mutual inductance of coaxial circles

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11: 18.30 Associated OP’s

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18.30.11 $$w^{\lambda }(x,c)=\frac{x^{\lambda }{\mathrm{e}}^{-x}}{{\left|U(c,1-\lambda ,x{\mathrm{e}}^{-\mathrm{i}\pi })\right|}^2}.$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $w(x)$: weight function and $x$: real variable

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http://dlmf.nist.gov/18.30.E11

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Annotations for §18.30(iii), §18.30 and Ch.18

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18.30.14 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{H}_n(x;c)H_m(x;c)w(x,c)dx=2^n{\pi }^{1/2}\mathrm{\Gamma }\left(n+c+1\right){\delta }_{n,m},$$ $c>-1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_{j,k}$: Kronecker delta, $H_n(x;c)$: associated Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $w(x)$: weight function, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

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http://dlmf.nist.gov/18.30.E14

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Annotations for §18.30(iv), §18.30 and Ch.18

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18.30.17 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{𝒫}_n^{\lambda }(x;\varphi ,c){𝒫}_m^{\lambda }(x;\varphi ,c)w^{(\lambda )}(x,\varphi ,c)dx=\frac{\mathrm{\Gamma }\left(n+c+2\lambda \right)\mathrm{\Gamma }\left(c+1\right)}{{\left(c+1\right)}_n}{\delta }_{n,m},$$ or ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_{j,k}$: Kronecker delta, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${𝒫}_n^{\lambda }(x;\varphi ,c)$: associated Meixner–Pollaczek polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $w(x)$: weight function, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Sources:

Askey and Wimp (1984, (1.9)); Pollaczek (1950, formula following (16))

Permalink:

http://dlmf.nist.gov/18.30.E17

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Annotations for §18.30(v), §18.30 and Ch.18

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18.30.20 $$H_n(x;c)={\left(c+1\right)}_n\underset{\lambda \to \mathrm{\infty }}{lim}{\lambda }^{-n/2}{𝒫}_n^{\lambda }(x{\lambda }^{1/2};\pi /2,c).$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $H_n(x;c)$: associated Hermite polynomial, ${𝒫}_n^{\lambda }(x;\varphi ,c)$: associated Meixner–Pollaczek polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer and $x$: real variable

Proof sketch:

Using the limit will convert (18.30.16) into (18.30.13). See also Askey and Wimp (1984, §4).

Permalink:

http://dlmf.nist.gov/18.30.E20

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Annotations for §18.30(v), §18.30 and Ch.18

… ►follows by induction on $n$. …

12: 17.2 Calculus

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17.2.6_1 $${(q;q)}_{\mathrm{\infty }}=\sqrt{\frac{2\pi }{t}}\exp \left(-\frac{{\pi }^2}{6t}+\frac{t}{24}\right){(\widehat{q};\widehat{q})}_{\mathrm{\infty }},$$ $\mathrm{\Re }t>0$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $\mathrm{\Re }$: real part and $q$: complex base

Referenced by:

§17.2(i), §17.2(i), Erratum (V1.1.5) for Additions

Permalink:

http://dlmf.nist.gov/17.2.E6_1

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Addition (effective with 1.1.5):

This equation was added.

See also:

Annotations for §17.2(i), §17.2 and Ch.17

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17.2.6_2 $${(-q;q)}_{\mathrm{\infty }}=\frac{1}{\sqrt{2}}\exp \left(\frac{{\pi }^2}{12t}+\frac{t}{24}\right){({\widehat{q}}^{\frac{1}{2}};\widehat{q})}_{\mathrm{\infty }},$$ $t>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial) and $q$: complex base

Referenced by:

§17.2(i), Erratum (V1.1.5) for Additions

Permalink:

http://dlmf.nist.gov/17.2.E6_2

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Addition (effective with 1.1.5):

This equation was added.

See also:

Annotations for §17.2(i), §17.2 and Ch.17

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13: Bibliography R

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The effective resistance and inductance of a concentric main, and methods of computing the $\mathrm{ber}$ and $\mathrm{bei}$ and allied functions. Philos. Mag. (6) 17, pp. 524–552.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.73(iii).

Encodings:

BibTeX

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14: 10.73 Physical Applications

… ►Consequently, Bessel functions $J_n\left(x\right)$, and modified Bessel functions $I_n\left(x\right)$, are central to the analysis of microwave and optical transmission in waveguides, including coaxial and fiber. …

15: 15.5 Derivatives and Contiguous Functions

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16: 4.22 Infinite Products and Partial Fractions

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4.22.1 $$\sin z=z\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{z^2}{n^2{\pi }^2}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.89

Permalink:

http://dlmf.nist.gov/4.22.E1

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Annotations for §4.22 and Ch.4

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4.22.2 $$\cos z=\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{4z^2}{{(2n-1)}^2{\pi }^2}\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.90

Permalink:

http://dlmf.nist.gov/4.22.E2

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Annotations for §4.22 and Ch.4

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4.22.3 $$\cot z=\frac{1}{z}+2z\sum _{n=1}^{\mathrm{\infty }}\frac{1}{z^2-n^2{\pi }^2},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.91

Referenced by:

§22.12

Permalink:

http://dlmf.nist.gov/4.22.E3

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Annotations for §4.22 and Ch.4

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4.22.4 $${\csc }^{2}z=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{{(z-n\pi )}^2},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.92

Permalink:

http://dlmf.nist.gov/4.22.E4

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Annotations for §4.22 and Ch.4

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4.22.5 $$\csc z=\frac{1}{z}+2z\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n}{z^2-n^2{\pi }^2}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.93

Referenced by:

§22.12

Permalink:

http://dlmf.nist.gov/4.22.E5

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Annotations for §4.22 and Ch.4

17: 4.36 Infinite Products and Partial Fractions

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4.36.1 $$\sinh z=z\prod _{n=1}^{\mathrm{\infty }}\left(1+\frac{z^2}{n^2{\pi }^2}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sinh z$: hyperbolic sine function, $n$: integer and $z$: complex variable

A&S Ref:

4.5.68

Permalink:

http://dlmf.nist.gov/4.36.E1

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Annotations for §4.36 and Ch.4

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4.36.2 $$\cosh z=\prod _{n=1}^{\mathrm{\infty }}\left(1+\frac{4z^2}{{(2n-1)}^2{\pi }^2}\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $n$: integer and $z$: complex variable

A&S Ref:

4.5.69

Permalink:

http://dlmf.nist.gov/4.36.E2

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Annotations for §4.36 and Ch.4

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4.36.3 $$\coth z=\frac{1}{z}+2z\sum _{n=1}^{\mathrm{\infty }}\frac{1}{z^2+n^2{\pi }^2},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\coth z$: hyperbolic cotangent function, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.36.E3

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Annotations for §4.36 and Ch.4

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4.36.4 $${\mathrm{csch}}^{2}z=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{{(z-n\pi \mathrm{i})}^2},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{csch}z$: hyperbolic cosecant function, $\mathrm{i}$: imaginary unit, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.36.E4

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Annotations for §4.36 and Ch.4

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4.36.5 $$\mathrm{csch}z=\frac{1}{z}+2z\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n}{z^2+n^2{\pi }^2}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{csch}z$: hyperbolic cosecant function, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.36.E5

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Annotations for §4.36 and Ch.4

18: 3.12 Mathematical Constants

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3.12.1 $$\pi =\mathrm{3.14159\hspace{0.33em}26535\hspace{0.33em}89793\hspace{0.33em}23846}\mathrm{\dots }$$

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Defines:

$\pi$: the ratio of the circumference of a circle to its diameter

Notes:

For more digits see OEIS Sequence A000796; see also Sloane (2003).

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http://dlmf.nist.gov/3.12.E1

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Annotations for §3.12 and Ch.3

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3.12.2 $$\pi =4{\int }_0^1\frac{dt}{1+t^2}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ and $\int$: integral

Permalink:

http://dlmf.nist.gov/3.12.E2

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Annotations for §3.12 and Ch.3

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19: 6.15 Sums

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6.15.1 $$\sum _{n=1}^{\mathrm{\infty }}\mathrm{Ci}\left(\pi n\right)=\frac{1}{2}(\ln 2-\gamma ),$$

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Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E1

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Annotations for §6.15 and Ch.6

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6.15.2 $$\sum _{n=1}^{\mathrm{\infty }}\frac{\mathrm{si}\left(\pi n\right)}{n}=\frac{1}{2}\pi (\ln \pi -1),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{si}\left(z\right)$: sine integral and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E2

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Annotations for §6.15 and Ch.6

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6.15.3 $$\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\mathrm{Ci}\left(2\pi n\right)=1-\ln 2-\frac{1}{2}\gamma ,$$

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Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E3

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Annotations for §6.15 and Ch.6

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6.15.4 $$\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\frac{\mathrm{si}\left(2\pi n\right)}{n}=\pi (\frac{3}{2}\ln 2-1).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{si}\left(z\right)$: sine integral and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E4

Encodings:

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See also:

Annotations for §6.15 and Ch.6

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20: 28.3 Graphics

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