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.


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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  (29)

Numerical Evaluation  (4)

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:

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, )}, {x, +, {ⅈ,  , y}}}, HypergeometricU]:

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

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:



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  (2)

Solve the confluent hypergeometric differential equation:

Borel summation of divergent series of gives HypergeometricU:

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  (1)

Visualize the confluency relation :

Introduced in 1988