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LommelT2

LommelT2

LommelT2[m,n,z]

gives the modified Lommel function of the second kind .

Details

LommelT2 is also known as a modified Lommel function of the second kind.
Lommel functions are typically used to represent particular solutions of Bessel-type differential equations.
Mathematical function, suitable for both symbolic and numerical manipulation.
LommelT2[m,n,z] has a branch cut discontinuity in the complex z plane running from to .
For certain special arguments, LommelT2 automatically evaluates to exact values.
LommelT2 can be evaluated to arbitrary numerical precision.
LommelT2 automatically threads over lists.

Examples

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Basic Examples (5)

Evaluate numerically:

Plot :

Plot over a subset of the complexes:

Expand LommelT2 in a Taylor series at the origin:

Series expansion at a generic point:

Scope (25)

Numerical Evaluation (4)

Evaluate numerically to high precision:

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

Evaluate for complex arguments and parameters:

LommelS2 threads elementwise over lists in the last argument:

Compute the matrix LommelT2 function using MatrixFunction:

Specific Values (2)

LommelT2 automatically evaluates to simpler functions for certain parameters:

The exact value of LommelT2 at unity for a specific set of parameters:

Visualization (3)

Plot the LommelT2 function for and varying :

Plot the LommelT2 function for and varying :

Plot the real part of :

Plot the imaginary part of :

Function Properties (3)

Real domain of LommelT2:

Complex domain of LommelT2:

Singularity and discontinuity information of LommelT2:

has a branch cut discontinuity for :

is a singular point for :

Differentiation (2)

First derivative:

Higher derivatives:

Plot higher derivatives for , :

Integration (4)

Indefinite integral of LommelT2:

Definite integral of LommelT2:

Integral involving a power function:

More integrals with LommelT2:

Series Expansions (2)

Taylor expansion for LommelT2:

Approximations for around :

Expand LommelT2 in a series around :

Integral Transforms (1)

HankelTransform:

Function Representations (4)

Relation to the HypergeometricPFQ and BesselI functions:

Relation to the HypergeometricPFQ and Hypergeometric0F1 functions:

LommelT2 can be represented in terms of MeijerG:

FoxH representation of the LommelT2 function:

Properties & Relations (2)

Use FunctionExpand to expand LommelT2 into hypergeometric and Bessel functions:

FunctionExpand for a specific set of parameters can generate simpler special functions:

Neat Examples (1)

Riemann surface of LommelT2:

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