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AppellF1
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AppellF1

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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

open allclose all

Basic Examples  (8)Summary of the most common use cases

Evaluate numerically:

In[1]:=1
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https://wolfram.com/xid/0cg43j0b0-tbhg
Out[1]=1

Evaluate symbolically:

In[1]:=1
✖
https://wolfram.com/xid/0cg43j0b0-dwr7ms
Out[1]=1

The defining sum:

In[1]:=1
✖
https://wolfram.com/xid/0cg43j0b0-vxycv
Out[1]=1

Plot over a subset of the reals:

In[1]:=1
✖
https://wolfram.com/xid/0cg43j0b0-m51
Out[1]=1

Plot over a subset of the complexes:

In[1]:=1
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https://wolfram.com/xid/0cg43j0b0-kiedlx
Out[1]=1

Series expansion at the origin:

In[1]:=1
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https://wolfram.com/xid/0cg43j0b0-f65ufv
Out[1]=1

Series expansion at Infinity:

In[1]:=1
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https://wolfram.com/xid/0cg43j0b0-fgrnr3
Out[1]=1

Series expansion at a singular point:

In[1]:=1
✖
https://wolfram.com/xid/0cg43j0b0-onayf6
Out[1]=1

Scope  (28)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

In[2]:=2
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https://wolfram.com/xid/0cg43j0b0-l274ju
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0cg43j0b0-cksbl4
Out[3]=3

Evaluate to high precision:

In[1]:=1
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https://wolfram.com/xid/0cg43j0b0-b0wt9
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0cg43j0b0-xth5g
Out[2]=2

Complex number inputs:

In[1]:=1
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https://wolfram.com/xid/0cg43j0b0-hfml09
Out[1]=1

Evaluate AppellF1 efficiently at high precision:

In[1]:=1
✖
https://wolfram.com/xid/0cg43j0b0-di5gcr
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0cg43j0b0-bq2c6r
Out[2]=2

Compute average-case statistical intervals using Around:

In[1]:=1
✖
https://wolfram.com/xid/0cg43j0b0-cw18bq
Out[1]=1

Compute the elementwise values of an array:

In[1]:=1
✖
https://wolfram.com/xid/0cg43j0b0-thgd2
Out[1]=1

Or compute the matrix AppellF1 function using MatrixFunction:

In[2]:=2
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https://wolfram.com/xid/0cg43j0b0-o5jpo
Out[2]=2

Specific Values  (4)

Values at fixed points:

In[1]:=1
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https://wolfram.com/xid/0cg43j0b0-o7spmt
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0cg43j0b0-bjvvy9
Out[2]=2

Evaluate symbolically:

In[1]:=1
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https://wolfram.com/xid/0cg43j0b0-nev5ew
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0cg43j0b0-ff32fh
Out[2]=2

Value at zero:

In[1]:=1
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https://wolfram.com/xid/0cg43j0b0-jevg27
Out[1]=1

For simple parameters, AppellF1 evaluates to simpler functions:

In[1]:=1
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https://wolfram.com/xid/0cg43j0b0-dk2yax
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0cg43j0b0-yc3mn
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0cg43j0b0-ih9u38
Out[3]=3

Visualization  (4)

Plot the AppellF1 function for various parameters:

In[1]:=1
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https://wolfram.com/xid/0cg43j0b0-ecj8m7
Out[1]=1

Plot AppellF1 as a function of its second parameter :

In[1]:=1
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https://wolfram.com/xid/0cg43j0b0-gq0e7
Out[1]=1

Plot the real part of :

In[1]:=1
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https://wolfram.com/xid/0cg43j0b0-8gdas
Out[1]=1

Plot the imaginary part of :

In[2]:=2
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https://wolfram.com/xid/0cg43j0b0-ceiagl
Out[2]=2

Plot the real part of F_1(2,1,4,3,0,z) in three dimensions:

In[1]:=1
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https://wolfram.com/xid/0cg43j0b0-btjb9w
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0cg43j0b0-bwytwm
Out[2]=2

Function Properties  (9)

Real domain of AppellF1:

In[1]:=1
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https://wolfram.com/xid/0cg43j0b0-cl7ele
Out[1]=1

Complex domain of AppellF1:

In[1]:=1
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https://wolfram.com/xid/0cg43j0b0-imtovy
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0cg43j0b0-fholy3
Out[2]=2

AppellF1 is not an analytic function:

In[1]:=1
✖
https://wolfram.com/xid/0cg43j0b0-h5x4l2
Out[1]=1

Has both singularities and discontinuities:

In[2]:=2
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https://wolfram.com/xid/0cg43j0b0-mdtl3h
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0cg43j0b0-mn5jws
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0cg43j0b0-nlz7s
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0cg43j0b0-poz8g
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0cg43j0b0-ctca0g
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0cg43j0b0-cxk3a6
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0cg43j0b0-frlnsr
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0cg43j0b0-kvwb5k
Out[1]=1

TemplateBox[{1, 1, 1, 1, 4, x}, AppellF1] is neither convex nor concave:

In[1]:=1
✖
https://wolfram.com/xid/0cg43j0b0-8kku21
Out[1]=1

TraditionalForm formatting:

In[1]:=1
✖
https://wolfram.com/xid/0cg43j0b0-im209a

Differentiation  (3)

First derivative with respect to y:

In[1]:=1
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https://wolfram.com/xid/0cg43j0b0-krpoah
Out[1]=1

Higher derivatives with respect to y:

In[1]:=1
✖
https://wolfram.com/xid/0cg43j0b0-z33jv
Out[1]=1

Plot the higher derivatives with respect to y when a=b1=b2=2, c=10 and x=1/2:

In[2]:=2
✖
https://wolfram.com/xid/0cg43j0b0-fxwmfc
Out[2]=2

Formula for the ^(th) derivative with respect to y:

In[1]:=1
✖
https://wolfram.com/xid/0cg43j0b0-cb5zgj
Out[1]=1

Series Expansions  (2)

Find the Taylor expansion using Series:

In[1]:=1
✖
https://wolfram.com/xid/0cg43j0b0-ewr1h8
Out[1]=1

Plots of the first three approximations around :

In[2]:=2
✖
https://wolfram.com/xid/0cg43j0b0-binhar
Out[3]=3

Taylor expansion at a generic point:

In[1]:=1
✖
https://wolfram.com/xid/0cg43j0b0-jwxla7
Out[1]=1

Applications  (1)Sample problems that can be solved with this function

The Appell function solves the following system of PDEs with polynomial coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0cg43j0b0-c7ls19

Check that is a solution:

In[2]:=2
✖
https://wolfram.com/xid/0cg43j0b0-gxo7wy
In[3]:=3
✖
https://wolfram.com/xid/0cg43j0b0-hxlylz
Out[3]=3

Properties & Relations  (2)Properties of the function, and connections to other functions

Evaluate integrals in terms of AppellF1:

In[1]:=1
✖
https://wolfram.com/xid/0cg43j0b0-bu0r44
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg43j0b0-npk18o
Out[2]=2

Use FullSimplify to simplify some expressions involving AppellF1:

In[1]:=1
✖
https://wolfram.com/xid/0cg43j0b0-f8hbfb
Out[1]=1

Neat Examples  (1)Surprising or curious use cases

Many elementary and special functions are special cases of AppellF1:

In[1]:=1
✖
https://wolfram.com/xid/0cg43j0b0-chq0g5
In[2]:=2
✖
https://wolfram.com/xid/0cg43j0b0-2lhh82
Wolfram Research (1999), AppellF1, Wolfram Language function, https://reference.wolfram.com/language/ref/AppellF1.html (updated 2023).
✖
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).

✖
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.

✖
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

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

BibTeX

@misc{reference.wolfram_2025_appellf1, author="Wolfram Research", title="{AppellF1}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/AppellF1.html}", note=[Accessed: 13-April-2025 ]}

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@misc{reference.wolfram_2025_appellf1, author="Wolfram Research", title="{AppellF1}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/AppellF1.html}", note=[Accessed: 13-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_appellf1, organization={Wolfram Research}, title={AppellF1}, year={2023}, url={https://reference.wolfram.com/language/ref/AppellF1.html}, note=[Accessed: 13-April-2025 ]}

✖
@online{reference.wolfram_2025_appellf1, organization={Wolfram Research}, title={AppellF1}, year={2023}, url={https://reference.wolfram.com/language/ref/AppellF1.html}, note=[Accessed: 13-April-2025 ]}