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 two‐variable complex space at Re(x)=1 and Re(y)=1, 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  (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 TemplateBox[{1, 1, 1, 2, x, y}, AppellF1]:

Complex domain of TemplateBox[{1, 1, 1, 2, x, y}, AppellF1]:

AppellF1 is not an analytic function:

It has both singularities and discontinuities:

TemplateBox[{1, 1, 1, 1, 4, x}, AppellF1] is neither nondecreasing nor nonincreasing:

TemplateBox[{1, 1, 1, 1, 4, x}, AppellF1] is injective:

TemplateBox[{1, 1, 1, 1, 4, x}, AppellF1] is not surjective:

TemplateBox[{1, 1, 1, 1, 4, x}, AppellF1] is neither non-negative nor non-positive:

TemplateBox[{1, 1, 1, 1, 4, x}, AppellF1] 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=5 and x=1/2:

Formula for the ^(th) 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:

Wolfram Research (1999), AppellF1, Wolfram Language function, https://reference.wolfram.com/language/ref/AppellF1.html.

Text

Wolfram Research (1999), AppellF1, Wolfram Language function, https://reference.wolfram.com/language/ref/AppellF1.html.

CMS

Wolfram Language. 1999. "AppellF1." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/AppellF1.html.

APA

Wolfram Language. (1999). AppellF1. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AppellF1.html

BibTeX

@misc{reference.wolfram_2022_appellf1, author="Wolfram Research", title="{AppellF1}", year="1999", howpublished="\url{https://reference.wolfram.com/language/ref/AppellF1.html}", note=[Accessed: 08-June-2023 ]}

BibLaTeX

@online{reference.wolfram_2022_appellf1, organization={Wolfram Research}, title={AppellF1}, year={1999}, url={https://reference.wolfram.com/language/ref/AppellF1.html}, note=[Accessed: 08-June-2023 ]}