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StruveH

gives the Struve function .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
for integer n satisfies the differential equation .
StruveH[n,z] has a branch cut discontinuity in the complex plane running from to .
For certain special arguments, StruveH automatically evaluates to exact values.
StruveH can be evaluated to arbitrary numerical precision.
StruveH 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 (43)

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 StruveH efficiently at high precision:

StruveH threads elementwise over lists:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix StruveH function using MatrixFunction:

Specific Values (4)

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

Limiting values at infinity:

Value of at a complex infinity is indeterminate:

Find a zero of :

Visualization (5)

Plot the StruveH function for :

Plot the StruveH function for negative integer values of :

Plot the StruveH 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 StruveH for half-integer :

Complex domain:

Approximate function range of :

Function range of :

Parity:

is analytic in the interior of its real domain:

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

is neither nondecreasing nor nonincreasing:

is not injective:

is not surjective:

is neither non-negative nor non-positive:

is neither convex nor concave:

Differentiation (3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Plot higher derivatives for :

Formula for the derivative:

Integration (4)

Indefinite integral:

Definite integral of StruveH:

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 StruveH at infinity:

StruveH can be applied to a power series:

Integral Transforms (2)

Compute the Hankel transform using HankelTransform:

Mellin transform for using MellinTransform:

Function Identities and Simplifications (2)

Argument simplifications:

Recurrence relation:

Function Representations (4)

Series representation:

Representation in terms of StruveL:

StruveH can be represented in terms of MeijerG:

TraditionalForm formatting:

Generalizations & Extensions (1)

StruveH can be applied to a power series:

Applications (2)

Solve the inhomogeneous Bessel differential equation:

The diffraction pattern from an infinitely long line source by a circular aperture:

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