Enable JavaScript to interact with content and submit forms on Wolfram websites. Learn how

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 $sum_(m=0)^inftysum_(n=0)^infty(TemplateBox[{a, {m, +, n}}, Pochhammer] TemplateBox[{{b, _, 1}, m}, Pochhammer] TemplateBox[{{b, _, 2}, n}, Pochhammer] )/(TemplateBox[{c, {m, +, n}}, Pochhammer]m! n!)x^m y^n$ , 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 two‐variable 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 (28)

Numerical Evaluation (6)

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:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix AppellF1 function using MatrixFunction:

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:

TraditionalForm formatting:

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:

Top