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  (34)

Numerical Evaluation  (4)

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:

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 TemplateBox[{1, {1, /, 2}, {sqrt(, 2, )}, {x, +, iy}}, Hypergeometric2F1Regularized]:

Plot the imaginary part of TemplateBox[{1, {1, /, 2}, {sqrt(, 2, )}, {x, +, iy}}, Hypergeometric2F1Regularized]:

Function Properties  (11)

Real domain of Hypergeometric2F1Regularized:

Complex domain:

Recurrence identity:

Hypergeometric2F1Regularized threads elementwise over lists:

TemplateBox[{{2, /, 3}, {3,  , {sqrt(, 2, )}}, 3, z}, Hypergeometric2F1Regularized] is analytic on its real domain:

It is neither analytic nor meromorphic in the complex plane:

TemplateBox[{{2, /, 3}, {3,  , {sqrt(, 2, )}}, 3, z}, Hypergeometric2F1Regularized] is non-decreasing on its real domain:

TemplateBox[{1, {1, /, 2}, 1, z}, Hypergeometric2F1Regularized] is injective:

TemplateBox[{1, {1, /, 2}, 1, z}, Hypergeometric2F1Regularized] is not surjective:

TemplateBox[{{2, /, 3}, {3,  , {sqrt(, 2, )}}, 3, z}, Hypergeometric2F1Regularized] is non-negative on its real domain:

TemplateBox[{a, {1, /, 2}, 1, z}, Hypergeometric2F1Regularized] has both singularity and discontinuity for :

TemplateBox[{1, {1, /, 2}, 1, z}, Hypergeometric2F1Regularized] is convex on its real domain:

TraditionalForm formatting:

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 ^(th) 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 alpha 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:

Wolfram Research (1996), Hypergeometric2F1Regularized, Wolfram Language function, https://reference.wolfram.com/language/ref/Hypergeometric2F1Regularized.html.

Text

Wolfram Research (1996), Hypergeometric2F1Regularized, Wolfram Language function, https://reference.wolfram.com/language/ref/Hypergeometric2F1Regularized.html.

BibTeX

@misc{reference.wolfram_2021_hypergeometric2f1regularized, author="Wolfram Research", title="{Hypergeometric2F1Regularized}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/Hypergeometric2F1Regularized.html}", note=[Accessed: 27-November-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_hypergeometric2f1regularized, organization={Wolfram Research}, title={Hypergeometric2F1Regularized}, year={1996}, url={https://reference.wolfram.com/language/ref/Hypergeometric2F1Regularized.html}, note=[Accessed: 27-November-2021 ]}

CMS

Wolfram Language. 1996. "Hypergeometric2F1Regularized." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Hypergeometric2F1Regularized.html.

APA

Wolfram Language. (1996). Hypergeometric2F1Regularized. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Hypergeometric2F1Regularized.html