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  (35)

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:

StruveH threads elementwise over lists:

Specific Values  (4)

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

Limiting values at infinity:

Value of TemplateBox[{{1, /, 2}, z}, StruveH] at a complex infinity is indeterminate:

Find a zero of TemplateBox[{0, x}, StruveH]:

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 TemplateBox[{0, {x, +, {ⅈ,  , y}}}, StruveH]:

Plot the imaginary part of TemplateBox[{0, {x, +, {ⅈ,  , y}}}, StruveH]:

Plot the real part of TemplateBox[{{-, 3}, {x, +, {ⅈ,  , y}}}, StruveH]:

Plot the imaginary part of TemplateBox[{{-, 3}, {x, +, {ⅈ,  , y}}}, StruveH]:

Function Properties  (3)

Function domain of StruveH for half-integer :

Complex domain:

Approximate function range of TemplateBox[{{-, {1, /, 2}}, x}, StruveH]:

Function range of TemplateBox[{{5, /, 2}, x}, StruveH]:

Parity:

Differentiation  (3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Plot higher derivatives for :

Formula for the ^(th) derivative:

Integration  (4)

Indefinite integral:

Definite integral of StruveH:

Definite integral of the odd integrand TemplateBox[{0, x}, StruveH] over an interval centered at the origin is 0:

Definite integral of the even integrand TemplateBox[{1, x}, StruveH] over an interval centered at the origin:

This is twice the integral over half the interval:

Series Expansions  (4)

Taylor expansion for TemplateBox[{0, x}, StruveH]:

Plot the first three approximations for TemplateBox[{0, x}, StruveH] around :

General term in the series expansion of TemplateBox[{0, x}, StruveH]:

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 TemplateBox[{0, x}, StruveH] 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:

Introduced in 1999
 (4.0)