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 TemplateBox[{x}, Abs]+TemplateBox[{y}, Abs]<1.
  • 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:

TraditionalForm formatting:

Scope  (17)

Numerical Evaluation  (6)

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:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix AppellF2 function using MatrixFunction:

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 ^(th) 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:

Wolfram Research (2023), AppellF2, Wolfram Language function, https://reference.wolfram.com/language/ref/AppellF2.html.

Text

Wolfram Research (2023), AppellF2, Wolfram Language function, https://reference.wolfram.com/language/ref/AppellF2.html.

CMS

Wolfram Language. 2023. "AppellF2." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/AppellF2.html.

APA

Wolfram Language. (2023). AppellF2. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AppellF2.html

BibTeX

@misc{reference.wolfram_2025_appellf2, author="Wolfram Research", title="{AppellF2}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/AppellF2.html}", note=[Accessed: 05-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_appellf2, organization={Wolfram Research}, title={AppellF2}, year={2023}, url={https://reference.wolfram.com/language/ref/AppellF2.html}, note=[Accessed: 05-July-2025 ]}