is the confluent hypergeometric function .
- Mathematical function, suitable for both symbolic and numerical manipulation.
- The function has the integral representation .
- HypergeometricU[a,b,z] has a branch cut discontinuity in the complex plane running from to .
- For certain special arguments, HypergeometricU automatically evaluates to exact values.
- HypergeometricU can be evaluated to arbitrary numerical precision.
- HypergeometricU automatically threads over lists.
- HypergeometricU can be used with Interval and CenteredInterval objects. »
Examplesopen allclose all
Basic Examples (5)
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 for complex arguments:
Evaluate HypergeometricU efficiently at high precision:
HypergeometricU threads elementwise over lists:
HypergeometricU can be used with Interval and CenteredInterval objects:
Specific Values (3)
HypergeometricU automatically evaluates to simpler functions for certain parameters:
Plot the HypergeometricU function:
Plot HypergeometricU as a function of its second parameter:
Function Properties (9)
Real domain of HypergeometricU:
Complex domain of HypergeometricU:
is neither non-decreasing nor non-increasing on its real domain:
is positive on its real domain:
has both singularity and discontinuity for z≤0:
Indefinite integral HypergeometricU:
Definite integral of HypergeometricU:
Series Expansions (3)
Series expansion for HypergeometricU:
Plot the first three approximations for around :
Expand HypergeometricU in series around infinity:
Apply HypergeometricU to a power series:
Integral Transforms (3)
Compute the Laplace transform using LaplaceTransform:
Function Representations (5)
Representation through Gamma and Hypergeometric1F1:
HypergeometricU can be represented in terms of MeijerG:
HypergeometricU can be represented as a DifferentialRoot:
Solve the confluent hypergeometric differential equation:
Borel summation of divergent series of gives HypergeometricU:
Define distribution for scaled condition number of a WishartMatrixDistribution:
Sample the scaled condition number of a large matrix and check that it agrees with asymptotic closed-form distribution:
The asymptotic scaled condition number distribution has infinite mean:
Properties & Relations (3)
Use FunctionExpand to expand HypergeometricU into simpler functions:
Integrate may give results involving HypergeometricU:
HypergeometricU can be represented as a DifferenceRoot:
Possible Issues (1)
The default setting of $MaxExtraPrecision can be insufficient to obtain requested precision:
A larger setting for $MaxExtraPrecision may be needed:
Wolfram Research (1988), HypergeometricU, Wolfram Language function, https://reference.wolfram.com/language/ref/HypergeometricU.html (updated 2022).
Wolfram Language. 1988. "HypergeometricU." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/HypergeometricU.html.
Wolfram Language. (1988). HypergeometricU. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HypergeometricU.html