Hypergeometric2F1

Hypergeometric2F1[a,b,c,z]

is the hypergeometric function .

Details

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  (43)

Numerical Evaluation  (4)

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 threads elementwise over lists:

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 TemplateBox[{{1, /, 3}, {1, /, 3}, {2, /, 3}, x}, Hypergeometric2F1]=2/3:

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 TemplateBox[{1, {1, /, 2}, {sqrt(, 2, )}, {x, +, {ⅈ,  , y}}}, Hypergeometric2F1]:

Plot the imaginary part of TemplateBox[{1, {1, /, 2}, {sqrt(, 2, )}, {x, +, {ⅈ,  , y}}}, Hypergeometric2F1]:

Function Properties  (9)

Real domain of Hypergeometric2F1:

Complex domain of Hypergeometric2F1:

TemplateBox[{{2, /, 3}, {3, {sqrt(, 2, )}}, 3, z}, Hypergeometric2F1] is an analytic function on its real domain:

It is neither analytic nor meromorphic in the complex plane:

TemplateBox[{{2, /, 3}, {3, {sqrt(, 2, )}}, 3, z}, Hypergeometric2F1] is non-decreasing on its real domain:

TemplateBox[{1, {1, /, 2}, 1, z}, Hypergeometric2F1] is injective:

TemplateBox[{1, {1, /, 2}, 1, z}, Hypergeometric2F1] is not surjective:

TemplateBox[{{2, /, 3}, {3, {sqrt(, 2, )}}, 3, z}, Hypergeometric2F1] is non-negative on its real domain:

TemplateBox[{a, {1, /, 2}, 1, z}, Hypergeometric2F1] has both singularity and discontinuity for :

TemplateBox[{a, {1, /, 2}, 1, z}, Hypergeometric2F1] is convex on its real domain:

TraditionalForm formatting:

Differentiation  (3)

First derivative:

Higher derivatives:

Plot higher derivatives for , and :

Formula for the ^(th) 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 TemplateBox[{{1, /, 3}, {1, /, 3}, {2, /, 3}, x}, Hypergeometric2F1] 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:

HankelTransform:

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:

TraditionalForm formatting:

Applications  (1)

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:

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)

TemplateBox[{a, b, c, x}, Hypergeometric2F1] 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 :

Wolfram Research (1988), Hypergeometric2F1, Wolfram Language function, https://reference.wolfram.com/language/ref/Hypergeometric2F1.html (updated 1999).

Text

Wolfram Research (1988), Hypergeometric2F1, Wolfram Language function, https://reference.wolfram.com/language/ref/Hypergeometric2F1.html (updated 1999).

BibTeX

@misc{reference.wolfram_2021_hypergeometric2f1, author="Wolfram Research", title="{Hypergeometric2F1}", year="1999", howpublished="\url{https://reference.wolfram.com/language/ref/Hypergeometric2F1.html}", note=[Accessed: 13-June-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_hypergeometric2f1, organization={Wolfram Research}, title={Hypergeometric2F1}, year={1999}, url={https://reference.wolfram.com/language/ref/Hypergeometric2F1.html}, note=[Accessed: 13-June-2021 ]}

CMS

Wolfram Language. 1988. "Hypergeometric2F1." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 1999. 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