AppellF2 [a,b1,b2,c1,c2,x,y]
is the Appell hypergeometric function of two variables .
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
open all close allBasic 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)
Visualization (3)
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 :
See Also
AppellF1 AppellF3 AppellF4 Hypergeometric2F1 Gamma Pochhammer
Related Guides
Related Links
History
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: 17-November-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: 17-November-2025]}