22 Jacobian Elliptic Functions Properties 22.10 Maclaurin Series 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

§22.11 Fourier and Hyperbolic Series

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Keywords:: Fourier series, Jacobian elliptic functions, for squares, hyperbolic series for squares, trigonometric series expansions
Notes:: See Walker (1996, §5.4), Whittaker and Watson (1927, §22.6). For (22.11.13) see Deconinck and Kutz (2006, Eq. (48)). The version of this formula in Byrd and Friedman (1971, p. 307, Eq. (911.01)) is incorrect; see Tang (1969).
Permalink:: http://dlmf.nist.gov/22.11
See also:: Annotations for Ch.22

Throughout this section and $\zeta$ are defined as in §22.2.

If $q\exp\left(2|\Im\zeta|\right)<1$ , then

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

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Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, : nome, $\sin\NVar{z}$ : sine function, : complex, : modulus and $\zeta$ : change of variable
A&S Ref:: 16.23.1
Permalink:: http://dlmf.nist.gov/22.11.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §22.11 and Ch.22

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

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Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\cos\NVar{z}$ : cosine function, : nome, : complex, : modulus and $\zeta$ : change of variable
A&S Ref:: 16.23.2
Permalink:: http://dlmf.nist.gov/22.11.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §22.11 and Ch.22

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

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Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\cos\NVar{z}$ : cosine function, : nome, : complex, : modulus and $\zeta$ : change of variable
A&S Ref:: 16.23.3
Permalink:: http://dlmf.nist.gov/22.11.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §22.11 and Ch.22

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

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Symbols:: $\operatorname{cd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\cos\NVar{z}$ : cosine function, : nome, : complex, : modulus and $\zeta$ : change of variable
A&S Ref:: 16.23.4
Permalink:: http://dlmf.nist.gov/22.11.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §22.11 and Ch.22

22.11.5 $\operatorname{sd}\left(z,k\right)=\frac{2\pi}{Kkk^{\prime}}\sum_{n=0}^{\infty}% \frac{(-1)^{n}q^{n+\frac{1}{2}}\sin\left((2n+1)\zeta\right)}{1+q^{2n+1}},$

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Symbols:: $\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, : nome, $\sin\NVar{z}$ : sine function, : complex, : 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:: TeX, pMML, png
See also:: Annotations for §22.11 and Ch.22

22.11.6 $\operatorname{nd}\left(z,k\right)=\frac{\pi}{2Kk^{\prime}}+\frac{2\pi}{Kk^{% \prime}}\sum_{n=1}^{\infty}\frac{(-1)^{n}q^{n}\cos\left(2n\zeta\right)}{1+q^{2% n}}.$

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Symbols:: $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\cos\NVar{z}$ : cosine function, : nome, : complex, : modulus, $k^{\prime}$ : complementary modulus and $\zeta$ : change of variable
A&S Ref:: 16.23.6
Permalink:: http://dlmf.nist.gov/22.11.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §22.11 and Ch.22

Next, if $q\exp\left(|\Im\zeta|\right)<1$ , then

22.11.7 $\operatorname{ns}\left(z,k\right)-\frac{\pi}{2K}\csc\zeta=\frac{2\pi}{K}\sum_{% n=0}^{\infty}\frac{q^{2n+1}\sin\left((2n+1)\zeta\right)}{1-q^{2n+1}},$

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Symbols:: $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\csc\NVar{z}$ : cosecant function, : nome, $\sin\NVar{z}$ : sine function, : complex, : modulus and $\zeta$ : change of variable
A&S Ref:: 16.23.10
Referenced by:: §22.11
Permalink:: http://dlmf.nist.gov/22.11.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §22.11 and Ch.22

22.11.8 $\operatorname{ds}\left(z,k\right)-\frac{\pi}{2K}\csc\zeta=-\frac{2\pi}{K}\sum_% {n=0}^{\infty}\frac{q^{2n+1}\sin\left((2n+1)\zeta\right)}{1+q^{2n+1}},$

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Symbols:: $\operatorname{ds}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\csc\NVar{z}$ : cosecant function, : nome, $\sin\NVar{z}$ : sine function, : complex, : modulus and $\zeta$ : change of variable
A&S Ref:: 16.23.11
Permalink:: http://dlmf.nist.gov/22.11.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §22.11 and Ch.22

22.11.9 $\operatorname{cs}\left(z,k\right)-\frac{\pi}{2K}\cot\zeta=-\frac{2\pi}{K}\sum_% {n=1}^{\infty}\frac{q^{2n}\sin\left(2n\zeta\right)}{1+q^{2n}},$

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Symbols:: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\cot\NVar{z}$ : cotangent function, : nome, $\sin\NVar{z}$ : sine function, : complex, : modulus and $\zeta$ : change of variable
A&S Ref:: 16.23.12
Permalink:: http://dlmf.nist.gov/22.11.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §22.11 and Ch.22

22.11.10 $\operatorname{dc}\left(z,k\right)-\frac{\pi}{2K}\sec\zeta=\frac{2\pi}{K}\sum_{% n=0}^{\infty}\frac{(-1)^{n}q^{2n+1}\cos\left((2n+1)\zeta\right)}{1-q^{2n+1}},$

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Symbols:: $\operatorname{dc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\cos\NVar{z}$ : cosine function, : nome, $\sec\NVar{z}$ : secant function, : complex, : modulus and $\zeta$ : change of variable
A&S Ref:: 16.23.7
Permalink:: http://dlmf.nist.gov/22.11.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §22.11 and Ch.22

22.11.11 $\operatorname{nc}\left(z,k\right)-\frac{\pi}{2Kk^{\prime}}\sec\zeta=-\frac{2% \pi}{Kk^{\prime}}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{2n+1}\cos\left((2n+1)% \zeta\right)}{1+q^{2n+1}},$

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Symbols:: $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\cos\NVar{z}$ : cosine function, : nome, $\sec\NVar{z}$ : secant function, : complex, : modulus, $k^{\prime}$ : complementary modulus and $\zeta$ : change of variable
A&S Ref:: 16.23.8
Permalink:: http://dlmf.nist.gov/22.11.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §22.11 and Ch.22

22.11.12 $\operatorname{sc}\left(z,k\right)-\frac{\pi}{2Kk^{\prime}}\tan\zeta=\frac{2\pi% }{Kk^{\prime}}\sum_{n=1}^{\infty}\frac{(-1)^{n}q^{2n}\sin\left(2n\zeta\right)}% {1+q^{2n}}.$

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Symbols:: $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, : nome, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function, : complex, : modulus, $k^{\prime}$ : complementary modulus and $\zeta$ : change of variable
A&S Ref:: 16.23.9
Referenced by:: §22.11
Permalink:: http://dlmf.nist.gov/22.11.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §22.11 and Ch.22

In (22.11.7)–(22.11.12) the left-hand sides are replaced by their limiting values at the poles of the Jacobian functions.

Next, with $E=E\left(k\right)$ denoting the complete elliptic integral of the second kind (§19.2(ii)) and $q\exp\left(2|\Im\zeta|\right)<1$ ,

22.11.13 ${\operatorname{sn}}^{2}\left(z,k\right)=\frac{1}{k^{2}}\left(1-\frac{E}{K}% \right)-\frac{2\pi^{2}}{k^{2}K^{2}}\sum_{n=1}^{\infty}\frac{nq^{n}}{1-q^{2n}}% \cos\left(2n\zeta\right).$

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Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $\cos\NVar{z}$ : cosine function, : nome, : complex, : modulus and $\zeta$ : change of variable
Referenced by:: §22.11
Permalink:: http://dlmf.nist.gov/22.11.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §22.11 and Ch.22

Similar expansions for ${\operatorname{cn}}^{2}\left(z,k\right)$ and ${\operatorname{dn}}^{2}\left(z,k\right)$ follow immediately from (22.6.1).

For further Fourier series see Oberhettinger (1973, pp. 23–27).

A related hyperbolic series is

22.11.14 $k^{2}{\operatorname{sn}}^{2}\left(z,k\right)=\frac{{E^{\prime}}}{{K^{\prime}}}% -\left(\frac{\pi}{2{K^{\prime}}}\right)^{2}\sum_{n=-\infty}^{\infty}\left({% \operatorname{sech}}^{2}\left(\frac{\pi}{2{K^{\prime}}}(z-2nK)\right)\right),$

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Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\pi$ : the ratio of the circumference of a circle to its diameter, ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, ${E^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the second kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, : complex and : modulus
Permalink:: http://dlmf.nist.gov/22.11.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §22.11 and Ch.22

where ${E^{\prime}}={E^{\prime}}\left(k\right)$ is defined by §19.2.9. Again, similar expansions for ${\operatorname{cn}}^{2}\left(z,k\right)$ and ${\operatorname{dn}}^{2}\left(z,k\right)$ may be derived via (22.6.1). See Dunne and Rao (2000).

22.10 Maclaurin Series 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

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