Struve functions

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1: 11.12 Physical Applications

§11.12 Physical Applications

►Applications of Struve functions occur in water-wave and surface-wave problems (Hirata (1975) and Ahmadi and Widnall (1985)), unsteady aerodynamics (Shaw (1985) and Wehausen and Laitone (1960)), distribution of fluid pressure over a vibrating disk (McLachlan (1934)), resistive MHD instability theory (Paris and Sy (1983)), and optical diffraction (Levine and Schwinger (1948)). More recently Struve functions have appeared in many particle quantum dynamical studies of spin decoherence (Shao and Hänggi (1998)) and nanotubes (Pedersen (2003)).

2: 11.8 Analogs to Kelvin Functions

§11.8 Analogs to Kelvin Functions

►For properties of Struve functions of argument $x{\mathrm{e}}^{\pm 3\pi \mathrm{i}/4}$ see McLachlan and Meyers (1936).

3: 11.3 Graphics

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Figure 11.3.1: ${𝐇}_{\nu }\left(x\right)$ for $0\le x\le 12$ and $\nu =0,\frac{1}{2},1,\frac{3}{2},2,3$. Magnify

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Figure 11.3.2: ${𝐊}_{\nu }\left(x\right)$ for and $\nu =0,\frac{1}{2},1,\frac{3}{2},2,3$. Magnify

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Figure 11.3.3: ${𝐇}_{\nu }\left(x\right)$ for $0\le x\le 12$ and $\nu =-3,-2,-\frac{3}{2},-1,-\frac{1}{2}$. Magnify

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Figure 11.3.4: ${𝐊}_{\nu }\left(x\right)$ for and $\nu =-4,-3,-2,-1,0$. … Magnify

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Figure 11.3.5: ${𝐇}_{\nu }\left(x\right)$ for $0\le x\le 8$ and $-4\le \nu \le 4$. Magnify 3D Help

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4: 11.1 Special Notation

§11.1 Special Notation

… ►For the functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$, $I_{\nu }\left(z\right)$, and $K_{\nu }\left(z\right)$ see §§10.2(ii), 10.25(ii). ►The functions treated in this chapter are the Struve functions ${𝐇}_{\nu }\left(z\right)$ and ${𝐊}_{\nu }\left(z\right)$, the modified Struve functions ${𝐋}_{\nu }\left(z\right)$ and ${𝐌}_{\nu }\left(z\right)$, the Lommel functions $s_{\mu ,\nu }\left(z\right)$ and $S_{\mu ,\nu }\left(z\right)$, the Anger function ${𝐉}_{\nu }\left(z\right)$, the Weber function ${𝐄}_{\nu }\left(z\right)$, and the associated Anger–Weber function ${𝐀}_{\nu }\left(z\right)$.

5: 11.7 Integrals and Sums

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§11.7(i) Indefinite Integrals

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§11.7(ii) Definite Integrals

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§11.7(iii) Laplace Transforms

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§11.7(v) Compendia

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6: 11.14 Tables

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§11.14(ii) Struve Functions

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§11.14(iii) Integrals

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§11.14(v) Incomplete Functions

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Agrest and Maksimov (1971, Chapter 11) defines incomplete Struve, Anger, and Weber functions and includes tables of an incomplete Struve function ${𝐇}_n\left(x,\alpha \right)$ for $n=0,1$, $x=0(.2)10$, and $\alpha =0(.2)1.4,\frac{1}{2}\pi$, together with surface plots.

7: 11.2 Definitions

§11.2 Definitions

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§11.2(i) Power-Series Expansions

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8: 11.4 Basic Properties

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§11.4(i) Half-Integer Orders

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§11.4(ii) Inequalities

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§11.4(iv) Expansions in Series of Bessel Functions

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§11.4(v) Recurrence Relations and Derivatives

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§11.4(vii) Zeros

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9: 11.15 Approximations

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§11.15(i) Expansions in Chebyshev Series

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10: 11.5 Integral Representations

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§11.5(i) Integrals Along the Real Line

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§11.5(ii) Contour Integrals

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Mellin–Barnes Integrals

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11.5.8 $${(\frac{1}{2}x)}^{-\nu -1}{𝐇}_{\nu }\left(x\right)=-\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\pi \csc \left(\pi s\right)}{\mathrm{\Gamma }\left(\frac{3}{2}+s\right)\mathrm{\Gamma }\left(\frac{3}{2}+\nu +s\right)}{(\frac{1}{4}{x}^2)}^{s}ds,$$ $x>0$, $\mathrm{\Re }\nu >-1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $\nu$: real or complex order

Keywords:

Mellin transform

Referenced by:

§11.5(ii), §11.5(ii)

Permalink:

http://dlmf.nist.gov/11.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §11.5(ii), §11.5(ii), §11.5 and Ch.11

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§11.5(iii) Compendia

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