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31: 22.11 Fourier and Hyperbolic Series

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22.11.6 $$\mathrm{nd}(z,k)=\frac{\pi }{2K{k}^{\prime }}+\frac{2\pi }{K{k}^{\prime }}\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^n\cos \left(2n\zeta \right)}{1+q^{2n}}.$$

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Symbols:

$\mathrm{nd}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cos z$: cosine function, $q$: nome, $z$: complex, $k$: modulus, $k^{\prime }$: complementary modulus and $\zeta$: change of variable

A&S Ref:

16.23.6

Permalink:

http://dlmf.nist.gov/22.11.E6

Encodings:

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See also:

Annotations for §22.11 and Ch.22

… ►Next, with $E=E\left(k\right)$ denoting the complete elliptic integral of the second kind (§19.2(ii)) and , …Similar expansions for ${\mathrm{cn}}^2(z,k)$ and ${\mathrm{dn}}^2(z,k)$ follow immediately from (22.6.1). … ►where $E^{\prime }=E^{\prime }\left(k\right)$ is defined by §19.2.9. Again, similar expansions for ${\mathrm{cn}}^2(z,k)$ and ${\mathrm{dn}}^2(z,k)$ may be derived via (22.6.1). …

32: 24.5 Recurrence Relations

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24.5.8 $$\sum _{k=0}^n\frac{2^{2k}{B}_{2k}}{(2k)!(2n+1-2k)!}=\frac{1}{(2n)!},$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $!$: factorial (as in $n!$), $k$: integer and $n$: integer

Referenced by:

§24.5(ii)

Permalink:

http://dlmf.nist.gov/24.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.5(ii), §24.5 and Ch.24

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$a_n={\displaystyle \sum _{k=0}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){\displaystyle \frac{b_{n-k}}{k+1}},$

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$b_n={\displaystyle \sum _{k=0}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)B_k{a}_{n-k}.$

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$a_n={\displaystyle \sum _{k=0}^{\lfloor n/2\rfloor }}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2k}}\right)b_{n-2k},$

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$b_n={\displaystyle \sum _{k=0}^{\lfloor n/2\rfloor }}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2k}}\right)E_{2k}{a}_{n-2k}.$

33: 32.3 Graphics

… ►Plots of solutions $w_k(x)$ of ${\text{P}}_{\text{I}}$ with $w_k(0)=0$ and $w_k^{\prime }(0)=k$ for various values of $k$, and the parabola $6w^2+x=0$. … ►

►►

Figure 32.3.3: $w_k(x)$ for $-12\le x\le 0.73$ and $k=\mathrm{1.85185\hspace{0.33em}3}$, $\mathrm{1.85185\hspace{0.33em}5}$. … Magnify

… ►Here $w_k(x)$ is the solution of ${\text{P}}_{\text{II}}$ with $\alpha =0$ and such that … ►

►►

Figure 32.3.5: $w_k(x)$ and $k\mathrm{Ai}\left(x\right)$ for $-10\le x\le 4$ with $k=0.5$. … Magnify

… ►Here $u=u_k(x;\nu )$ is the solution of …

34: 29.1 Special Notation

… ►The main functions treated in this chapter are the eigenvalues $a_{\nu }^{2m}\left(k^2\right)$, $a_{\nu }^{2m+1}\left(k^2\right)$, $b_{\nu }^{2m+1}\left(k^2\right)$, $b_{\nu }^{2m+2}\left(k^2\right)$, the Lamé functions ${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)$, ${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)$, and the Lamé polynomials ${\mathrm{𝑢𝐸}}_{2n}^m(z,k^2)$, ${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$, ${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$, ${\mathrm{𝑑𝐸}}_{2n+1}^m(z,k^2)$, ${\mathrm{𝑠𝑐𝐸}}_{2n+2}^m(z,k^2)$, ${\mathrm{𝑠𝑑𝐸}}_{2n+2}^m(z,k^2)$, ${\mathrm{𝑐𝑑𝐸}}_{2n+2}^m(z,k^2)$, ${\mathrm{𝑠𝑐𝑑𝐸}}_{2n+3}^m(z,k^2)$. … ►Other notations that have been used are as follows: Ince (1940a) interchanges $a_{\nu }^{2m+1}\left(k^2\right)$ with $b_{\nu }^{2m+1}\left(k^2\right)$. The relation to the Lamé functions $L_{c\nu }^{(m)}$, $L_{s\nu }^{(m)}$of Jansen (1977) is given by …where $\psi =\mathrm{am}(z,k)$; see §22.16(i). …where the positive factors $c_{\nu }^m(k^2)$ and $s_{\nu }^m(k^2)$ are determined by …

35: 32.5 Integral Equations

… ►Let $K(z,\zeta )$ be the solution of ►

32.5.1 $$K(z,\zeta )=k\mathrm{Ai}\left(\frac{z+\zeta }{2}\right)+\frac{k^2}{4}{\int }_z^{\mathrm{\infty }}{\int }_z^{\mathrm{\infty }}K(z,s)\mathrm{Ai}\left(\frac{s+t}{2}\right)\mathrm{Ai}\left(\frac{t+\zeta }{2}\right)dsdt,$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $dx$: differential of $x$ , $\int$: integral, $z$: real, $k$: real and $K(z,\zeta )$: solution

Permalink:

http://dlmf.nist.gov/32.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §32.5 and Ch.32

►where $k$ is a real constant, and $\mathrm{Ai}\left(z\right)$ is defined in §9.2. … ►

32.5.2 $$w(z)=K(z,z),$$

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Symbols:

$z$: real and $K(z,\zeta )$: solution

Permalink:

http://dlmf.nist.gov/32.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §32.5 and Ch.32

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32.5.3 $$w(z)\sim k\mathrm{Ai}\left(z\right),$$ $z\to +\mathrm{\infty }$.

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: asymptotic equality, $z$: real and $k$: real

Permalink:

http://dlmf.nist.gov/32.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §32.5 and Ch.32

36: 10.53 Power Series

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10.53.1 $${𝗃}_n\left(z\right)=z^n\sum _{k=0}^{\mathrm{\infty }}\frac{{(-\frac{1}{2}{z}^2)}^k}{k!(2n+2k+1)!!},$$

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Symbols:

$!!$: double factorial, $!$: factorial (as in $n!$), ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

10.1.2

Referenced by:

§10.53

Permalink:

http://dlmf.nist.gov/10.53.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.53 and Ch.10

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10.53.2 $${𝗒}_n\left(z\right)=-\frac{1}{z^{n+1}}\sum _{k=0}^n\frac{(2n-2k-1)!!{(\frac{1}{2}{z}^2)}^k}{k!}+\frac{{(-1)}^{n+1}}{z^{n+1}}\sum _{k=n+1}^{\mathrm{\infty }}\frac{{(-\frac{1}{2}{z}^2)}^k}{k!(2k-2n-1)!!}.$$

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Symbols:

$!!$: double factorial, $!$: factorial (as in $n!$), ${𝗒}_n\left(z\right)$: spherical Bessel function of the second kind, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

10.1.3

Referenced by:

§10.53

Permalink:

http://dlmf.nist.gov/10.53.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.53 and Ch.10

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10.53.3 $${𝗂}_n^{(1)}\left(z\right)=z^n\sum _{k=0}^{\mathrm{\infty }}\frac{{(\frac{1}{2}{z}^2)}^k}{k!(2n+2k+1)!!},$$

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Symbols:

$!!$: double factorial, $!$: factorial (as in $n!$), ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

10.2.5

Referenced by:

§10.53

Permalink:

http://dlmf.nist.gov/10.53.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.53 and Ch.10

►

10.53.4 $${𝗂}_n^{(2)}\left(z\right)=\frac{{(-1)}^n}{z^{n+1}}\sum _{k=0}^n\frac{(2n-2k-1)!!{(-\frac{1}{2}{z}^2)}^k}{k!}+\frac{1}{z^{n+1}}\sum _{k=n+1}^{\mathrm{\infty }}\frac{{(\frac{1}{2}{z}^2)}^k}{k!(2k-2n-1)!!}.$$

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Symbols:

$!!$: double factorial, $!$: factorial (as in $n!$), ${𝗂}_n^{(2)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

10.2.6

Referenced by:

§10.53

Permalink:

http://dlmf.nist.gov/10.53.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.53 and Ch.10

… ►For ${𝗄}_n\left(z\right)$ combine (10.47.11), (10.53.3), and (10.53.4).

37: 19.38 Approximations

… ►Minimax polynomial approximations (§3.11(i)) for $K\left(k\right)$ and $E\left(k\right)$ in terms of $m=k^2$ with can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸. Approximations of the same type for $K\left(k\right)$ and $E\left(k\right)$ for are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. … ►They are valid over parts of the complex $k$ and $\varphi$ planes. …

38: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

… ►

22.12.1 $$\tau =\mathrm{i}{K}^{\prime }\left(k\right)/K\left(k\right),$$

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Symbols:

$K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $k$: modulus and $\tau$: ratio

Permalink:

http://dlmf.nist.gov/22.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

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22.12.2 $$2Kk\mathrm{sn}(2Kt,k)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\pi }{\sin \left(\pi (t-(n+\frac{1}{2})\tau )\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^m}{t-m-(n+\frac{1}{2})\tau }\right),$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: sine function, $k$: modulus and $\tau$: ratio

Referenced by:

§22.12, §22.12

Permalink:

http://dlmf.nist.gov/22.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

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22.12.6 $$-2\mathrm{i}Kk{k}^{\prime }\mathrm{sd}(2Kt,k)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^n\pi }{\sin \left(\pi (t+\frac{1}{2}-(n+\frac{1}{2})\tau )\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^{m+n}}{t+\frac{1}{2}-m-(n+\frac{1}{2})\tau }\right),$$

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Symbols:

$\mathrm{sd}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $k$: modulus, $k^{\prime }$: complementary modulus and $\tau$: ratio

Permalink:

http://dlmf.nist.gov/22.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

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22.12.8 $$2K\mathrm{dc}(2Kt,k)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\pi }{\sin \left(\pi (t+\frac{1}{2}-n\tau )\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^m}{t+\frac{1}{2}-m-n\tau }\right),$$

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Symbols:

$\mathrm{dc}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: sine function, $k$: modulus and $\tau$: ratio

Permalink:

http://dlmf.nist.gov/22.12.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

… ►

22.12.11 $$2K\mathrm{ns}(2Kt,k)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\pi }{\sin \left(\pi (t-n\tau )\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^m}{t-m-n\tau }\right),$$

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Symbols:

$\mathrm{ns}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\sin z$: sine function, $k$: modulus and $\tau$: ratio

Permalink:

http://dlmf.nist.gov/22.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

…

39: 9.9 Zeros

… ►They are denoted by $a_k$, $a_k^{\prime }$, $b_k$, $b_k^{\prime }$, respectively, arranged in ascending order of absolute value for $k=1,2,\mathrm{\dots }.$ … ►

§9.9(iii) Derivatives With Respect to $k$

►If $k$ is regarded as a continuous variable, then … ►For large $k$ … ►For error bounds for the asymptotic expansions of $a_k$, $b_k$, $a_k^{\prime }$, and $b_k^{\prime }$ see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999). …

40: 10.44 Sums

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10.44.1 $${𝒵}_{\nu }\left(\lambda z\right)={\lambda }^{\pm \nu }\sum _{k=0}^{\mathrm{\infty }}\frac{{({\lambda }^2-1)}^k{(\frac{1}{2}z)}^k}{k!}{𝒵}_{\nu \pm k}\left(z\right),$$ .

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Symbols:

$!$: factorial (as in $n!$), ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.51

Referenced by:

§10.44(i), §10.66

Permalink:

http://dlmf.nist.gov/10.44.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.44(i), §10.44 and Ch.10

… ►

$I_{\nu }\left(z\right)={\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\displaystyle \frac{z^k}{k!}}{J}_{\nu +k}\left(z\right),$

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$J_{\nu }\left(z\right)={\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{(-1)}^k{\displaystyle \frac{z^k}{k!}}{I}_{\nu +k}\left(z\right).$

… ►

10.44.4 $${\left(\frac{1}{2}z\right)}^{\nu }=\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{(\nu +2k)\mathrm{\Gamma }\left(\nu +k\right)}{k!}{I}_{\nu +2k}\left(z\right),$$ $\nu \ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.44.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.44(iii), §10.44 and Ch.10

… ►

10.44.6 $$K_n\left(z\right)=\frac{n!{(\frac{1}{2}z)}^{-n}}{2}\sum _{k=0}^{n-1}{(-1)}^k\frac{{(\frac{1}{2}z)}^k{I}_k\left(z\right)}{k!(n-k)}+{(-1)}^{n-1}\left(\ln \left(\frac{1}{2}z\right)-\psi \left(n+1\right)\right)I_n\left(z\right)+{(-1)}^n\sum _{k=1}^{\mathrm{\infty }}\frac{(n+2k)I_{n+2k}\left(z\right)}{k(n+k)},$$

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Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

9.6.53

Permalink:

http://dlmf.nist.gov/10.44.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.44(iii), §10.44 and Ch.10

…