Heine transformations (first, second, third)

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21: 10.4 Connection Formulas

… ►Other solutions of (10.2.1) include $J_{-\nu }\left(z\right)$, $Y_{-\nu }\left(z\right)$, $H_{-\nu }^{(1)}\left(z\right)$, and $H_{-\nu }^{(2)}\left(z\right)$. … ►

$$H_{-n}^{(1)}\left(z\right)={(-1)}^n{H}_n^{(1)}\left(z\right),$$

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$$H_{-n}^{(2)}\left(z\right)={(-1)}^n{H}_n^{(2)}\left(z\right).$$

… ►

10.4.7 $$H_{\nu }^{(1)}\left(z\right)=\mathrm{i}\csc \left(\nu \pi \right)\left({\mathrm{e}}^{-\nu \pi \mathrm{i}}{J}_{\nu }\left(z\right)-J_{-\nu }\left(z\right)\right)=\csc \left(\nu \pi \right)\left(Y_{-\nu }\left(z\right)-{\mathrm{e}}^{-\nu \pi \mathrm{i}}{Y}_{\nu }\left(z\right)\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.4, §10.8

Permalink:

http://dlmf.nist.gov/10.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.4 and Ch.10

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10.4.8 $$H_{\nu }^{(2)}\left(z\right)=\mathrm{i}\csc \left(\nu \pi \right)\left(J_{-\nu }\left(z\right)-{\mathrm{e}}^{\nu \pi \mathrm{i}}{J}_{\nu }\left(z\right)\right)=\csc \left(\nu \pi \right)\left(Y_{-\nu }\left(z\right)-{\mathrm{e}}^{\nu \pi \mathrm{i}}{Y}_{\nu }\left(z\right)\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.4, §10.8

Permalink:

http://dlmf.nist.gov/10.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.4 and Ch.10

…

22: 19.15 Advantages of Symmetry

… ►Symmetry in $x,y,z$ of $R_F(x,y,z)$, $R_G(x,y,z)$, and $R_J(x,y,z,p)$ replaces the five transformations (19.7.2), (19.7.4)–(19.7.7) of Legendre’s integrals; compare (19.25.17). Symmetry unifies the Landen transformations of §19.8(ii) with the Gauss transformations of §19.8(iii), as indicated following (19.22.22) and (19.36.9). (19.21.12) unifies the three transformations in §19.7(iii) that change the parameter of Legendre’s third integral. ►Symmetry allows the expansion (19.19.7) in a series of elementary symmetric functions that gives high precision with relatively few terms and provides the most efficient method of computing the incomplete integral of the third kind (§19.36(i)). …

23: 2.5 Mellin Transform Methods

§2.5 Mellin Transform Methods

… ►The Mellin transform of $f(t)$ is defined by …The inversion formula is given by … ► … ►

§2.5(iii) Laplace Transforms with Small Parameters

…

24: 19.36 Methods of Computation

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§19.36(ii) Quadratic Transformations

… ►Thompson (1997, pp. 499, 504) uses descending Landen transformations for both $F(\varphi ,k)$ and $E(\varphi ,k)$. … ►Descending Gauss transformations of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ (see (19.8.20)) are used in Fettis (1965) to compute a large table (see §19.37(iii)). …If ${\alpha }^2=k^2$, then the method fails, but the function can be expressed by (19.6.13) in terms of $E(\varphi ,k)$, for which Neuman (1969b) uses ascending Landen transformations. … ►Bulirsch (1969a, b) extend Bartky’s transformation to $\mathrm{el3}(x,k_c,p)$ by expressing it in terms of the first incomplete integral, a complete integral of the third kind, and a more complicated integral to which Bartky’s method can be applied. …

25: 10.9 Integral Representations

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10.9.23 $$J_{\nu }\left(z\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }-\mathrm{i}c}^{-\mathrm{\infty }+\mathrm{i}c}\frac{\mathrm{\Gamma }\left(t\right)}{\mathrm{\Gamma }\left(\nu -t+1\right)}{(\frac{1}{2}z)}^{\nu -2t}dt,$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable, $\nu$: complex parameter and $c$: constant

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/10.9.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

… ►

10.9.24 $$H_{\nu }^{(1)}\left(z\right)=-\frac{{\mathrm{e}}^{-\frac{1}{2}\nu \pi \mathrm{i}}}{2{\pi }^2}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(t\right)\mathrm{\Gamma }\left(t-\nu \right){(-\frac{1}{2}\mathrm{i}z)}^{\nu -2t}dt,$$ ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: complex parameter and $c$: constant

Keywords:

Mellin transform

Referenced by:

§10.9(ii)

Permalink:

http://dlmf.nist.gov/10.9.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

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10.9.25 $$H_{\nu }^{(2)}\left(z\right)=\frac{{\mathrm{e}}^{\frac{1}{2}\nu \pi \mathrm{i}}}{2{\pi }^2}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(t\right)\mathrm{\Gamma }\left(t-\nu \right){(\frac{1}{2}\mathrm{i}z)}^{\nu -2t}dt,$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: complex parameter and $c$: constant

Keywords:

Mellin transform

Referenced by:

§10.9(ii), §10.9(ii)

Permalink:

http://dlmf.nist.gov/10.9.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

… ►For the function $I_{\nu }$ see §10.25(ii). … ►For the function $K_0$ see §10.25(ii). …

26: 19.25 Relations to Other Functions

… ►The transformations in §19.7(ii) result from the symmetry and homogeneity of functions on the right-hand sides of (19.25.5), (19.25.7), and (19.25.14). …Thus the five permutations induce five transformations of Legendre’s integrals (and also of the Jacobian elliptic functions). ►The three changes of parameter of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ in §19.7(iii) are unified in (19.21.12) by way of (19.25.14). … ►Inversions of 12 elliptic integrals of the first kind, producing the 12 Jacobian elliptic functions, are combined and simplified by using the properties of $R_F(x,y,z)$. …For analogous integrals of the second kind, which are not invertible in terms of single-valued functions, see (19.29.20) and (19.29.21) and compare with Gradshteyn and Ryzhik (2000, §3.153,1–10 and §3.156,1–9). …

27: 15.17 Mathematical Applications

… ►The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … ►First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL$(2,ℝ)$, and spherical functions on certain nonsymmetric Gelfand pairs. Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. … ►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). … ►By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group. …

28: 14.17 Integrals

… ►In (14.17.1)–(14.17.4), $𝖯$ may be replaced by $𝖰$, and in (14.17.3) and (14.17.4), $𝖰$ may be replaced by $𝖯$. … ►

§14.17(v) Laplace Transforms

►For Laplace transforms and inverse Laplace transforms involving associated Legendre functions, see Erdélyi et al. (1954a, pp. 179–181, 270–272), Oberhettinger and Badii (1973, pp. 113–118, 317–324), Prudnikov et al. (1992a, §§3.22, 3.32, and 3.33), and Prudnikov et al. (1992b, §§3.20, 3.30, and 3.31). ►

§14.17(vi) Mellin Transforms

►For Mellin transforms involving associated Legendre functions see Oberhettinger (1974, pp. 69–82) and Marichev (1983, pp. 247–283), and for inverse transforms see Oberhettinger (1974, pp. 205–215).

29: 35.2 Laplace Transform

§35.2 Laplace Transform

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Definition

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Inversion Formula

… ►

Convolution Theorem

►If $g_j$ is the Laplace transform of $f_j$, $j=1,2$, then $g_1{g}_2$ is the Laplace transform of the convolution $f_1\ast f_2$, where …

30: 14.31 Other Applications

… ►

§14.31(ii) Conical Functions

►The conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^m\left(x\right)$ appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). The conical functions and Mehler–Fock transform generalize to Jacobi functions and the Jacobi transform; see Koornwinder (1984a) and references therein. … ►Legendre functions $P_{\nu }\left(x\right)$ of complex degree $\nu$ appear in the application of complex angular momentum techniques to atomic and molecular scattering (Connor and Mackay (1979)). …