Jacobi function

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1: 22.16 Related Functions

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§22.16(i) Jacobi’s Amplitude ($\mathrm{am}$) Function

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§22.16(ii) Jacobi’s Epsilon Function

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Relation to Theta Functions

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§22.16(iii) Jacobi’s Zeta Function

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Definition

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2: 22.6 Elementary Identities

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22.6.1 $${\mathrm{sn}}^2(z,k)+{\mathrm{cn}}^2(z,k)=k^2{\mathrm{sn}}^2(z,k)+{\mathrm{dn}}^2(z,k)=1,$$

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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 and $k$: modulus

A&S Ref:

16.9.1

Referenced by:

§22.11, §22.11, §22.20(ii), §22.6(ii)

Permalink:

http://dlmf.nist.gov/22.6.E1

Encodings:

pMML, png, TeX

See also:

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

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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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22.6.5 $$\mathrm{sn}(2z,k)=\frac{2\mathrm{sn}(z,k)\mathrm{cn}(z,k)\mathrm{dn}(z,k)}{1-k^2{\mathrm{sn}}^4(z,k)},$$

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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 and $k$: modulus

A&S Ref:

16.18.1

Permalink:

http://dlmf.nist.gov/22.6.E5

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(iv) Rotation of Argument (Jacobi’s Imaginary Transformation)

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3: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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Jacobi Form

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35.7.2 $$P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)=\frac{{\mathrm{\Gamma }}_m\left(\gamma +\nu +\frac{1}{2}(m+1)\right)}{{\mathrm{\Gamma }}_m\left(\gamma +\frac{1}{2}(m+1)\right)}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-\nu ,\gamma +\delta +\nu +\frac{1}{2}(m+1)}{\gamma +\frac{1}{2}(m+1)};𝐓),$$ ; $\gamma ,\delta ,\nu \in ℂ$; $\mathrm{\Re }\left(\gamma \right)>-1$.

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Defines:

$P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)$: Jacobi function of matrix argument

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $ℂ$: complex plane, $\in$: element of, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, : less-than for matrices, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Permalink:

http://dlmf.nist.gov/35.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(i), §35.7(i), §35.7 and Ch.35

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4: 22.8 Addition Theorems

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22.8.1 $$\mathrm{sn}(u+v)=\frac{\mathrm{sn}u\mathrm{cn}v\mathrm{dn}v+\mathrm{sn}v\mathrm{cn}u\mathrm{dn}u}{1-k^2{\mathrm{sn}}^{2}u{\mathrm{sn}}^{2}v},$$

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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, $u$: complex, $v$: complex and $k$: modulus

A&S Ref:

16.17.1

Referenced by:

§22.18(iv)

Permalink:

http://dlmf.nist.gov/22.8.E1

Encodings:

pMML, png, TeX

See also:

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

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22.8.2 $$\mathrm{cn}(u+v)=\frac{\mathrm{cn}u\mathrm{cn}v-\mathrm{sn}u\mathrm{dn}u\mathrm{sn}v\mathrm{dn}v}{1-k^2{\mathrm{sn}}^{2}u{\mathrm{sn}}^{2}v},$$

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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, $u$: complex, $v$: complex and $k$: modulus

A&S Ref:

16.17.2

Referenced by:

§22.6(ii)

Permalink:

http://dlmf.nist.gov/22.8.E2

Encodings:

pMML, png, TeX

See also:

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

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22.8.3 $$\mathrm{dn}(u+v)=\frac{\mathrm{dn}u\mathrm{dn}v-k^2\mathrm{sn}u\mathrm{cn}u\mathrm{sn}v\mathrm{cn}v}{1-k^2{\mathrm{sn}}^{2}u{\mathrm{sn}}^{2}v}.$$

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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, $u$: complex, $v$: complex and $k$: modulus

A&S Ref:

16.17.3

Permalink:

http://dlmf.nist.gov/22.8.E3

Encodings:

pMML, png, TeX

See also:

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

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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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22.8.6 $$\mathrm{nd}(u+v)=\frac{\mathrm{nd}u\mathrm{nd}v+k^2\mathrm{sd}u\mathrm{cd}u\mathrm{sd}v\mathrm{cd}v}{1+k^{2}k_{}^{\prime }{}_{}{}^2{\mathrm{sd}}^{2}u{\mathrm{sd}}^{2}v},$$

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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.E6

Encodings:

pMML, png, TeX

See also:

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

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5: 14.31 Other Applications

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§14.31(ii) Conical Functions

… ►The conical functions and Mehler–Fock transform generalize to Jacobi functions and the Jacobi transform; see Koornwinder (1984a) and references therein. …

6: 15.17 Mathematical Applications

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§15.17(iii) Group Representations

►For harmonic analysis it is more natural to represent hypergeometric functions as a Jacobi function (§15.9(ii)). …Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. …

7: 22.14 Integrals

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22.14.1 $$\int \mathrm{sn}(x,k)dx=k^{-1}\ln \left(\mathrm{dn}(x,k)-k\mathrm{cn}(x,k)\right),$$

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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, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $x$: real and $k$: modulus

A&S Ref:

16.24.1

Permalink:

http://dlmf.nist.gov/22.14.E1

Encodings:

pMML, png, TeX

See also:

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

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22.14.3 $$\int \mathrm{dn}(x,k)dx=\mathrm{Arcsin}\left(\mathrm{sn}(x,k)\right)=\mathrm{am}(x,k).$$

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Symbols:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{Arcsin}z$: general arcsine function, $x$: real and $k$: modulus

A&S Ref:

16.24.3

Permalink:

http://dlmf.nist.gov/22.14.E3

Encodings:

pMML, png, TeX

See also:

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

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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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22.14.7 $$\int \mathrm{dc}(x,k)dx=\ln \left(\mathrm{nc}(x,k)+\mathrm{sc}(x,k)\right),$$

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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, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $x$: real and $k$: modulus

A&S Ref:

16.24.7

Permalink:

http://dlmf.nist.gov/22.14.E7

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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8: 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 $z=2mK+(2n+1)\mathrm{i}{K}^{\prime }$. … ► … ►Then: (a) In any lattice unit cell $\mathrm{p}\mathrm{q}(z,k)$ has a simple zero at $z=\text{p}$ and a simple pole at $z=\text{q}$. (b) The difference between p and the nearest q is a half-period of $\mathrm{p}\mathrm{q}(z,k)$. … ►For example, $\mathrm{sn}(z+K,k)=\mathrm{cd}(z,k)$. …

9: 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, $\mathrm{sn}(\frac{1}{2}K,k)={(1+k^{\prime })}^{-1/2}$. … ►In these cases the elliptic functions degenerate into elementary trigonometric and hyperbolic functions, respectively. … ►

10: 22.15 Inverse Functions

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22.15.1 $$\mathrm{sn}(\xi ,k)=x,$$ $-1\le x\le 1$,

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $x$: real, $k$: modulus and $\xi$: solution

Referenced by:

§22.15(i), §22.15(i)

Permalink:

http://dlmf.nist.gov/22.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.15(i), §22.15 and Ch.22

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22.15.2 $$\mathrm{cn}(\eta ,k)=x,$$ $-1\le x\le 1$,

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $x$: real, $k$: modulus and $\eta$: solution

Referenced by:

§22.15(i)

Permalink:

http://dlmf.nist.gov/22.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.15(i), §22.15 and Ch.22

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22.15.3 $$\mathrm{dn}(\zeta ,k)=x,$$ $k^{\prime }\le x\le 1$,

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $x$: real, $k$: modulus, $k^{\prime }$: complementary modulus and $\zeta$: solution

Referenced by:

§22.15(i)

Permalink:

http://dlmf.nist.gov/22.15.E3

Encodings:

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See also:

Annotations for §22.15(i), §22.15 and Ch.22

►are denoted respectively by … ►

22.15.11 $$x={\int }_0^{\mathrm{sn}(x,k)}\frac{dt}{\sqrt{(1-t^2)(1-k^2{t}^2)}},$$ $-1\le x\le 1$, $0\le k\le 1$.

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $x$: real and $k$: modulus

Referenced by:

§19.25(v)

Permalink:

http://dlmf.nist.gov/22.15.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §22.15(i), §22.15 and Ch.22

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