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 max(TemplateBox[{x}, Abs],TemplateBox[{y}, Abs])<1.
  • 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 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

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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  (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 F_1(2,1,4,3,0,z) in three dimensions:

Plot the imaginary part of F_1(2,1,4,3,0,z) 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:

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=10 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 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:

Wolfram Research (1999), AppellF1, Wolfram Language function, https://reference.wolfram.com/language/ref/AppellF1.html (updated 2023).

Text

Wolfram Research (1999), AppellF1, Wolfram Language function, https://reference.wolfram.com/language/ref/AppellF1.html (updated 2023).

CMS

Wolfram Language. 1999. "AppellF1." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. 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_2023_appellf1, author="Wolfram Research", title="{AppellF1}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/AppellF1.html}", note=[Accessed: 06-December-2023 ]}

BibLaTeX

@online{reference.wolfram_2023_appellf1, organization={Wolfram Research}, title={AppellF1}, year={2023}, url={https://reference.wolfram.com/language/ref/AppellF1.html}, note=[Accessed: 06-December-2023 ]}