Hypergeometric2F1Regularized [a,b,c,z]
is the regularized hypergeometric function TemplateBox[{a, b, c, z}, Hypergeometric2F1]/TemplateBox[{c}, Gamma].
Hypergeometric2F1Regularized
Hypergeometric2F1Regularized [a,b,c,z]
is the regularized hypergeometric function TemplateBox[{a, b, c, z}, Hypergeometric2F1]/TemplateBox[{c}, Gamma].
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- Hypergeometric2F1Regularized [a,b,c,z] is finite for all finite values of a, b, c, and z so long as .
- For certain special arguments, Hypergeometric2F1Regularized automatically evaluates to exact values.
- Hypergeometric2F1Regularized can be evaluated to arbitrary numerical precision.
- Hypergeometric2F1Regularized automatically threads over lists.
- Hypergeometric2F1Regularized can be used with Interval and CenteredInterval objects. »
Examples
open all close allBasic 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 (36)
Numerical Evaluation (6)
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:
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 Hypergeometric2F1Regularized function using MatrixFunction :
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, )}, z}, Hypergeometric2F1Regularized]:
Plot the imaginary part of TemplateBox[{1, {1, /, 2}, {sqrt(, 2, )}, z}, 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:
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 :
See Also
Tech Notes
Related Guides
Related Links
History
Introduced in 1996 (3.0) | Updated in 2021 (13.0) ▪ 2022 (13.1)
Text
Wolfram Research (1996), Hypergeometric2F1Regularized, Wolfram Language function, https://reference.wolfram.com/language/ref/Hypergeometric2F1Regularized.html (updated 2022).
CMS
Wolfram Language. 1996. "Hypergeometric2F1Regularized." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. 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
BibTeX
@misc{reference.wolfram_2025_hypergeometric2f1regularized, author="Wolfram Research", title="{Hypergeometric2F1Regularized}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/Hypergeometric2F1Regularized.html}", note=[Accessed: 04-January-2026]}
BibLaTeX
@online{reference.wolfram_2025_hypergeometric2f1regularized, organization={Wolfram Research}, title={Hypergeometric2F1Regularized}, year={2022}, url={https://reference.wolfram.com/language/ref/Hypergeometric2F1Regularized.html}, note=[Accessed: 04-January-2026]}