# HypergeometricU

HypergeometricU[a,b,z]

is the confluent hypergeometric function .

# Details • 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. »

# 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 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 :

### Visualization(3)

Plot the HypergeometricU function:

Plot HypergeometricU as a function of its second parameter:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(9)

Real domain of HypergeometricU:

Complex domain of HypergeometricU: is not an analytic function: is neither non-decreasing nor non-increasing on its real domain: is injective: is not surjective: is positive on its real domain: has both singularity and discontinuity for z0: is convex on its real domain:

### Differentiation(3)

First derivative:

Higher derivatives:

Plot higher derivatives for and :

Formula for the  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 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:

## 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 :