The best-known properties and formulas for Struve functions
Real values for real arguments
For real values of parameter and positive argument , the values of the Struve functions and are real.
Simple values at zero
The Struve functions and have rather simple values for the argument :
Specific values for specialized parameter
In the cases when parameter is equal to , the Struve functions and can be expressed through the sine and cosine (or hyperbolic sine and cosine) multiplied by rational and sqrt functions, for example:
The previous formulas are the particular cases of the following general formulas:
The Struve functions and are defined for all complex values of their parameter and variable . They are analytical functions of and over the whole complex ‐ and ‐planes excluding the branch cuts. For fixed integer , the functions and are entire functions of . For fixed , the functions and are entire functions of .
Poles and essential singularities
For fixed , the functions and have an essential singularity at . At the same time, the point is a branch point (except cases for integer ).
With respect to , the Struve functions have only one essential singular point at .
Branch points and branch cuts.
For fixed noninteger , the functions and have two branch points: and .
If functions and have branch cuts, they are single‐valued functions on the ‐plane cut along the interval , where they are continuous from above:
From below, functions have discontinuities that are described by the formulas:
The Struve functions and do not have periodicity.
Parity and symmetry
The Struve functions and have mirror symmetry (except on the branch cut interval (-∞, 0)):
The Struve functions and have generalized parity (either odd or even) with respect to variable :
The Struve functions and have the following series expansions through series that converge on the whole ‐plane:
Interestingly, closed-form expressions for the truncated version of the Taylor series at the origin can be expressed through the generalized hypergeometric function , for example:
Asymptotic series expansions
The asymptotic behavior of the Struve functions and can be described by the following formulas (only the main terms of asymptotic expansion are given):
The previous formulas are valid in any directions approaching point to infinity (z∞). In particular cases when or , these formulas can be simplified to the following relations:
The Struve functions and have simple integral representations through the sine (or hyperbolic sine) and power functions:
Arguments of the Struve functions and with square root arguments can sometimes be simplified:
The Struve functions and satisfy the following recurrence identities:
The previous identities can be generalized to the following recurrence identities with a jump of length n:
Simple representations of derivatives
The derivatives of the Struve functions and have simple representations that can also be expressed through Struve functions with different indices:
The symbolic -order derivatives have the following representations:
The Struve functions and appeared as special solutions of the special inhomogeneous Bessel second-order linear differential equations:
where and are arbitrary constants and , , , and are Bessel functions.
The previous equations are very similar and can be converted into each other by changing to .