives with respect to y:

Plot the higher derivatives with respect to y when a=b1=b2=2, c=10 and x=1/2:

Formula for the  derivative with respect to y:

### Series Expansions(2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

## Applications(1)

The Appell function solves the following system of PDEs with polynomial coefficients:

Check that is a solution:

## Properties & Relations(2)

Evaluate integrals in terms of AppellF1:

Use FullSimplify to simplify some expressions involving AppellF1:

## Neat Examples(1)

Many elementary and special functions are special cases of AppellF1: