DLMF: Untitled Document

… ►For example, at

z=K+iK^{\prime}

,

\operatorname{sn}\left(z,k\right)=1/k

,

\ifrac{\mathrm{d}\operatorname{sn}\left(z,k\right)}{\mathrm{d}z}=0

. … ►For example,

\operatorname{sn}\left(\frac{1}{2}K,k\right)=(1+k^{\prime})^{-1/2}

. … ►

§22.5(ii) Limiting Values of $k$

►If

k\to 0+

, then

K\to\pi/2

and

K^{\prime}\to\infty

; if

k\to 1-

, then

K\to\infty

and

K^{\prime}\to\pi/2

. … ►

… ►

22.7.1

k_{1}=\frac{1-k^{\prime}}{1+k^{\prime}},

ⓘ

Defines:: $k_{1}$ : change of variable (locally)
Symbols:: $k^{\prime}$ : complementary modulus
A&S Ref:: 16.12.1
Referenced by:: §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(i), §22.7 and Ch.22

►

22.7.2

\operatorname{sn}\left(z,k\right)=\frac{(1+k_{1})\operatorname{sn}\left(z/(1+k% _{1}),k_{1}\right)}{1+k_{1}{\operatorname{sn}}^{2}\left(z/(1+k_{1}),k_{1}% \right)},

ⓘ

Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k_{1}$ : change of variable
A&S Ref:: 16.12.2
Referenced by:: §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(i), §22.7 and Ch.22

►

22.7.3

\operatorname{cn}\left(z,k\right)=\frac{\operatorname{cn}\left(z/(1+k_{1}),k_{% 1}\right)\operatorname{dn}\left(z/(1+k_{1}),k_{1}\right)}{1+k_{1}{% \operatorname{sn}}^{2}\left(z/(1+k_{1}),k_{1}\right)},

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k_{1}$ : change of variable
A&S Ref:: 16.12.3
Referenced by:: §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(i), §22.7 and Ch.22

►

22.7.4

\operatorname{dn}\left(z,k\right)=\frac{{\operatorname{dn}}^{2}\left(z/(1+k_{1% }),k_{1}\right)-(1-k_{1})}{1+k_{1}-{\operatorname{dn}}^{2}\left(z/(1+k_{1}),k_% {1}\right)}.

ⓘ

Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex, $k$ : modulus and $k_{1}$ : change of variable
A&S Ref:: 16.12.4
Referenced by:: §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.7.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §22.7(i), §22.7 and Ch.22

… ►

k_{2}^{\prime}=\frac{1-k}{1+k},

…

… ►Abramowitz and Stegun (1964, Chapter 23) includes exact values of

\sum_{k=1}^{m}k^{n}

,

m=1(1)100

,

n=1(1)10

;

\sum_{k=1}^{\infty}k^{-n}

,

\sum_{k=1}^{\infty}(-1)^{k-1}k^{-n}

,

\sum_{k=0}^{\infty}(2k+1)^{-n}

,

n=1,2,\dotsc

, 20D;

\sum_{k=0}^{\infty}(-1)^{k}(2k+1)^{-n}

,

n=1,2,\dotsc

, 18D. …

… ►

k_{1}=\frac{k}{\sqrt{1+k^{2}}},

►

k_{1}k_{1}^{\prime}=\frac{k}{1+k^{2}},

… ►

22.17.7

\operatorname{cn}\left(z,ik\right)=\operatorname{cd}\left(z/k_{1}^{\prime},k_{% 1}\right),

ⓘ

Symbols:: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathrm{i}$ : imaginary unit, $z$ : complex, $k$ : modulus, $k_{1}$ : change of variable and $k_{1}^{\prime}$ : change of variable
A&S Ref:: 16.10.3 (modified)
Permalink:: http://dlmf.nist.gov/22.17.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §22.17(i), §22.17 and Ch.22

►

22.17.8

\operatorname{dn}\left(z,ik\right)=\operatorname{nd}\left(z/k_{1}^{\prime},k_{% 1}\right).

ⓘ

Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathrm{i}$ : imaginary unit, $z$ : complex, $k$ : modulus, $k_{1}$ : change of variable and $k_{1}^{\prime}$ : change of variable
A&S Ref:: 16.10.4 (modified)
Permalink:: http://dlmf.nist.gov/22.17.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §22.17(i), §22.17 and Ch.22

… ►In particular, the Landen transformations in §§22.7(i) and 22.7(ii) are valid for all complex values of

k

, irrespective of which values of

\sqrt{k}

and

k^{\prime}=\sqrt{1-k^{2}}

are chosen—as long as they are used consistently. …

… ►

e_{j}^{k}=e_{j-1}^{k+1}+q_{j}^{k+1}-q_{j}^{k},

j\geq 1

,

k\geq 0

,

►

q_{j+1}^{k}=q_{j}^{k+1}e_{j}^{k+1}/e_{j}^{k},

j\geq 1

,

k\geq 0

.

… ►

u_{k}=b_{k}+\frac{a_{k+1}}{u_{k+1}}

,

k=n-1,n-2,\dots,0

.

… ►

t_{k}=\rho_{k}t_{k-1},

… ►

\rho_{k}=\frac{a_{k}(1+\rho_{k-1})}{1-a_{k}(1+\rho_{k-1})}

,

k=1,2,3,\dots

.

…

… ►If

S=\sum_{k=0}^{\infty}(-1)^{k}a_{k}

is a convergent series, then ►

3.9.2

S=\sum_{k=0}^{\infty}(-1)^{k}2^{-k-1}\Delta^{k}a_{0},

ⓘ

Symbols:: $\Delta$ : forward difference operator and $S$ : a convergent series
Permalink:: http://dlmf.nist.gov/3.9.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §3.9(ii), §3.9 and Ch.3

… ►

3.9.3

\Delta^{k}a_{0}=\Delta^{k-1}a_{1}-\Delta^{k-1}a_{0},

k=1,2,\dotsc

.

ⓘ

Symbols:: $\Delta$ : forward difference operator
A&S Ref:: 3.6.27
Permalink:: http://dlmf.nist.gov/3.9.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §3.9(ii), §3.9 and Ch.3

… ►Aitken’s

\Delta^{2}

-process is the case

k=1

. … ►

3.9.14

c_{j,k,n}=\frac{(n+j+1)^{k-1}}{(n+k+1)^{k-1}}.

ⓘ

Defines:: $c_{j,k,n}$ : coefficient (locally)
Permalink:: http://dlmf.nist.gov/3.9.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §3.9(v), §3.9 and Ch.3

…

… ►

24.6.1

B_{2n}=\sum_{k=2}^{2n+1}\frac{(-1)^{k-1}}{k}{2n+1\choose k}\sum_{j=1}^{k-1}j^{% 2n},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $j$ : integer, $k$ : integer and $n$ : integer
Referenced by:: §24.6
Permalink:: http://dlmf.nist.gov/24.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

… ►

24.6.3

B_{2n}=\sum_{k=1}^{n}\frac{(k-1)!k!}{(2k+1)!}\*\sum_{j=1}^{k}(-1)^{j-1}{2k% \choose k+j}j^{2n}.

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $!$ : factorial (as in $n!$ ), $j$ : integer, $k$ : integer and $n$ : integer
Referenced by:: §24.6
Permalink:: http://dlmf.nist.gov/24.6.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

►

24.6.4

E_{2n}=\sum_{k=1}^{n}\frac{1}{2^{k-1}}\sum_{j=1}^{k}(-1)^{j}{2k\choose k-j}j^{% 2n},

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $j$ : integer, $k$ : integer and $n$ : integer
Referenced by:: §24.6
Permalink:: http://dlmf.nist.gov/24.6.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

… ►

24.6.6

E_{2n}=\sum_{k=1}^{2n}\frac{(-1)^{k}}{2^{k-1}}{2n+1\choose k+1}\*\sum_{j=0}^{% \left\lfloor\tfrac{1}{2}k-\tfrac{1}{2}\right\rfloor}{k\choose j}(k-2j)^{2n}.

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $j$ : integer, $k$ : integer and $n$ : integer
Referenced by:: §24.6
Permalink:: http://dlmf.nist.gov/24.6.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

… ►

24.6.9

B_{n}=\sum_{k=0}^{n}\frac{1}{k+1}\sum_{j=0}^{k}(-1)^{j}{k\choose j}j^{n},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $j$ : integer, $k$ : integer and $n$ : integer
Referenced by:: §24.6
Permalink:: http://dlmf.nist.gov/24.6.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

…

… ►where

\mathsf{P}^{m}_{n+2k}\left(x\right)

is the Ferrers function of the first kind (§14.3(i)),

R=\left\lfloor\frac{1}{2}(n-m)\right\rfloor

, and the coefficients

a^{m}_{n,k}(\gamma^{2})

are given by …Then the set of coefficients

a^{m}_{n,k}(\gamma^{2})

,

k=-R,-R+1,-R+2,\dots

is the solution of the difference equation … ►For

k=-N,-N+1,\dots,-R-1

they are determined from (30.8.4) by forward recursion using

a^{m}_{n,-N-1}(\gamma^{2})=0

. The set of coefficients

{a^{\prime}}^{m}_{n,k}(\gamma^{2})

,

k=-N-1,-N-2,\dots

, is the recessive solution of (30.8.4) as

k\to-\infty

that is normalized by …It should be noted that if the forward recursion (30.8.4) beginning with

f_{-N-1}=0

,

f_{-N}=1

leads to

f_{-R}=0

, then

a^{m}_{n,k}(\gamma^{2})

is undefined for

n<-R

and

\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right)

does not exist. …

… ►

13.2.27

\sum_{k=1}^{n}\frac{n!(k-1)!}{(n-k)!{\left(1-a\right)_{k}}}z^{-k}-\sum_{k=0}^{% \infty}\frac{{\left(a\right)_{k}}}{{\left(n+1\right)_{k}}k!}z^{k}\left(\ln z+% \psi\left(a+k\right)-\psi\left(1+k\right)-\psi\left(n+k+1\right)\right),

ⓘ

Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §13.2(v)
Permalink:: http://dlmf.nist.gov/13.2.E27
Encodings:: pMML, png, TeX
See also:: Annotations for §13.2(v), §13.2 and Ch.13

… ►

13.2.28

\sum_{k=1}^{n}\frac{n!(k-1)!}{(n-k)!{\left(1-a\right)_{k}}}z^{-k}-\sum_{k=0}^{% -a}\frac{{\left(a\right)_{k}}}{{\left(n+1\right)_{k}}k!}z^{k}\left(\ln z+\psi% \left(1-a-k\right)-\psi\left(1+k\right)-\psi\left(n+k+1\right)\right)+(-1)^{1-% a}(-a)!\sum_{k=1-a}^{\infty}\frac{(k-1+a)!}{{\left(n+1\right)_{k}}k!}z^{k},

ⓘ

Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.2.E28
Encodings:: pMML, png, TeX
See also:: Annotations for §13.2(v), §13.2 and Ch.13

… ►

13.2.30

\sum_{k=1}^{n+1}\frac{(n+1)!(k-1)!}{(n-k+1)!{\left(-a-n\right)_{k}}}z^{n-k+1}-% \sum_{k=0}^{\infty}\frac{{\left(a+n+1\right)_{k}}}{{\left(n+2\right)_{k}}k!}z^% {n+k+1}\left(\ln z+\psi\left(a+n+k+1\right)-\psi\left(1+k\right)-\psi\left(n+k% +2\right)\right),

ⓘ

Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.2.E30
Encodings:: pMML, png, TeX
See also:: Annotations for §13.2(v), §13.2 and Ch.13

… ►

13.2.31

\sum_{k=1}^{n+1}\frac{(n+1)!(k-1)!}{(n-k+1)!{\left(-a-n\right)_{k}}}z^{n-k+1}-% \sum_{k=0}^{-a-n-1}\frac{{\left(a+n+1\right)_{k}}}{{\left(n+2\right)_{k}}k!}z^% {n+k+1}\left(\ln z+\psi\left(-a-n-k\right)-\psi\left(1+k\right)-\psi\left(n+k+% 2\right)\right)+(-1)^{n-a}{(-a-n-1)!}\sum_{k=-a-n}^{\infty}\frac{(k+a+n)!}{{% \left(n+2\right)_{k}}k!}z^{n+k+1},

ⓘ

Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.2.E31
Encodings:: pMML, png, TeX
See also:: Annotations for §13.2(v), §13.2 and Ch.13

… ►

13.2.32

\sum_{k=a+n+1}^{n+1}\frac{(k-1)!}{(n-k+1)!(k-a-n-1)!}z^{n-k+1},

ⓘ

Symbols:: $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §13.2(v)
Permalink:: http://dlmf.nist.gov/13.2.E32
Encodings:: pMML, png, TeX
See also:: Annotations for §13.2(v), §13.2 and Ch.13

…

… ►Here

f_{0}=1

,

g_{0}=0

,

\widehat{f}_{0}=0

,

\widehat{g}_{0}=1-(\eta/\rho)

, and for

k=0,1,2,\dots

, … ►

\widehat{f}_{k+1}=\lambda_{k}\widehat{f}_{k}-\mu_{k}\widehat{g}_{k}-(f_{k+1}/% \rho),

►

\widehat{g}_{k+1}=\lambda_{k}\widehat{g}_{k}+\mu_{k}\widehat{f}_{k}-(g_{k+1}/% \rho),

… ►

\lambda_{k}=\frac{(2k+1)\eta}{(2k+2)\rho},

►

\mu_{k}=\frac{\ell(\ell+1)-k(k+1)+\eta^{2}}{(2k+2)\rho}.

limiting forms as k→0 or k→1

1—10 of 448 matching pages

1: 22.5 Special Values

§22.5(ii) Limiting Values of $k$

2: 22.7 Landen Transformations

3: 24.20 Tables

4: 22.17 Moduli Outside the Interval [0,1]

5: 3.10 Continued Fractions

6: 3.9 Acceleration of Convergence

7: 24.6 Explicit Formulas

8: 30.8 Expansions in Series of Ferrers Functions

9: 13.2 Definitions and Basic Properties

10: 33.11 Asymptotic Expansions for Large $\rho$

limiting forms as k→0 or k→1

1—10 of 448 matching pages

1: 22.5 Special Values

§22.5(ii) Limiting Values of k

2: 22.7 Landen Transformations

3: 24.20 Tables

4: 22.17 Moduli Outside the Interval [0,1]

5: 3.10 Continued Fractions

6: 3.9 Acceleration of Convergence

7: 24.6 Explicit Formulas

8: 30.8 Expansions in Series of Ferrers Functions

9: 13.2 Definitions and Basic Properties

10: 33.11 Asymptotic Expansions for Large ρ

§22.5(ii) Limiting Values of $k$

10: 33.11 Asymptotic Expansions for Large $\rho$