Jacobi?s

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1: 22.8 Addition Theorems

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22.8.5 $$\mathrm{sd}(u+v)=\frac{\mathrm{sd}u\mathrm{cd}v\mathrm{nd}v+\mathrm{sd}v\mathrm{cd}u\mathrm{nd}u}{1+k^{2}k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^{2}u{\mathrm{sd}}^{2}v},$$

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Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{nd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $u$: complex, $v$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.8(i), §22.8 and Ch.22

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22.8.9 $$\mathrm{sc}(u+v)=\frac{\mathrm{sc}u\mathrm{dc}v\mathrm{nc}v+\mathrm{sc}v\mathrm{dc}u\mathrm{nc}u}{1-k_{}^{\prime }{}_{}{}^2{\mathrm{sc}}^{2}u{\mathrm{sc}}^{2}v},$$

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Symbols:

$\mathrm{dc}(z,k)$: Jacobian elliptic function, $\mathrm{nc}(z,k)$: Jacobian elliptic function, $\mathrm{sc}(z,k)$: Jacobian elliptic function, $u$: complex, $v$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §22.8(i), §22.8 and Ch.22

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22.8.10 $$\mathrm{ns}(u+v)=\frac{\mathrm{ns}u\mathrm{ds}v\mathrm{cs}v-\mathrm{ns}v\mathrm{ds}u\mathrm{cs}u}{{\mathrm{cs}}^{2}v-{\mathrm{cs}}^{2}u},$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{ds}(z,k)$: Jacobian elliptic function, $\mathrm{ns}(z,k)$: Jacobian elliptic function, $u$: complex, $v$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.8.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §22.8(i), §22.8 and Ch.22

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22.8.11 $$\mathrm{ds}(u+v)=\frac{\mathrm{ds}u\mathrm{cs}v\mathrm{ns}v-\mathrm{ds}v\mathrm{cs}u\mathrm{ns}u}{{\mathrm{cs}}^{2}v-{\mathrm{cs}}^{2}u},$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{ds}(z,k)$: Jacobian elliptic function, $\mathrm{ns}(z,k)$: Jacobian elliptic function, $u$: complex, $v$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.8.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §22.8(i), §22.8 and Ch.22

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22.8.12 $$\mathrm{cs}(u+v)=\frac{\mathrm{cs}u\mathrm{ds}v\mathrm{ns}v-\mathrm{cs}v\mathrm{ds}u\mathrm{ns}u}{{\mathrm{cs}}^{2}v-{\mathrm{cs}}^{2}u}.$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{ds}(z,k)$: Jacobian elliptic function, $\mathrm{ns}(z,k)$: Jacobian elliptic function, $u$: complex, $v$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.8.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §22.8(i), §22.8 and Ch.22

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2: 22.6 Elementary Identities

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22.6.2 $$1+{\mathrm{cs}}^2(z,k)=k^2+{\mathrm{ds}}^2(z,k)={\mathrm{ns}}^2(z,k),$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{ds}(z,k)$: Jacobian elliptic function, $\mathrm{ns}(z,k)$: Jacobian elliptic function, $z$: complex and $k$: modulus

A&S Ref:

16.9.4

Permalink:

http://dlmf.nist.gov/22.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.6(i), §22.6 and Ch.22

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22.6.9 $$\mathrm{sd}(2z,k)=\frac{2\mathrm{sd}(z,k)\mathrm{cd}(z,k)\mathrm{nd}(z,k)}{1+k^{2}k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^4(z,k)},$$

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Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{nd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §22.6(ii), §22.6 and Ch.22

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22.6.14 $$\mathrm{ns}(2z,k)=\frac{{\mathrm{ns}}^4(z,k)-k^2}{2\mathrm{cs}(z,k)\mathrm{ds}(z,k)\mathrm{ns}(z,k)},$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{ds}(z,k)$: Jacobian elliptic function, $\mathrm{ns}(z,k)$: Jacobian elliptic function, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.6.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §22.6(ii), §22.6 and Ch.22

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22.6.15 $$\mathrm{ds}(2z,k)=\frac{k^{2}k_{}^{\prime }{}_{}{}^2+{\mathrm{ds}}^4(z,k)}{2\mathrm{cs}(z,k)\mathrm{ds}(z,k)\mathrm{ns}(z,k)},$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{ds}(z,k)$: Jacobian elliptic function, $\mathrm{ns}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.6.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §22.6(ii), §22.6 and Ch.22

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22.6.16 $$\mathrm{cs}(2z,k)=\frac{{\mathrm{cs}}^4(z,k)-k_{}^{\prime }{}_{}{}^2}{2\mathrm{cs}(z,k)\mathrm{ds}(z,k)\mathrm{ns}(z,k)}.$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{ds}(z,k)$: Jacobian elliptic function, $\mathrm{ns}(z,k)$: Jacobian elliptic function, $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.6.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §22.6(ii), §22.6 and Ch.22

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3: 22.13 Derivatives and Differential Equations

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22.13.5 $${\left(\frac{d}{dz}\mathrm{sd}(z,k)\right)}^2=\left(1-k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^2(z,k)\right)\left(1+k^2{\mathrm{sd}}^2(z,k)\right),$$

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Symbols:

$\mathrm{sd}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(ii), §22.13 and Ch.22

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22.13.9 $${\left(\frac{d}{dz}\mathrm{sc}(z,k)\right)}^2=\left(1+{\mathrm{sc}}^2(z,k)\right)\left(1+k_{}^{\prime }{}_{}{}^2{\mathrm{sc}}^2(z,k)\right),$$

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Symbols:

$\mathrm{sc}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.13.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(ii), §22.13 and Ch.22

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22.13.12 $${\left(\frac{d}{dz}\mathrm{cs}(z,k)\right)}^2=\left(1+{\mathrm{cs}}^2(z,k)\right)\left(k_{}^{\prime }{}_{}{}^2+{\mathrm{cs}}^2(z,k)\right).$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Referenced by:

§22.13(iii)

Permalink:

http://dlmf.nist.gov/22.13.E12

Encodings:

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See also:

Annotations for §22.13(ii), §22.13 and Ch.22

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22.13.17 $$\frac{d^2}{{dz}^2}\mathrm{sd}(z,k)=(k^2-k_{}^{\prime }{}_{}{}^2)\mathrm{sd}(z,k)-2k^{2}k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^3(z,k),$$

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Symbols:

$\mathrm{sd}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.13.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(iii), §22.13 and Ch.22

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22.13.21 $$\frac{d^2}{{dz}^2}\mathrm{sc}(z,k)=(1+k_{}^{\prime }{}_{}{}^2)\mathrm{sc}(z,k)+2k_{}^{\prime }{}_{}{}^2{\mathrm{sc}}^3(z,k),$$

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Symbols:

$\mathrm{sc}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.13.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §22.13(iii), §22.13 and Ch.22

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4: 22.4 Periods, Poles, and Zeros

… ►For example, the poles of $\mathrm{sn}(z,k)$, abbreviated as $\mathrm{sn}$ in the following tables, are at $z=2mK+(2n+1)\mathrm{i}{K}^{\prime }$. … ►For example, $\mathrm{sn}(z+K,k)=\mathrm{cd}(z,k)$. … ►

Table 22.4.3: Half- or quarter-period shifts of variable for the Jacobian elliptic functions.

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	$u$
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$\mathrm{sn}u$	$\mathrm{cd}z$	$k^{-1}\mathrm{dc}z$	$k^{-1}\mathrm{ns}z$	$-\mathrm{sn}z$	$-\mathrm{sn}z$	$\mathrm{sn}z$
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$\mathrm{ns}u$	$\mathrm{dc}z$	$k\mathrm{cd}z$	$k\mathrm{sn}z$	$-\mathrm{ns}z$	$-\mathrm{ns}z$	$\mathrm{ns}z$
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$\mathrm{cs}u$	$-k^{\prime }\mathrm{sc}z$	$-\mathrm{i}{k}^{\prime }\mathrm{nd}z$	$-\mathrm{i}\mathrm{dn}z$	$\mathrm{cs}z$	$-\mathrm{cs}z$	$-\mathrm{cs}z$

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5: 27.9 Quadratic Characters

… ►If an odd integer $P$ has prime factorization $P={\prod }_{r=1}^{\nu \left(n\right)}{p}_r^{a_r}$, then the Jacobi symbol $(n|P)$ is defined by $(n|P)={\prod }_{r=1}^{\nu \left(n\right)}{(n|p_r)}^{a_r}$, with $(n|1)=1$. The Jacobi symbol $(n|P)$ is a Dirichlet character (mod $P$). …

6: 22.14 Integrals

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22.14.4 $$\int \mathrm{cd}(x,k)dx=k^{-1}\ln \left(\mathrm{nd}(x,k)+k\mathrm{sd}(x,k)\right),$$

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Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{nd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $x$: real and $k$: modulus

A&S Ref:

16.24.4

Permalink:

http://dlmf.nist.gov/22.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

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22.14.5 $$\int \mathrm{sd}(x,k)dx={(k{k}^{\prime })}^{-1}\mathrm{Arcsin}\left(-k\mathrm{cd}(x,k)\right),$$

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Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{Arcsin}z$: general arcsine function, $x$: real, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.24.5

Permalink:

http://dlmf.nist.gov/22.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

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22.14.10 $$\int \mathrm{ns}(x,k)dx=\ln \left(\mathrm{ds}(x,k)-\mathrm{cs}(x,k)\right),$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{ds}(z,k)$: Jacobian elliptic function, $\mathrm{ns}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $x$: real and $k$: modulus

A&S Ref:

16.24.10

Permalink:

http://dlmf.nist.gov/22.14.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

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22.14.11 $$\int \mathrm{ds}(x,k)dx=\ln \left(\mathrm{ns}(x,k)-\mathrm{cs}(x,k)\right),$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{ds}(z,k)$: Jacobian elliptic function, $\mathrm{ns}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $x$: real and $k$: modulus

A&S Ref:

16.24.11

Permalink:

http://dlmf.nist.gov/22.14.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

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22.14.12 $$\int \mathrm{cs}(x,k)dx=\ln \left(\mathrm{ns}(x,k)-\mathrm{ds}(x,k)\right).$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{ds}(z,k)$: Jacobian elliptic function, $\mathrm{ns}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $x$: real and $k$: modulus

A&S Ref:

16.24.12

Permalink:

http://dlmf.nist.gov/22.14.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

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7: 22.5 Special Values

… ►For example, at $z=K+\mathrm{i}{K}^{\prime }$, $\mathrm{sn}(z,k)=1/k$, $d\mathrm{sn}(z,k)/dz=0$. … ►Table 22.5.2 gives $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, $\mathrm{dn}(z,k)$ for other special values of $z$. For example, $\mathrm{sn}(\frac{1}{2}K,k)={(1+k^{\prime })}^{-1/2}$. … ►

Table 22.5.3: Limiting forms of Jacobian elliptic functions as $k\to 0$.

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$\mathrm{sn}(z,k)$ $\to$	$\sin z$	$\mathrm{cd}(z,k)$ $\to$	$\cos z$	$\mathrm{dc}(z,k)$ $\to$	$\sec z$	$\mathrm{ns}(z,k)$ $\to$	$\csc z$
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$\mathrm{dn}(z,k)$ $\to$	$1$	$\mathrm{nd}(z,k)$ $\to$	$1$	$\mathrm{sc}(z,k)$ $\to$	$\tan z$	$\mathrm{cs}(z,k)$ $\to$	$\cot z$

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Table 22.5.4: Limiting forms of Jacobian elliptic functions as $k\to 1$.

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$\mathrm{sn}(z,k)$ $\to$	$\tanh z$	$\mathrm{cd}(z,k)$ $\to$	$1$	$\mathrm{dc}(z,k)$ $\to$	$1$	$\mathrm{ns}(z,k)$ $\to$	$\coth z$
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$\mathrm{dn}(z,k)$ $\to$	$\mathrm{sech}z$	$\mathrm{nd}(z,k)$ $\to$	$\cosh z$	$\mathrm{sc}(z,k)$ $\to$	$\sinh z$	$\mathrm{cs}(z,k)$ $\to$	$\mathrm{csch}z$

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8: 22.1 Special Notation

… ►The functions treated in this chapter are the three principal Jacobian elliptic functions $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, $\mathrm{dn}(z,k)$; the nine subsidiary Jacobian elliptic functions $\mathrm{cd}(z,k)$, $\mathrm{sd}(z,k)$, $\mathrm{nd}(z,k)$, $\mathrm{dc}(z,k)$, $\mathrm{nc}(z,k)$, $\mathrm{sc}(z,k)$, $\mathrm{ns}(z,k)$, $\mathrm{ds}(z,k)$, $\mathrm{cs}(z,k)$; the amplitude function $\mathrm{am}(x,k)$; Jacobi’s epsilon and zeta functions $ℰ(x,k)$ and $\mathrm{Z}\left(x|k\right)$. … ►The notation $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, $\mathrm{dn}(z,k)$ is due to Gudermann (1838), following Jacobi (1827); that for the subsidiary functions is due to Glaisher (1882). Other notations for $\mathrm{sn}(z,k)$ are $\mathrm{sn}(z|m)$ and $\mathrm{sn}(z,m)$ with $m=k^2$; see Abramowitz and Stegun (1964) and Walker (1996). …

9: 22.3 Graphics

… ►Line graphs of the functions $\mathrm{sn}(x,k)$, $\mathrm{cn}(x,k)$, $\mathrm{dn}(x,k)$, $\mathrm{cd}(x,k)$, $\mathrm{sd}(x,k)$, $\mathrm{nd}(x,k)$, $\mathrm{dc}(x,k)$, $\mathrm{nc}(x,k)$, $\mathrm{sc}(x,k)$, $\mathrm{ns}(x,k)$, $\mathrm{ds}(x,k)$, and $\mathrm{cs}(x,k)$ for representative values of real $x$ and real $k$ illustrating the near trigonometric ($k=0$), and near hyperbolic ($k=1$) limits. … ► $\mathrm{sn}(x,k)$, $\mathrm{cn}(x,k)$, and $\mathrm{dn}(x,k)$ as functions of real arguments $x$ and $k$. … ►

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Figure 22.3.13: $\mathrm{sn}(x,k)$ for $k=1-{\mathrm{e}}^{-n}$, $n=0$ to 20, $-5\pi \le x\le 5\pi$. Magnify 3D Help

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Figure 22.3.21: $\mathrm{ns}(x+\mathrm{i}y,k)$ for $k=0.99$, $-3K\le x\le 3K$, $0\le y\le 4K^{\prime }$. … Magnify 3D Help

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Figure 22.3.22: $\mathrm{\Re }\mathrm{sn}(x,k)$, $x=120$, as a function of $k^2=\mathrm{i}{\kappa }^2$, $0\le \kappa \le 4$. Magnify

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10: 27.1 Special Notation

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$d,k,m,n$	positive integers (unless otherwise indicated).
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$(n|P)$	Jacobi symbol; see §27.9.
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