# StruveH

StruveH[n,z]

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(41)

### 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 StruveH efficiently at high precision:

### 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:

## 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:

Wolfram Research (1999), StruveH, Wolfram Language function, https://reference.wolfram.com/language/ref/StruveH.html.

#### Text

Wolfram Research (1999), StruveH, Wolfram Language function, https://reference.wolfram.com/language/ref/StruveH.html.

#### CMS

Wolfram Language. 1999. "StruveH." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/StruveH.html.

#### APA

Wolfram Language. (1999). StruveH. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/StruveH.html

#### BibTeX

@misc{reference.wolfram_2023_struveh, author="Wolfram Research", title="{StruveH}", year="1999", howpublished="\url{https://reference.wolfram.com/language/ref/StruveH.html}", note=[Accessed: 27-February-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2023_struveh, organization={Wolfram Research}, title={StruveH}, year={1999}, url={https://reference.wolfram.com/language/ref/StruveH.html}, note=[Accessed: 27-February-2024 ]}