is the Appell hypergeometric function of two variables .
- Mathematical function, suitable for both symbolic and numerical manipulation.
- has series expansion .
- 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 and , and has branch cut discontinuities along the rays from to in and .
- FullSimplify and FunctionExpand include transformation rules for AppellF1.
Examplesopen allclose all
Basic Examples (8)
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:
Numerical Evaluation (4)
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:
Value at zero:
For simple parameters, AppellF1 evaluates to simpler functions:
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 :
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=5 and x=1/2:
Formula for the 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:
The Appell system of PDEs for the Picard modular function associated with :
Check that is a solution:
Properties & Relations (2)