# Hypergeometric2F1

Hypergeometric2F1[a,b,c,z]

is the hypergeometric function .

# 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:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix Hypergeometric2F1 function using MatrixFunction:

### 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 simplify 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(3)

An expression for the 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:

An approximation for the 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? Compute the probability that the first player gets paid:

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 :

Riemann's differential equation with three regular singularities at and exponent parameters , subject to the constraint :

Construct two linearly independent solutions in terms of Hypergeometric2F1:

Verify that the solutions satisfy Riemann's equation:

## 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 in terms of hypergeometric functions:

The energy depends on :

Finite norm states exist for an attractive potential with and :

Wolfram Research (1988), Hypergeometric2F1, Wolfram Language function, https://reference.wolfram.com/language/ref/Hypergeometric2F1.html (updated 2022).

#### Text

Wolfram Research (1988), Hypergeometric2F1, Wolfram Language function, https://reference.wolfram.com/language/ref/Hypergeometric2F1.html (updated 2022).

#### CMS

Wolfram Language. 1988. "Hypergeometric2F1." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/Hypergeometric2F1.html.

#### APA

Wolfram Language. (1988). Hypergeometric2F1. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Hypergeometric2F1.html

#### BibTeX

@misc{reference.wolfram_2024_hypergeometric2f1, author="Wolfram Research", title="{Hypergeometric2F1}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/Hypergeometric2F1.html}", note=[Accessed: 13-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_hypergeometric2f1, organization={Wolfram Research}, title={Hypergeometric2F1}, year={2022}, url={https://reference.wolfram.com/language/ref/Hypergeometric2F1.html}, note=[Accessed: 13-August-2024 ]}