gives the modified Struve function .


  • 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.


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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:

StruveL threads elementwise over lists:

Specific Values  (4)

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

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

Limiting value at infinity:

Find a root of TemplateBox[{0, x}, StruveL]=1:

Visualization  (4)

Plot the StruveL function for integer :

Plot the StruveL function for half-integer values of :

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

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

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

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

Function Properties  (3)

Function domain of StruveL for half-integer :

Complex domain:

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


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:

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

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

This is twice the integral over half the interval:

Series Expansions  (4)

Taylor expansion for TemplateBox[{1, x}, StruveL]:

Plot the first three approximations for TemplateBox[{1, x}, StruveL] around :

General term in the series expansion of TemplateBox[{1, x}, StruveL]:

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:

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:

Introduced in 1999