# AppellF2

AppellF2[a,b1,b2,c1,c2,x,y]

is the Appell hypergeometric function of two variables .

# Details • AppellF2 belongs to the family of Appell functions that generalize 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 F2 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, AppellF2 automatically evaluates to exact values.
• AppellF2 can be evaluated to arbitrary numerical precision.

# Examples

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## Basic Examples(7)

Evaluate numerically:

The defining sum:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Plot a family of AppellF2 functions:

Series expansion at the origin:

## Scope(15)

### Numerical Evaluation(4)

Evaluate numerically:

Evaluate to high precision:

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

Complex number inputs:

Evaluate AppellF2 efficiently at high precision:

### Specific Values(3)

Values at fixed points:

Simplify to Hypergeometric2F1 functions:

Value at zero:

### Visualization(3)

Plot the AppellF2 function for various parameters:

Plot AppellF2 as a function of its second parameter :

Plot the real part of :

Plot the imaginary part of :

### Differentiation(4)

First derivative with respect to x:

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, c1=c2=5 and x=1/5:

Formula for the  derivative with respect to y:

### Series Expansions(1)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

## Applications(1)

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

Check that is a solution:

## Neat Examples(1)

Many elementary and special functions are special cases of AppellF2: