Built-in Mathematica Symbol

HypergeometricU

HypergeometricU[a, b, z]
is the confluent hypergeometric function

.

MORE INFORMATION

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.

EXAMPLES

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Basic Examples (3)

Evaluate numerically:

Plot

:

Series expansion at the origin:

Evaluate numerically:

In[1]:=

Out[1]=

Plot

:

In[1]:=

Out[1]=

Series expansion at the origin:

In[1]:=

Out[1]=

Scope (6)

Evaluate for complex arguments:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

HypergeometricU threads element-wise over lists:

HypergeometricU automatically evaluates to simpler functions for certain parameters:

TraditionalForm formatting:

Generalizations & Extensions (1)

Expand HypergeometricU in series around infinity:

Applications (2)

Solve the confluent hypergeometric differential equation:

Borel summation of divergent series of

gives HypergeometricU:

Properties & Relations (2)

Use FunctionExpand to expand HypergeometricU into simpler functions:

Integrate may give results involving HypergeometricU:

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

: