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StruveL

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 $TemplateBox[{n, {i, , z}}, StruveL]=-ie^(-inpi/2)TemplateBox[{n, z}, StruveH]$ .
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 (42)

Numerical Evaluation (6)

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:

StruveL threads elementwise over lists:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix StruveL function using MatrixFunction:

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 (9)

Function domain of StruveL for half-integer :

Complex domain:

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

Parity:

is analytic in the interior of its real domain:

It is not analytic everywhere, as it has both singularities and discontinuities:

is nondecreasing on its real domain:

is injective:

is not surjective:

is non-negative on its real domain:

is convex on its real domain:

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:

HankelTransform:

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:

TraditionalForm formatting:

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:

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