# Hypergeometric2F1 Hypergeometric2F1[a,b,c,z]

is the hypergeometric function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• The function has the series expansion .
• For certain special arguments, Hypergeometric2F1 automatically evaluates to exact values.
• Hypergeometric2F1 can be evaluated to arbitrary numerical precision.
• Hypergeometric2F1 automatically threads over lists.
• Hypergeometric2F1[a,b,c,z] has a branch cut discontinuity in the complex plane running from to .
• FullSimplify and FunctionExpand include transformation rules for Hypergeometric2F1.
• Hypergeometric2F1 can be used with Interval and CenteredInterval objects. »

# Examples

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## Basic Examples(7)

Evaluate numerically:

Evaluate symbolically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Expand Hypergeometric2F1 in a Taylor series at the origin:

Series expansion at Infinity:

Series expansion at a singular point:

## Scope(44)

### Numerical Evaluation(5)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate Hypergeometric2F1 efficiently at high precision:

Hypergeometric2F1 can be used with Interval and CenteredInterval objects:

### Specific Values(6)

Hypergeometric2F1 automatically evaluates to simpler functions for certain parameters:

Exact value of Hypergeometric2F1 at unity:

Hypergeometric series terminates if either of the first two parameters is a negative integer:

Find a value of satisfying the equation :

Permutation symmetry:

Heun functions can be reduced to hypergeometric functions:

### Visualization(3)

Plot the Hypergeometric2F1 function:

Plot Hypergeometric2F1 as a function of its third parameter :

Plot the real part of :

Plot the imaginary part of :

### Function Properties(9)

Real domain of Hypergeometric2F1:

Complex domain of Hypergeometric2F1: is an analytic function 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(3)

First derivative:

Higher derivatives:

Plot higher derivatives for , and :

Formula for the  derivative:

### Integration(3)

Indefinite integral of Hypergeometric2F1:

Definite integral of Hypergeometric2F1:

Integral involving a power function:

### Series Expansions(6)

Taylor expansion for Hypergeometric2F1:

Plot the first three approximations for around :

General term in the series expansion of Hypergeometric2F1:

Expand Hypergeometric2F1 in a series near :

Expand Hypergeometric2F1 in a series around :

Give the result for an arbitrary symbolic direction :

Apply Hypergeometric2F1 to a power series:

### Integral Transforms(2)

Compute the Laplace transform using LaplaceTransform:

### Function Identities and Simplifications(2)

Argument simplification:

Recurrence identities:

### Function Representations(5)

Basic definition:

Relation to the JacobiP polynomial:

Hypergeometric2F1 can be represented as a DifferentialRoot:

Hypergeometric2F1 can be represented in terms of MeijerG:

## Applications(2)

Force acting on an electric point charge outside a neutral dielectric sphere of radius :

The limit of infinite dielectric constant, corresponding to an uncharged insulated conducting sphere:

Force at a large distance from the sphere:

Two players roll dice. If the total of both numbers is less than 10, the second player is paid 4 cents; otherwise the first player is paid 9 cents. Is the game fair?:

The game is not fair, since mean scores per game are not equal:

Find the probability that after n games the player at the disadvantage scores more:

The probability exhibits oscillations:

The maximum probability is attained at :

## Properties & Relations(2)

Use FunctionExpand to expand Hypergeometric2F1 into other functions:

Find limits of Hypergeometric2F1 from below and above the branch cut:

## Possible Issues(1) is equivalent to for generic :

However, if is a negative integer, Hypergeometric2F1 returns a polynomial:

## Neat Examples(1)

The discrete Kepler problem with initial conditions and can be solved as a hypergeometric function:

The energy depends on :

Finite norm states exist for an attractive potential with and :