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AppellF4

See Also
- AppellF1
- AppellF2
- AppellF3
- Hypergeometric2F1
- Gamma
- Pochhammer
Related Guides
- Hypergeometric Functions
- See Also
  - AppellF1
  - AppellF2
  - AppellF3
  - Hypergeometric2F1
  - Gamma
  - Pochhammer
- Related Guides
  - Hypergeometric Functions

AppellF4

AppellF4[a,b,c₁,c₂,x,y]

is the Appell hypergeometric function of two variables .

Details

AppellF4 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 .
The region of convergence of the Appell F4 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, AppellF4 automatically evaluates to exact values.
AppellF4 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 AppellF4 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 AppellF4 efficiently at high precision:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix AppellF4 function using MatrixFunction:

Specific Values (3)

Values at fixed points:

Simplify to Hypergeometric2F1 functions:

Value at zero:

Visualization (3)

Plot the AppellF4 function for various parameters:

Plot AppellF4 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=1/2, b=3/2, c₁=1/3, c₂=4 and x=1/5:

Formula for the 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 AppellF4:

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