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31: 10.43 Integrals

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§10.43(i) Indefinite Integrals

… ►For the modified Struve function ${𝐋}_{\nu }\left(z\right)$ see §11.2(i). … ► … ► ►

§10.43(v) Kontorovich–Lebedev Transform

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32: 10.76 Approximations

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§10.76(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions

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Real Variable; Imaginary Order

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33: 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 ${𝐀}_{\nu }\left(z\right)$.

34: 11.14 Tables

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§11.14(ii) Struve Functions

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Abramowitz and Stegun (1964, Chapter 12) tabulates ${𝐇}_n\left(x\right)$, ${𝐇}_n\left(x\right)-Y_n\left(x\right)$, and $I_n\left(x\right)-{𝐋}_n\left(x\right)$ for $n=0,1$ and $x=0(.1)5$, $x^{-1}=0(.01)0.2$ to 6D or 7D.

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Agrest et al. (1982) tabulates ${𝐇}_n\left(x\right)$ and ${\mathrm{e}}^{-x}{𝐋}_n\left(x\right)$ for $n=0,1$ and $x=0(.001)5(.005)15(.01)100$ to 11D.

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§11.14(iii) Integrals

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§11.14(v) Incomplete Functions

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35: 10.32 Integral Representations

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§10.32(i) Integrals along the Real Line

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Basset’s Integral

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§10.32(ii) Contour Integrals

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§10.32(iii) Products

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§10.32(iv) Compendia

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36: 10.40 Asymptotic Expansions for Large Argument

§10.40 Asymptotic Expansions for Large Argument

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§10.40(i) Hankel’s Expansions

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Products

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§10.40(iv) Exponentially-Improved Expansions

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37: 11.15 Approximations

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§11.15(i) Expansions in Chebyshev Series

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Luke (1975, pp. 416–421) gives Chebyshev-series expansions for ${𝐇}_n\left(x\right)$, ${𝐋}_n\left(x\right)$, $0\le \left|x\right|\le 8$, and ${𝐇}_n\left(x\right)-Y_n\left(x\right)$, $x\ge 8$, for $n=0,1$; ${\int }_0^x{t}^{-m}{𝐇}_0\left(t\right)dt$, ${\int }_0^x{t}^{-m}{𝐋}_0\left(t\right)dt$, $0\le \left|x\right|\le 8$, $m=0,1$ and ${\int }_0^x({𝐇}_0\left(t\right)-Y_0\left(t\right))dt$, ${\int }_x^{\mathrm{\infty }}{t}^{-1}({𝐇}_0\left(t\right)-Y_0\left(t\right))dt$, $x\ge 8$; the coefficients are to 20D.

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MacLeod (1993) gives Chebyshev-series expansions for ${𝐋}_0\left(x\right)$, ${𝐋}_1\left(x\right)$, $0\le x\le 16$, and $I_0\left(x\right)-{𝐋}_0\left(x\right)$, $I_1\left(x\right)-{𝐋}_1\left(x\right)$, $x\ge 16$; the coefficients are to 20D.

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38: 11.2 Definitions

§11.2 Definitions

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§11.2(i) Power-Series Expansions

… ► … ►Particular solutions: … ►

Modified Struve’s Equation

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39: 28.26 Asymptotic Approximations for Large $q$

§28.26 Asymptotic Approximations for Large $q$

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28.26.1 $${\mathrm{Mc}}_m^{(3)}(z,h)=\frac{{\mathrm{e}}^{\mathrm{i}\varphi }}{{(\pi h\cosh z)}^{1/2}}\left({\mathrm{Fc}}_m(z,h)-\mathrm{i}{\mathrm{Gc}}_m(z,h)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.7 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

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28.26.2 $$\mathrm{i}{\mathrm{Ms}}_{m+1}^{(3)}(z,h)=\frac{{\mathrm{e}}^{\mathrm{i}\varphi }}{{(\pi h\cosh z)}^{1/2}}\left({\mathrm{Fs}}_m(z,h)-\mathrm{i}{\mathrm{Gs}}_m(z,h)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.7 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

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§28.26(ii) Uniform Approximations

… ►For asymptotic approximations for ${\mathrm{M}}_{\nu }^{(3,4)}(z,h)$ see also Naylor (1984, 1987, 1989).

40: 28.33 Physical Applications

§28.33 Physical Applications

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28.33.2 $$V_n(\xi ,\eta )=\left(c_n{\mathrm{M}}_n^{(1)}(\xi ,\sqrt{q})+d_n{\mathrm{M}}_n^{(2)}(\xi ,\sqrt{q})\right){\mathrm{me}}_n(\eta ,q),$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $q=h^2$: parameter, $n$: integer, $V(\xi ,\eta )$: solution, $d_n$: constants, $\eta$: variable, $\xi$: variable and $c_n$: constants

Permalink:

http://dlmf.nist.gov/28.33.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.33(ii), §28.33 and Ch.28

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28.33.3 $${\mathrm{M}}_n^{(1)}({\xi }_0,\sqrt{q}){\mathrm{M}}_n^{(2)}({\xi }_1,\sqrt{q})-{\mathrm{M}}_n^{(1)}({\xi }_1,\sqrt{q}){\mathrm{M}}_n^{(2)}({\xi }_0,\sqrt{q})=0.$$

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Symbols:

${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $q=h^2$: parameter, $n$: integer and $\xi$: variable

Referenced by:

§28.33(ii)

Permalink:

http://dlmf.nist.gov/28.33.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.33(ii), §28.33 and Ch.28

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Torres-Vega et al. (1998) for Mathieu functions in phase space.