HypergeometricU

HypergeometricU[a,b,z]

is the Tricomi confluent hypergeometric function TemplateBox[{a, b, z}, HypergeometricU].

Details

Examples

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Basic Examples  (5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Scope  (39)

Numerical Evaluation  (5)

Evaluate to high precision:

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:

Limiting value at infinity:

Find a value of satisfying the equation TemplateBox[{3, 2, x}, HypergeometricU]=1:

Visualization  (3)

Plot the HypergeometricU function:

Plot HypergeometricU as a function of its second parameter:

Plot the real part of TemplateBox[{{1, /, 2}, {sqrt(, 3, )}, z}, HypergeometricU]:

Plot the imaginary part of TemplateBox[{{1, /, 2}, {sqrt(, 3, )}, z}, HypergeometricU]:

Function Properties  (9)

Real domain of HypergeometricU:

Complex domain of HypergeometricU:

TemplateBox[{{sqrt(, 3, )}, {sqrt(, 2, )}, z}, HypergeometricU] is not an analytic function:

TemplateBox[{{sqrt(, 3, )}, {sqrt(, 2, )}, z}, HypergeometricU] is neither non-decreasing nor non-increasing on its real domain:

TemplateBox[{{sqrt(, 3, )}, {sqrt(, 2, )}, z}, HypergeometricU] is injective:

TemplateBox[{{sqrt(, 3, )}, {sqrt(, 2, )}, z}, HypergeometricU] is not surjective:

TemplateBox[{{sqrt(, 3, )}, {sqrt(, 2, )}, z}, HypergeometricU] is positive on its real domain:

TemplateBox[{{sqrt(, 3, )}, {sqrt(, 2, )}, z}, HypergeometricU] has both singularity and discontinuity for z0:

TemplateBox[{{sqrt(, 3, )}, {sqrt(, 2, )}, z}, HypergeometricU] 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 HypergeometricU:

Definite integral of HypergeometricU:

More integrals:

Series Expansions  (3)

Series expansion for HypergeometricU:

Plot the first three approximations for TemplateBox[{{1, /, 2}, {sqrt(, 3, )}, x}, HypergeometricU] around :

Expand HypergeometricU in series around infinity:

Apply HypergeometricU to a power series:

Integral Transforms  (3)

Compute the Laplace transform using LaplaceTransform:

MellinTransform:

HankelTransform:

Function Identities and Simplifications  (2)

Argument simplification:

Recurrence identities:

Function Representations  (5)

Primary definition:

Representation through Gamma and Hypergeometric1F1:

HypergeometricU can be represented in terms of MeijerG:

HypergeometricU can be represented as a DifferentialRoot:

TraditionalForm formatting:

Applications  (3)

Solve the confluent hypergeometric differential equation:

Borel summation of the divergent series for the function 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:

Neat Examples  (2)

Visualize the confluency relation TemplateBox[{TemplateBox[{a, {a, -, b, +, 1}, c, {1, -, {c, /, z}}}, Hypergeometric2F1], c, infty}, Limit2Arg]z^(-a)=TemplateBox[{a, b, z}, HypergeometricU]:

Plot the Riemann surface of TemplateBox[{{1, /, 2}, {2, /, 3}, z}, HypergeometricU]:

Wolfram Research (1988), HypergeometricU, Wolfram Language function, https://reference.wolfram.com/language/ref/HypergeometricU.html (updated 2022).

Text

Wolfram Research (1988), HypergeometricU, Wolfram Language function, https://reference.wolfram.com/language/ref/HypergeometricU.html (updated 2022).

CMS

Wolfram Language. 1988. "HypergeometricU." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/HypergeometricU.html.

APA

Wolfram Language. (1988). HypergeometricU. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HypergeometricU.html

BibTeX

@misc{reference.wolfram_2023_hypergeometricu, author="Wolfram Research", title="{HypergeometricU}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/HypergeometricU.html}", note=[Accessed: 19-March-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_hypergeometricu, organization={Wolfram Research}, title={HypergeometricU}, year={2022}, url={https://reference.wolfram.com/language/ref/HypergeometricU.html}, note=[Accessed: 19-March-2024 ]}