AppellF1

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

is the Appell hypergeometric function of two variables .

Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• has series expansion .
• 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

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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(25)

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(3)

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 :

Function Properties(9)

Real domain of :

Complex domain of :

AppellF1 is not an analytic function:

It 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=5 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 system of PDEs for the Picard modular function associated with :

Check that is a solution:

Properties & Relations(2)

Evaluate integrals in terms of AppellF1:

Use FullSimplify to simplify some expressions involving AppellF1: