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21: 10.32 Integral Representations

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10.32.19 $$K_{\mu }\left(z\right)K_{\nu }\left(z\right)=\frac{1}{8\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(t+\frac{1}{2}\mu +\frac{1}{2}\nu \right)\mathrm{\Gamma }\left(t+\frac{1}{2}\mu -\frac{1}{2}\nu \right)\mathrm{\Gamma }\left(t-\frac{1}{2}\mu +\frac{1}{2}\nu \right)\mathrm{\Gamma }\left(t-\frac{1}{2}\mu -\frac{1}{2}\nu \right)}{\mathrm{\Gamma }\left(2t\right)}{(\frac{1}{2}z)}^{-2t}dt,$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $z$: complex variable and $\nu$: complex parameter

Keywords:

Mellin transform

Referenced by:

§10.32(iii)

Permalink:

http://dlmf.nist.gov/10.32.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(iii), §10.32(iii), §10.32 and Ch.10

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22: 34.12 Physical Applications

§34.12 Physical Applications

►The angular momentum coupling coefficients ($\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ symbols) are essential in the fields of nuclear, atomic, and molecular physics. …$\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols are also found in multipole expansions of solutions of the Laplace and Helmholtz equations; see Carlson and Rushbrooke (1950) and Judd (1976).

23: 16.7 Relations to Other Functions

… ►For $\mathit{3}j$, $\mathit{6}j$, $\mathit{9}j$ symbols see Chapter 34. Further representations of special functions in terms of ${}_p{}^{}F_q^{}$ functions are given in Luke (1969a, §§6.2–6.3), and an extensive list of ${}_{q+1}{}^{}F_q^{}$ functions with rational numbers as parameters is given in Krupnikov and Kölbig (1997).

24: 28.2 Definitions and Basic Properties

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$z\to z\pm \pi ;$

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$z\to z\pm \frac{1}{2}\pi ,q\to -q;$

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28.2.7 $$w_{\text{I}}(z\pm \pi ;a,q)=w_{\text{I}}(\pi ;a,q)w_{\text{I}}(z;a,q)\pm w_{\text{I}}^{\prime }(\pi ;a,q)w_{\text{II}}(z;a,q),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $q=h^2$: parameter, $z$: complex variable, $a$: parameter, $w(z)$: Mathieu’s equation solution, $w_{\text{I}}(z;a,q)$: fundamental solution and $w_{\text{II}}(z;a,q)$: fundamental solution

Referenced by:

§28.2(iv)

Permalink:

http://dlmf.nist.gov/28.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(ii), §28.2 and Ch.28

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28.2.8 $$w_{\text{II}}(z\pm \pi ;a,q)=\pm w_{\text{II}}(\pi ;a,q)w_{\text{I}}(z;a,q)+w_{\text{II}}^{\prime }(\pi ;a,q)w_{\text{II}}(z;a,q),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $q=h^2$: parameter, $z$: complex variable, $a$: parameter, $w(z)$: Mathieu’s equation solution, $w_{\text{I}}(z;a,q)$: fundamental solution and $w_{\text{II}}(z;a,q)$: fundamental solution

Permalink:

http://dlmf.nist.gov/28.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(ii), §28.2 and Ch.28

… ►The general solution of (28.2.16) is $\nu =\pm \widehat{\nu }+2n$, where $n\in \mathbb{Z}$. …

25: 15.3 Graphics

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Figure 15.3.1: $F(\frac{4}{3},\frac{9}{16};\frac{14}{5};x),-100\le x\le 1$. Magnify

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Figure 15.3.2: $F(5,-10;1;x),-0.023\le x\le 1$. Magnify

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Figure 15.3.3: $F(1,-10;10;x),-3\le x\le 1$. Magnify

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Figure 15.3.4: $F(5,10;1;x),-1\le x\le 0.022$. Magnify

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Figure 15.3.5: $F(\frac{4}{3},\frac{9}{16};\frac{14}{5};x+\mathrm{i}y),0\le x\le 2,-0.5\le y\le 0.5$. … Magnify 3D Help

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26: 16.24 Physical Applications

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§16.24(iii) $\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ Symbols

… ►They can be expressed as ${}_3{}^{}F_2^{}$ functions with unit argument. …These are balanced ${}_4{}^{}F_3^{}$ functions with unit argument. Lastly, special cases of the $\mathit{9}j$ symbols are ${}_5{}^{}F_4^{}$ functions with unit argument. …

27: 9 Airy and Related Functions

Chapter 9 Airy and Related Functions

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28: 34.14 Tables

§34.14 Tables

►Tables of exact values of the squares of the $\mathit{3}j$ and $\mathit{6}j$ symbols in which all parameters are $\le 8$ are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols on pp. … ►Some selected $\mathit{9}j$ symbols are also given. … 16-17; for $\mathit{9}j$ symbols on p. … ► 310–332, and for the $\mathit{9}j$ symbols on pp. …

29: 1.9 Calculus of a Complex Variable

… ►A domain $D$, say, is an open set in $ℂ$ that is connected, that is, any two points can be joined by a polygonal arc (a finite chain of straight-line segments) lying in the set. Any point whose neighborhoods always contain members and nonmembers of $D$ is a boundary point of $D$. … ►A function $f(z)$ is analytic in a domain $D$ if it is analytic at each point of $D$. … ►If $f(z)$ is analytic in an open domain $D$, then each of its derivatives $f^{\prime }(z)$, $f^{\prime \prime }(z)$, $\mathrm{\dots }$ exists and is analytic in $D$. … ►If $f(z)=u(x,y)+\mathrm{i}v(x,y)$ is analytic in an open domain $D$, then $u$ and $v$ are harmonic in $D$, that is, …

30: 34 3j, 6j, 9j Symbols

Chapter 34 $\mathit{3}j,\mathit{6}j,\mathit{9}j$ Symbols

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