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11: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta \right|$

… ►

$$F_{\mathrm{\ell }}(\eta ,\rho )=\sin \left({\theta }_{\mathrm{\ell }}(\eta ,\rho )\right)+o\left(1\right),$$

… ►where ${\theta }_{\mathrm{\ell }}(\eta ,\rho )$ is defined by (33.2.9). … ►In particular, for $\mathrm{\ell }=0$, … ►

$$G_{\mathrm{\ell }}(\eta ,\rho )=-\frac{\pi {(-2\eta )}^{\mathrm{\ell }}}{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}\left({(-2\eta \rho )}^{1/2}{Y}_{2\mathrm{\ell }+1}\left({(-8\eta \rho )}^{1/2}\right)+o\left({\left|\eta \right|}^{1/4}\right)\right).$$

►In particular, for $\mathrm{\ell }=0$, …

12: 24.16 Generalizations

… ►For $\mathrm{\ell }=0,1,2,\mathrm{\dots }$, Bernoulli and Euler polynomials of order $\mathrm{\ell }$ are defined respectively by …When $x=0$ they reduce to the Bernoulli and Euler numbers of order $\mathrm{\ell }$ : …Also for $\mathrm{\ell }=1,2,3,\mathrm{\dots }$, … ►For extensions of $B_n^{(\mathrm{\ell })}\left(x\right)$ to complex values of $x$, $n$, and $\mathrm{\ell }$, and also for uniform asymptotic expansions for large $x$ and large $n$, see Temme (1995b) and López and Temme (1999b, 2010b). … ►(This notation is consistent with (24.16.3) when $x=\mathrm{\ell }$.) …

13: 24.10 Arithmetic Properties

… ►where $n\ge 2$, and $\mathrm{\ell }(\ge 1)$ is an arbitrary integer such that $(p-1)p^{\mathrm{\ell }}|2n$. … ►valid when $m\equiv n(mod(p-1)p^{\mathrm{\ell }})$ and $n\cancel{\equiv }0(modp-1)$, where $\mathrm{\ell }(\ge 0)$ is a fixed integer. …valid for fixed integers $\mathrm{\ell }(\ge 0)$, and for all $n(\ge 0)$ and $w(\ge 0)$ such that $2^{\mathrm{\ell }}|w$. … ►valid for fixed integers $\mathrm{\ell }(\ge 1)$, and for all $n(\ge 1)$ such that $2n\cancel{\equiv }0$ $(modp-1)$ and $p^{\mathrm{\ell }}|2n$. …valid for fixed integers $\mathrm{\ell }(\ge 1)$ and for all $n(\ge 1)$ such that $(p-1)p^{\mathrm{\ell }-1}|2n$.

14: 33.17 Recurrence Relations and Derivatives

… ►

33.17.1 $$(\mathrm{\ell }+1)rf(ϵ,\mathrm{\ell }-1;r)-(2\mathrm{\ell }+1)\left(\mathrm{\ell }(\mathrm{\ell }+1)-r\right)f(ϵ,\mathrm{\ell };r)+\mathrm{\ell }\left(1+{(\mathrm{\ell }+1)}^2ϵ\right)rf(ϵ,\mathrm{\ell }+1;r)=0,$$

ⓘ

Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

►

33.17.2 $$(\mathrm{\ell }+1)\left(1+{\mathrm{\ell }}^2ϵ\right)rh(ϵ,\mathrm{\ell }-1;r)-(2\mathrm{\ell }+1)\left(\mathrm{\ell }(\mathrm{\ell }+1)-r\right)h(ϵ,\mathrm{\ell };r)+\mathrm{\ell }rh(ϵ,\mathrm{\ell }+1;r)=0,$$

ⓘ

Symbols:

$h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

►

33.17.3 $$(\mathrm{\ell }+1)r{f}^{\prime }(ϵ,\mathrm{\ell };r)=\left({(\mathrm{\ell }+1)}^2-r\right)f(ϵ,\mathrm{\ell };r)-\left(1+{(\mathrm{\ell }+1)}^2ϵ\right)rf(ϵ,\mathrm{\ell }+1;r),$$

ⓘ

Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

►

33.17.4 $$(\mathrm{\ell }+1)r{h}^{\prime }(ϵ,\mathrm{\ell };r)=\left({(\mathrm{\ell }+1)}^2-r\right)h(ϵ,\mathrm{\ell };r)-rh(ϵ,\mathrm{\ell }+1;r).$$

ⓘ

Symbols:

$h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

15: 10.42 Zeros

… ►For example, if $\nu$ is real, then the zeros of $I_{\nu }\left(z\right)$ are all complex unless for some positive integer $\mathrm{\ell }$, in which event $I_{\nu }\left(z\right)$ has two real zeros. ►The distribution of the zeros of $K_n\left(nz\right)$ in the sector $-\frac{3}{2}\pi \le \mathrm{ph}z\le \frac{1}{2}\pi$ in the cases $n=1,5,10$ is obtained on rotating Figures 10.21.2, 10.21.4, 10.21.6, respectively, through an angle $-\frac{1}{2}\pi$ so that in each case the cut lies along the positive imaginary axis. The zeros in the sector $-\frac{1}{2}\pi \le \mathrm{ph}z\le \frac{3}{2}\pi$ are their conjugates. ► $K_n\left(z\right)$ has no zeros in the sector $|\mathrm{ph}z|\le \frac{1}{2}\pi$; this result remains true when $n$ is replaced by any real number $\nu$. For the number of zeros of $K_{\nu }\left(z\right)$ in the sector $|\mathrm{ph}z|\le \pi$, when $\nu$ is real, see Watson (1944, pp. 511–513). …

16: 28.23 Expansions in Series of Bessel Functions

… ►

28.23.7 $${\mathrm{Mc}}_{2m}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{ce}}_{2m}(\frac{1}{2}\pi ,h^2)\right)}^{-1}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{A}_{2\mathrm{\ell }}^{2m}(h^2){𝒞}_{2\mathrm{\ell }}^{(j)}(2h\sinh z),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sinh z$: hyperbolic sine function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.23.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

… ►

28.23.9 $${\mathrm{Mc}}_{2m+1}^{(j)}(z,h)={(-1)}^{m+1}{\left({\mathrm{ce}}_{2m+1}^{\prime }(\frac{1}{2}\pi ,h^2)\right)}^{-1}\coth z\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}(2\mathrm{\ell }+1)A_{2\mathrm{\ell }+1}^{2m+1}(h^2){𝒞}_{2\mathrm{\ell }+1}^{(j)}(2h\sinh z),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.23.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

… ►

28.23.11 $${\mathrm{Ms}}_{2m+1}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{se}}_{2m+1}(\frac{1}{2}\pi ,h^2)\right)}^{-1}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{B}_{2\mathrm{\ell }+1}^{2m+1}(h^2){𝒞}_{2\mathrm{\ell }+1}^{(j)}(2h\sinh z),$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sinh z$: hyperbolic sine function, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $B_m(q)$: Fourier coefficient

A&S Ref:

20.6.13

Permalink:

http://dlmf.nist.gov/28.23.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

… ►

28.23.13 $${\mathrm{Ms}}_{2m+2}^{(j)}(z,h)={(-1)}^{m+1}{\left({\mathrm{se}}_{2m+2}^{\prime }(\frac{1}{2}\pi ,h^2)\right)}^{-1}\coth z\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}(2\mathrm{\ell }+2)B_{2\mathrm{\ell }+2}^{2m+2}(h^2){𝒞}_{2\mathrm{\ell }+2}^{(j)}(2h\sinh z).$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $B_m(q)$: Fourier coefficient

Referenced by:

§28.23

Permalink:

http://dlmf.nist.gov/28.23.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

… ►When $j=2,3,4$ the series in the even-numbered equations converge for $\mathrm{\Re }z>0$ and $|\cosh z|>1$, and the series in the odd-numbered equations converge for $\mathrm{\Re }z>0$ and $|\sinh z|>1$. …

17: 33.4 Recurrence Relations and Derivatives

… ►For $\mathrm{\ell }=1,2,3,\mathrm{\dots }$, let …Then, with $X_{\mathrm{\ell }}$ denoting any of $F_{\mathrm{\ell }}(\eta ,\rho )$, $G_{\mathrm{\ell }}(\eta ,\rho )$, or $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$, ►

33.4.2 $$R_{\mathrm{\ell }}{X}_{\mathrm{\ell }-1}-T_{\mathrm{\ell }}{X}_{\mathrm{\ell }}+R_{\mathrm{\ell }+1}{X}_{\mathrm{\ell }+1}=0,$$ $\mathrm{\ell }\ge 1$,

ⓘ

Symbols:

$\mathrm{\ell }$: nonnegative integer, $R_{\mathrm{\ell }}$: factor, $T_{\mathrm{\ell }}$: factor and $X_{\mathrm{\ell }}$: any Coulomb function

A&S Ref:

14.2.3

Permalink:

http://dlmf.nist.gov/33.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.4 and Ch.33

►

33.4.3 $$X_{\mathrm{\ell }}^{\prime }=R_{\mathrm{\ell }}{X}_{\mathrm{\ell }-1}-S_{\mathrm{\ell }}{X}_{\mathrm{\ell }},$$ $\mathrm{\ell }\ge 1$,

ⓘ

Symbols:

$\mathrm{\ell }$: nonnegative integer, $R_{\mathrm{\ell }}$: factor, $S_{\mathrm{\ell }}$: factor and $X_{\mathrm{\ell }}$: any Coulomb function

A&S Ref:

14.2.1

Referenced by:

§33.6

Permalink:

http://dlmf.nist.gov/33.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.4 and Ch.33

►

33.4.4 $$X_{\mathrm{\ell }}^{\prime }=S_{\mathrm{\ell }+1}{X}_{\mathrm{\ell }}-R_{\mathrm{\ell }+1}{X}_{\mathrm{\ell }+1},$$ $\mathrm{\ell }\ge 0$.

ⓘ

Symbols:

$\mathrm{\ell }$: nonnegative integer, $R_{\mathrm{\ell }}$: factor, $S_{\mathrm{\ell }}$: factor and $X_{\mathrm{\ell }}$: any Coulomb function

A&S Ref:

14.2.2

Referenced by:

§33.6

Permalink:

http://dlmf.nist.gov/33.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.4 and Ch.33

18: 32.14 Combinatorics

… ►Let $S_N$ be the group of permutations $𝝅$ of the numbers $1,2,\mathrm{\dots },N$ (§26.2). With , $𝝅(m_1),𝝅(m_2),\mathrm{\dots },𝝅(m_n)$ is said to be an increasing subsequence of $𝝅$ of length $n$ when . Let ${\mathrm{\ell }}_N(𝝅)$ be the length of the longest increasing subsequence of $𝝅$. … ►

32.14.1 $$\underset{N\to \mathrm{\infty }}{lim}\mathrm{Prob}\left(\frac{{\mathrm{\ell }}_N(𝝅)-2\sqrt{N}}{N^{1/6}}\le s\right)=F(s),$$

ⓘ

Symbols:

$𝝅(m)$: permutations, ${\mathrm{\ell }}_N(𝝅)$: length and $F(s)$: distribution function

Permalink:

http://dlmf.nist.gov/32.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §32.14 and Ch.32

…

19: 18.18 Sums

… ►

18.18.9 $$P_n\left(\cos {\theta }_1\cos {\theta }_2+\sin {\theta }_1\sin {\theta }_2\cos \varphi \right)=P_n\left(\cos {\theta }_1\right)P_n\left(\cos {\theta }_2\right)+2\sum _{\mathrm{\ell }=1}^n\frac{(n-\mathrm{\ell })!(n+\mathrm{\ell })!}{2^{2\mathrm{\ell }}{(n!)}^2}{(\sin {\theta }_1)}^{\mathrm{\ell }}{P}_{n-\mathrm{\ell }}^{(\mathrm{\ell },\mathrm{\ell })}\left(\cos {\theta }_1\right){(\sin {\theta }_2)}^{\mathrm{\ell }}{P}_{n-\mathrm{\ell }}^{(\mathrm{\ell },\mathrm{\ell })}\left(\cos {\theta }_2\right)\cos \left(\mathrm{\ell }\varphi \right).$$

ⓘ

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\sin z$: sine function, $\mathrm{\ell }$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§14.30(iii), §18.17(ii), §18.18(ii), §18.18(ii), §18.18(ii)

Permalink:

http://dlmf.nist.gov/18.18.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(ii), §18.18(ii), §18.18 and Ch.18

… ►

18.18.19 $$x^n={\left(\alpha +1\right)}_n\sum _{\mathrm{\ell }=0}^n\frac{{\left(-n\right)}_{\mathrm{\ell }}}{{\left(\alpha +1\right)}_{\mathrm{\ell }}}{L}_{\mathrm{\ell }}^{(\alpha )}\left(x\right).$$

ⓘ

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.18(iv)

Permalink:

http://dlmf.nist.gov/18.18.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(iv), §18.18(iv), §18.18 and Ch.18

… ►

18.18.22 $$C_m^{(\lambda )}\left(x\right)C_n^{(\lambda )}\left(x\right)=\sum _{\mathrm{\ell }=0}^{\min (m,n)}\frac{(m+n+\lambda -2\mathrm{\ell })(m+n-2\mathrm{\ell })!}{(m+n+\lambda -\mathrm{\ell })\mathrm{\ell }!(m-\mathrm{\ell })!(n-\mathrm{\ell })!}\frac{{\left(\lambda \right)}_{\mathrm{\ell }}{\left(\lambda \right)}_{m-\mathrm{\ell }}{\left(\lambda \right)}_{n-\mathrm{\ell }}{\left(2\lambda \right)}_{m+n-\mathrm{\ell }}}{{\left(\lambda \right)}_{m+n-\mathrm{\ell }}{\left(2\lambda \right)}_{m+n-2\mathrm{\ell }}}{C}_{m+n-2\mathrm{\ell }}^{(\lambda )}\left(x\right).$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $\mathrm{\ell }$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

(18.17.41_5), §18.17(viii), §18.18(v)

Permalink:

http://dlmf.nist.gov/18.18.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(v), §18.18(v), §18.18 and Ch.18

… ►

18.18.23 $$H_m\left(x\right)H_n\left(x\right)=\sum _{\mathrm{\ell }=0}^{\min (m,n)}\left(\genfrac{}{}{0.0pt}{}{m}{\mathrm{\ell }}\right)\left(\genfrac{}{}{0.0pt}{}{n}{\mathrm{\ell }}\right)2^{\mathrm{\ell }}\mathrm{\ell }!H_{m+n-2\mathrm{\ell }}\left(x\right).$$

ⓘ

Symbols:

$H_n\left(x\right)$: Hermite polynomial, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(viii), §18.17(v), §18.18(v)

Permalink:

http://dlmf.nist.gov/18.18.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(v), §18.18(v), §18.18 and Ch.18

… ►

18.18.24 $$b_{n,\mathrm{\ell }}=\left(\genfrac{}{}{0.0pt}{}{n}{\mathrm{\ell }}\right)\frac{{\left(n+\alpha +\beta +1\right)}_{\mathrm{\ell }}{\left(-\beta -n\right)}_{n-\mathrm{\ell }}}{2^{\mathrm{\ell }}{\left(\alpha +1\right)}_n},$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\mathrm{\ell }$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.18.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(vi), §18.18(vi), §18.18 and Ch.18

…

20: 20.10 Integrals

… ►

20.10.1 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}{\theta }_2\left(0\right|\mathrm{i}{x}^2)dx=2^s(1-2^{-s}){\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>1$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E1

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>1$ because ${\theta }_2\left(0\right|\mathrm{i}{x}^2)=x^{-1}{\theta }_4\left(0\right|\mathrm{i}{x}^{-2})\sim x^{-1}$ as $x\to 0+$, which follows from (20.7.31) and (20.4.5).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

… ►Let $s$, $\mathrm{\ell }$, and $\beta$ be constants such that $\mathrm{\Re }s>0$, $\mathrm{\ell }>0$, and $\left|\mathrm{\Re }\beta \right|+\left|\mathrm{\Im }\beta \right|\le \mathrm{\ell }$. Then ►

20.10.4 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_1\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_2\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=-\frac{\mathrm{\ell }}{\sqrt{s}}\sinh \left(\beta \sqrt{s}\right)\mathrm{sech}\left(\mathrm{\ell }\sqrt{s}\right),$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(ii), §20.10 and Ch.20

►

20.10.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_3\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_4\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=\frac{\mathrm{\ell }}{\sqrt{s}}\cosh \left(\beta \sqrt{s}\right)\mathrm{csch}\left(\mathrm{\ell }\sqrt{s}\right).$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(ii), §20.10 and Ch.20

…