Struve Function

Abramowitz and Stegun (1972, pp. 496-499) define the Struve function as

$${\mathcal H}_\nu(z) = ({\textstyle{1\over 2}}z)^{\nu+1} \sum... ...+{\textstyle{3\over 2}})\Gamma(k+\nu+{\textstyle{3\over 2}})},$$ (1)

where $\Gamma(z)$ is the Gamma Function. Watson (1966, p. 338) defines the Struve function as

$${\mathcal H}_\nu(z) \equiv{2({\textstyle{1\over 2}}z)^\nu\ov... ...textstyle{1\over 2}})} \int_0^1 (1-t^2)^{\nu-1/2}\sin(zt)\,dt.$$ (2)

The series expansion is

$${\mathcal H}_\nu(z) =\sum_{m=0}^\infty (-1)^m {({\textstyle{... ...+{\textstyle{3\over 2}})\Gamma(\nu+m+{\textstyle{3\over 2}})}.$$ (3)

For half integer orders,

$${\mathcal H}_{n+1/2}(z) = Y_{n+1/2}(z)+{1\over\pi}\sum_{m=0}^n {\Gamma(m+{\textstyle{1\over 2}})({\textstyle{1\over 2}}z)^{-2m+n-1/2}\over\Gamma(n+1-m)}$$ (4)

$${\mathcal H}_{-(n+1/2)}(z) = (-1)^n J_{n+1/2}(z).$$ (5)

The Struve function and its derivatives satisfy

$${\mathcal H}_{\nu-1}(z)-{\mathcal H}_{\nu+1}(z)=2{\mathcal H... ...}}z)^\nu\over\sqrt{\pi}\, \Gamma(\nu+{\textstyle{3\over 2}})}.$$ (6)

See also Anger Function, Bessel Function, Modified Struve Function, Weber Functions

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Struve Function ${\bf H}_\nu(x)$.'' §12.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 496-498, 1972.

Spanier, J. and Oldham, K. B. ``The Struve Function.'' Ch. 57 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 563-571, 1987.

Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.