The modified Struve functionsLα(x) are equal to −ie−iαπ / 2Hα(ix), are solutions y(x) of the non-homogeneous Bessel's differential equation:
And further defined its second-kind version as .
Definitions
Since this is a non-homogeneous equation, solutions can be constructed from a single particular solution by adding the solutions of the homogeneous problem. In this case, the homogeneous solutions are the Bessel functions, and the particular solution may be chosen as the corresponding Struve function.
Power series expansion
Struve functions, denoted as Hα(z) have the power series form
The modified Struve functions, denoted Lα(z), have the following power series form
Integral form
Another definition of the Struve function, for values of α satisfying Re(α) > − 1/2, is possible expressing in term of the Poisson's integral representation:
Asymptotic forms
For small x, the power series expansion is given above.
The Struve functions satisfy the following recurrence relations:
Relation to other functions
Struve functions of integer order can be expressed in terms of Weber functionsEn and vice versa: if n is a non-negative integer then
Struve functions of order n + 1/2 where n is an integer can be expressed in terms of elementary functions. In particular if n is a non-negative integer then
The Struve and Weber functions were shown to have an application to beamforming in.[1], and in describing the effect of confining interface on Brownian motion of colloidal particles at low Reynolds numbers.[2]
References
^K. Buchanan, C. Flores, S. Wheeland, J. Jensen, D. Grayson and G. Huff, "Transmit beamforming for radar applications using circularly tapered random arrays," 2017 IEEE Radar Conference (RadarConf), 2017, pp. 0112-0117, doi: 10.1109/RADAR.2017.7944181
^B. U. Felderhof, "Effect of the wall on the velocity autocorrelation function and long-time tail of Brownian motion." The Journal of Physical Chemistry B 109.45, 2005, pp. 21406-21412
R. M. Aarts and Augustus J. E. M. Janssen (2003). "Approximation of the Struve function H1 occurring in impedance calculations". J. Acoust. Soc. Am. 113 (5): 2635–2637. Bibcode:2003ASAJ..113.2635A. doi:10.1121/1.1564019. PMID12765381.