change of

SearchAdvancedHelp

(0.002 seconds)

1—10 of 120 matching pages

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

§22.17 Moduli Outside the Interval [0,1]

… ►

$k_1={\displaystyle \frac{k}{\sqrt{1+k^2}}},$

►

$k_1{k}_1^{\prime }={\displaystyle \frac{k}{1+k^2}},$

… ►

22.17.7 $$\mathrm{cn}(z,\mathrm{i}k)=\mathrm{cd}(z/k_1^{\prime },k_1),$$

ⓘ

Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{cn}(z,k)$: 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

… ►For proofs of these results and further information see Walker (2003).

2: 22.7 Landen Transformations

… ►

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

… ►

$k_2={\displaystyle \frac{2\sqrt{k}}{1+k}},$

►

$k_2^{\prime }={\displaystyle \frac{1-k}{1+k}},$

►

22.7.6 $$\mathrm{sn}(z,k)=\frac{(1+k_2^{\prime })\mathrm{sn}(z/(1+k_2^{\prime }),k_2)\mathrm{cn}(z/(1+k_2^{\prime }),k_2)}{\mathrm{dn}(z/(1+k_2^{\prime }),k_2)},$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus, $k_2$: change of variable and $k_2^{\prime }$: change of variable

A&S Ref:

16.14.2

Permalink:

http://dlmf.nist.gov/22.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §22.7(ii), §22.7 and Ch.22

… ►

22.7.8 $$\mathrm{dn}(z,k)=\frac{(1-k_2^{\prime })({\mathrm{dn}}^2(z/(1+k_2^{\prime }),k_2)+k_2^{\prime })}{k_2^2\mathrm{dn}(z/(1+k_2^{\prime }),k_2)}.$$

ⓘ

Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus, $k_2$: change of variable and $k_2^{\prime }$: change of variable

A&S Ref:

16.14.4

Permalink:

http://dlmf.nist.gov/22.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.7(ii), §22.7 and Ch.22

…

3: Sidebar 5.SB1: Gamma & Digamma Phase Plots

… ►In the upper half of the image, the poles of $\mathrm{\Gamma }\left(z\right)$ are clearly visible at negative integer values of $z$: the phase changes by $2\pi$ around each pole, showing a full revolution of the color wheel. … ►Phase changes around the zeros are of opposite sign to those around the poles. …

4: 19.7 Connection Formulas

… ►

§19.7(ii) Change of Modulus and Amplitude

… ►

$\kappa ={\displaystyle \frac{k}{\sqrt{1+k^2}}},$

►

${\kappa }^{\prime }={\displaystyle \frac{1}{\sqrt{1+k^2}}},$

… ►With $\sinh \varphi =\tan \psi$, … ►

§19.7(iii) Change of Parameter of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$

…

5: 19.8 Quadratic Transformations

… ►

$$k_1=\frac{1-k^{\prime }}{1+k^{\prime }},$$

… ►

$c_1={\csc }^2{\varphi }_1.$

… ►

$$k_2=2\sqrt{k}/(1+k),$$

… ►

$k_1=(1-k^{\prime })/(1+k^{\prime }),$

… ►

$c={\csc }^2\varphi .$

…

6: 7.16 Generalized Error Functions

… ►These functions can be expressed in terms of the incomplete gamma function $\gamma (a,z)$ (§8.2(i)) by change of integration variable.

7: 27 Functions of Number Theory

…

8: 23.21 Physical Applications

… ►Another form is obtained by identifying $e_1$, $e_2$, $e_3$ as lattice roots (§23.3(i)), and setting … ►

23.21.5 $$\left(\mathrm{\wp }\left(v\right)-\mathrm{\wp }\left(w\right)\right)\left(\mathrm{\wp }\left(w\right)-\mathrm{\wp }\left(u\right)\right)\left(\mathrm{\wp }\left(u\right)-\mathrm{\wp }\left(v\right)\right){\nabla }^2=\left(\mathrm{\wp }\left(w\right)-\mathrm{\wp }\left(v\right)\right)\frac{{\partial }^2}{{\partial u}^2}+\left(\mathrm{\wp }\left(u\right)-\mathrm{\wp }\left(w\right)\right)\frac{{\partial }^2}{{\partial v}^2}+\left(\mathrm{\wp }\left(v\right)-\mathrm{\wp }\left(u\right)\right)\frac{{\partial }^2}{{\partial w}^2}.$$

ⓘ

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $𝕃$: lattice, $u$: change of variable, $v$: change of variable and $w$: change of variable

Permalink:

http://dlmf.nist.gov/23.21.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §23.21(iii), §23.21 and Ch.23

…

9: Foreword

… ►Much has changed in the years since A&S was published. …However, we have also seen the birth of a new age of computing technology, which has not only changed how we utilize special functions, but also how we communicate technical information. …

10: 22.11 Fourier and Hyperbolic Series

… ►

22.11.1 $$\mathrm{sn}(z,k)=\frac{2\pi }{Kk}\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n+\frac{1}{2}}\sin \left((2n+1)\zeta \right)}{1-q^{2n+1}},$$

ⓘ

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, $q$: nome, $\sin z$: sine function, $z$: complex, $k$: modulus and $\zeta$: change of variable

A&S Ref:

16.23.1

Permalink:

http://dlmf.nist.gov/22.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

►

22.11.2 $$\mathrm{cn}(z,k)=\frac{2\pi }{Kk}\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n+\frac{1}{2}}\cos \left((2n+1)\zeta \right)}{1+q^{2n+1}},$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: 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 and $\zeta$: change of variable

A&S Ref:

16.23.2

Permalink:

http://dlmf.nist.gov/22.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

►

22.11.3 $$\mathrm{dn}(z,k)=\frac{\pi }{2K}+\frac{2\pi }{K}\sum _{n=1}^{\mathrm{\infty }}\frac{q^n\cos \left(2n\zeta \right)}{1+q^{2n}}.$$

ⓘ

Symbols:

$\mathrm{dn}(z,k)$: 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 and $\zeta$: change of variable

A&S Ref:

16.23.3

Permalink:

http://dlmf.nist.gov/22.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

►

22.11.4 $$\mathrm{cd}(z,k)=\frac{2\pi }{Kk}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{n+\frac{1}{2}}\cos \left((2n+1)\zeta \right)}{1-q^{2n+1}},$$

ⓘ

Symbols:

$\mathrm{cd}(z,k)$: 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 and $\zeta$: change of variable

A&S Ref:

16.23.4

Permalink:

http://dlmf.nist.gov/22.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

►

22.11.5 $$\mathrm{sd}(z,k)=\frac{2\pi }{Kk{k}^{\prime }}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{n+\frac{1}{2}}\sin \left((2n+1)\zeta \right)}{1+q^{2n+1}},$$

ⓘ

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, $q$: nome, $\sin z$: sine function, $z$: complex, $k$: modulus, $k^{\prime }$: complementary modulus and $\zeta$: change of variable

A&S Ref:

16.23.5

Permalink:

http://dlmf.nist.gov/22.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

…