change of variables

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1: 22.7 Landen Transformations

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22.7.1 $$k_1=\frac{1-k^{\prime }}{1+k^{\prime }},$$

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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

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

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

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

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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

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

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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

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

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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

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3: 22.17 Moduli Outside the Interval [0,1]

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

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

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22.17.6 $$\mathrm{sn}(z,\mathrm{i}k)=k_1^{\prime }\mathrm{sd}(z/k_1^{\prime },k_1),$$

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Symbols:

$\mathrm{sd}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: 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:

16.10.2 (modified)

Permalink:

http://dlmf.nist.gov/22.17.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §22.17(i), §22.17 and Ch.22

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

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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

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22.17.8 $$\mathrm{dn}(z,\mathrm{i}k)=\mathrm{nd}(z/k_1^{\prime },k_1).$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{nd}(z,k)$: 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:

16.10.4 (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

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4: 19.8 Quadratic Transformations

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

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

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

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

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

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5: 19.7 Connection Formulas

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$\mathrm{\Pi }(\varphi ,{\alpha }^2,k_1)=k\mathrm{\Pi }(\beta ,k^2{\alpha }^2,k),$ $k_1=1/k$, $\sin \beta =k_1\sin \varphi \le 1$.

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

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

… ►With $\sinh \varphi =\tan \psi$, … ►The first of the three relations maps each circular region onto itself and each hyperbolic region onto the other; in particular, it gives the Cauchy principal value of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ when ${\alpha }^2>{\csc }^2\varphi$ (see (19.6.5) for the complete case). …

6: 22.11 Fourier and Hyperbolic Series

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

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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, $q$: 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

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

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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

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

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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

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

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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

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

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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, $q$: 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

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7: 1.13 Differential Equations

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§1.13(iv) Change of Variables

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Transformation of the Point at Infinity

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Elimination of First Derivative by Change of Dependent Variable

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Elimination of First Derivative by Change of Independent Variable

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Liouville Transformation

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8: 9.7 Asymptotic Expansions

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9.7.1 $$\zeta =\frac{2}{3}{z}^{3/2}.$$

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Defines:

$\zeta$: change of variable (locally)

Symbols:

$z$: complex variable

Permalink:

http://dlmf.nist.gov/9.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(i), §9.7 and Ch.9

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9.7.18 $$\mathrm{Ai}\left(z\right)=\frac{{\mathrm{e}}^{-\zeta }}{2\sqrt{\pi }{z}^{1/4}}\left(\sum _{k=0}^{n-1}{(-1)}^k\frac{u_k}{{\zeta }^k}+R_n(z)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer, $z$: complex variable, $\zeta$: change of variable, $n$: index, $R_n$: remainder function and $u_s$: expansion coefficient

Proof sketch:

Use the methods of Olver (1991b, 1993a).

Permalink:

http://dlmf.nist.gov/9.7.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(v), §9.7 and Ch.9

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9.7.19 $${\mathrm{Ai}}^{\prime }\left(z\right)=-\frac{z^{1/4}{\mathrm{e}}^{-\zeta }}{2\sqrt{\pi }}\left(\sum _{k=0}^{n-1}{(-1)}^k\frac{v_k}{{\zeta }^k}+S_n(z)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer, $z$: complex variable, $\zeta$: change of variable, $n$: index, $S_n$: remainder function and $v_s$: expansion coefficient

Proof sketch:

Use the methods of Olver (1991b, 1993a).

Permalink:

http://dlmf.nist.gov/9.7.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(v), §9.7 and Ch.9

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9.7.20 $$R_n(z)={(-1)}^n\sum _{k=0}^{m-1}{(-1)}^k{u}_k\frac{G_{n-k}\left(2\zeta \right)}{{\zeta }^k}+R_{m,n}(z),$$

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Defines:

$R_n$: remainder function (locally)

Symbols:

$G_p\left(z\right)$: rescaled terminant function, $k$: nonnegative integer, $z$: complex variable, $\zeta$: change of variable, $m$: index, $n$: index and $u_s$: expansion coefficient

Proof sketch:

Use the methods of Olver (1991b, 1993a).

Permalink:

http://dlmf.nist.gov/9.7.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(v), §9.7 and Ch.9

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9.7.21 $$S_n(z)={(-1)}^{n-1}\sum _{k=0}^{m-1}{(-1)}^k{v}_k\frac{G_{n-k}\left(2\zeta \right)}{{\zeta }^k}+S_{m,n}(z),$$

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Defines:

$S_n$: remainder function (locally)

Symbols:

$G_p\left(z\right)$: rescaled terminant function, $k$: nonnegative integer, $z$: complex variable, $\zeta$: change of variable, $m$: index, $n$: index and $v_s$: expansion coefficient

Proof sketch:

Use the methods of Olver (1991b, 1993a).

Permalink:

http://dlmf.nist.gov/9.7.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(v), §9.7 and Ch.9

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9: 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.

10: 7.5 Interrelations

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7.5.7 $$\zeta =\frac{1}{2}\sqrt{\pi }(1\mp \mathrm{i})z,$$

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Defines:

$\zeta$: change of variable (locally)

Symbols:

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

Referenced by:

§7.12(ii), §7.5

Permalink:

http://dlmf.nist.gov/7.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.8 $$C\left(z\right)\pm \mathrm{i}S\left(z\right)=\frac{1}{2}(1\pm \mathrm{i})\mathrm{erf}\zeta .$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\mathrm{erf}z$: error function, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

Referenced by:

§7.5, §7.5, §7.6(i)

Permalink:

http://dlmf.nist.gov/7.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.9 $$C\left(z\right)\pm \mathrm{i}S\left(z\right)=\frac{1}{2}(1\pm \mathrm{i})\left(1-{\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}{z}^2}w\left(\mathrm{i}\zeta \right)\right).$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.10 $$\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right)=\frac{1}{2}(1\pm \mathrm{i}){\mathrm{e}}^{{\zeta }^2}\mathrm{erfc}\zeta .$$

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

A&S Ref:

7.3.22 (in different form)

Referenced by:

§7.12(ii), §7.13(iv), §7.5, §7.5

Permalink:

http://dlmf.nist.gov/7.5.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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