# Hypergeometric2F1Regularized

Hypergeometric2F1Regularized[a,b,c,z]

is the regularized hypergeometric function .

# Details # Examples

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## Basic Examples(7)

Evaluate numerically:

Regularize Hypergeometric2F1 for negative integer values of the parameter :

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:

## Scope(35)

### Numerical Evaluation(5)

Evaluate numerically:

Evaluate to high precision:

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

Complex number inputs:

Evaluate efficiently at high precision:

Hypergeometric2F1Regularized can be used with Interval and CenteredInterval objects:

### Specific Values(7)

Hypergeometric2F1Regularized for symbolic a and b:

Limiting values at infinity:

Values at zero:

Find a value of x for which Hypergeometric2F1Regularized[2,1,2,x ]=0.4:

Evaluate symbolically for integer parameters:

Evaluate symbolically for half-integer parameters:

Hypergeometric2F1Regularized automatically evaluates to simpler functions for certain parameters:

### Visualization(3)

Plot the Hypergeometric2F1Regularized function:

Plot Hypergeometric2F1Regularized as a function of its second parameter :

Plot the real part of :

Plot the imaginary part of :

### Function Properties(11)

Real domain of Hypergeometric2F1Regularized:

Complex domain:

Recurrence identity: is analytic on its real domain:

It is neither analytic nor meromorphic in the complex plane: is non-decreasing on its real domain: is injective: is not surjective: is non-negative on its real domain: has both singularity and discontinuity for : is convex on its real domain:

### Differentiation(2)

First derivative with respect to z when a=1, b=2, c=3:

Higher derivatives with respect to z when a=1, b=1/2, c=1/3:

Plot the higher derivatives with respect to z when a=1, b=1/2, c=1/3:

Formula for the  derivative with respect to z:

### Integration(3)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

### Series Expansions(4)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Find the series expansion at Infinity:

Find series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

## Applications(1)

Define the fractional derivative of EllipticK:

Check that for integer order it coincides with the ordinary derivative:

Evaluate derivative of order 1/2:

## Properties & Relations(5)

Evaluate symbolically for numeric third argument:

Use FunctionExpand to expand Hypergeometric2F1Regularized into other functions:

Integrate may give results involving Hypergeometric2F1Regularized:

Hypergeometric2F1Regularized can be represented as a DifferentialRoot:

Hypergeometric2F1Regularized can be represented in terms of MeijerG: