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LommelT1

LommelT1

LommelT1[m,n,z]

gives the modified Lommel function of the first kind .

Details

LommelT1 is also known as a modified Lommel function of the first kind.
Lommel functions are typically used to represent particular solutions of Bessel-type differential equations.
Mathematical function, suitable for both symbolic and numerical manipulation.
satisfies the differential equation . »
LommelT1[m,n,z] has a branch cut discontinuity in the complex z plane running from to .
For certain special arguments, LommelT1 automatically evaluates to exact values.
LommelT1 can be evaluated to arbitrary numerical precision.
LommelT1 automatically threads over lists.

Examples

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Basic Examples (6)

Evaluate numerically:

Evaluate symbolically:

Plot :

Plot over a subset of the complexes:

Expand LommelT1 in a Taylor series at the origin:

Series expansion at a generic point:

Scope (26)

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:

LommelT1 threads elementwise over lists in the last argument:

Compute the matrix LommelT1 function using MatrixFunction:

Specific Values (3)

LommelT1 automatically evaluates to simpler functions for certain parameters:

The exact value of LommelT1 at unity for a specific set of parameters:

LommelT1 is singular whenever or is a non-positive odd integer:

Visualization (3)

Plot the LommelT1 function for and varying :

Plot the LommelT1 function for and varying :

Plot the real part of :

Plot the imaginary part of :

Function Properties (3)

Real domain of LommelT1:

Complex domain of LommelT1:

Singularity and discontinuity information of LommelT1:

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 LommelT1:

Definite integral of LommelT1:

Integral involving a power function:

More integrals with LommelT1:

Series Expansions (2)

Taylor expansion for LommelT1:

Approximations for around :

Expand LommelT1 in a series around :

Integral Transforms (2)

Compute the Laplace transform using LaplaceTransform:

HankelTransform:

Function Representations (3)

Relation to the HypergeometricPFQ function:

LommelT1 can be represented in terms of MeijerG:

FoxH representation of LommelT1 function:

Applications (1)

Solve the inhomogeneous version of the modified Bessel equation:

Properties & Relations (2)

Use FunctionExpand to expand LommelT1 into hypergeometric functions:

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

Neat Examples (1)

Riemann surface of LommelT1:

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