is the Appell hypergeometric function of two variables .


  • AppellF3 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 max(TemplateBox[{x}, Abs],TemplateBox[{y}, Abs])<1.
  • The region of convergence of the Appell F3 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, AppellF3 automatically evaluates to exact values.
  • AppellF3 can be evaluated to arbitrary numerical precision.


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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 AppellF3 functions:

Series expansion at the origin:

TraditionalForm formatting:

Scope  (15)

Numerical Evaluation  (4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate AppellF3 efficiently at high precision:

Specific Values  (3)

Values at fixed points:

Simplify to Hypergeometric2F1 functions:

Value at zero:

Visualization  (3)

Plot the AppellF3 function for various parameters:

Plot AppellF3 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 a1=a2=2, b1=b2=5, c=1/2 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 AppellF3:

Wolfram Research (2023), AppellF3, Wolfram Language function,


Wolfram Research (2023), AppellF3, Wolfram Language function,


Wolfram Language. 2023. "AppellF3." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2023). AppellF3. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2023_appellf3, author="Wolfram Research", title="{AppellF3}", year="2023", howpublished="\url{}", note=[Accessed: 04-October-2023 ]}


@online{reference.wolfram_2023_appellf3, organization={Wolfram Research}, title={AppellF3}, year={2023}, url={}, note=[Accessed: 04-October-2023 ]}