# StruveL

StruveL[n,z]

gives the modified Struve function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• for integer is related to the ordinary Struve function by .
• StruveL[n,z] has a branch cut discontinuity in the complex plane running from to .
• For certain special arguments, StruveL automatically evaluates to exact values.
• StruveL can be evaluated to arbitrary numerical precision.
• StruveL automatically threads over lists.

# Examples

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## Basic Examples(5)

Evaluate numerically:

Plot :

Plot over a subset of the complexes:

Series expansion at the origin:

Asymptotic expansion at Infinity:

## Scope(34)

### 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:

Evaluate StruveL efficiently at high precision:

### Specific Values(4)

For half-integer indices, StruveL evaluates to elementary functions:

Value of at a complex infinity is indeterminate:

Limiting value at infinity:

Find a root of :

### Visualization(4)

Plot the StruveL function for integer :

Plot the StruveL function for half-integer values of :

Plot the real part of :

Plot the imaginary part of :

Plot the real part of :

Plot the imaginary part of :

### Function Properties(3)

Function domain of StruveL for half-integer :

Complex domain:

Approximate function range of StruveL for half-integer values of :

Parity:

### Differentiation(3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Plot higher derivatives for :

Formula for the  derivative:

### Integration(4)

Indefinite integral:

Definite integral:

Definite integral of the odd integrand over an interval centered at the origin is 0:

Definite integral of the even integrand over an interval centered at the origin:

This is twice the integral over half the interval:

### Series Expansions(4)

Taylor expansion for :

Plot the first three approximations for around :

General term in the series expansion of :

Series expansion of StruveL at infinity:

StruveL can be applied to a power series:

### Integral Transforms(2)

Compute the Laplace transform using LaplaceTransform:

### Function Identities and Simplifications(2)

Argument simplifications:

Recurrence relation:

### Function Representations(4)

Series representation:

Representation in terms of StruveH:

StruveL can be represented in terms of MeijerG:

## Generalizations & Extensions(1)

StruveL can be applied to a power series:

## Applications(3)

Solve the inhomogeneous Bessel differential equation:

3D relativistic, non-Markovian transition PDF that has the Gaussian non-relativistic limit:

Its normalization is computed after a change of variables :

The mean saddle order in the mean-field -trigonometric model as a function of temperature: