is the regularized hypergeometric function .
- 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. »
Examplesopen allclose all
Basic Examples (7)
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:
Numerical Evaluation (5)
The precision of the output tracks the precision of the input:
Evaluate efficiently at high precision:
Hypergeometric2F1Regularized can be used with Interval and CenteredInterval objects:
Specific Values (7)
Hypergeometric2F1Regularized for symbolic a and b:
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:
Plot the Hypergeometric2F1Regularized function:
Plot Hypergeometric2F1Regularized as a function of its second parameter :
Function Properties (11)
Real domain of Hypergeometric2F1Regularized:
Hypergeometric2F1Regularized threads elementwise over lists:
is analytic on its real domain:
It is neither analytic nor meromorphic in the complex plane:
is non-decreasing on its real domain:
is non-negative on its real domain:
has both singularity and discontinuity for :
Compute the indefinite integral using Integrate:
Define the fractional derivative of EllipticK:
Check that for integer order it coincides with the ordinary derivative:
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 (updated 2022).
Wolfram Language. 1996. "Hypergeometric2F1Regularized." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/Hypergeometric2F1Regularized.html.
Wolfram Language. (1996). Hypergeometric2F1Regularized. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Hypergeometric2F1Regularized.html