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

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).
Keywords:: Fourier series, Jacobian elliptic functions
Permalink:: http://dlmf.nist.gov/22.11

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

If $q\mathop{\exp\/}\nolimits\!\left(2|\imagpart{\zeta}|\right)<1$ , then

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

Symbols:: $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : nome, $\mathop{\sin\/}\nolimits z$ : 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

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

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\cos\/}\nolimits z$ : 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

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

Symbols:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\cos\/}\nolimits z$ : 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

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

Symbols:: $\mathop{\mathrm{cd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\cos\/}\nolimits z$ : 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

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

Symbols:: $\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : nome, $\mathop{\sin\/}\nolimits z$ : 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

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

Symbols:: $\mathop{\mathrm{nd}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\cos\/}\nolimits z$ : 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

Next, if $q\mathop{\exp\/}\nolimits\!\left(|\imagpart{\zeta}|\right)<1$ , then

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

Symbols:: $\mathop{\mathrm{ns}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\csc\/}\nolimits z$ : cosecant function, : nome, $\mathop{\sin\/}\nolimits z$ : 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

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

Symbols:: $\mathop{\mathrm{ds}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\csc\/}\nolimits z$ : cosecant function, : nome, $\mathop{\sin\/}\nolimits z$ : 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

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

Symbols:: $\mathop{\mathrm{cs}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\cot\/}\nolimits z$ : cotangent function, : nome, $\mathop{\sin\/}\nolimits z$ : 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

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

Symbols:: $\mathop{\mathrm{dc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, : nome, $\mathop{\sec\/}\nolimits z$ : 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

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

Symbols:: $\mathop{\mathrm{nc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, : nome, $\mathop{\sec\/}\nolimits z$ : 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

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

Symbols:: $\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : nome, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\tan\/}\nolimits z$ : 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

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 $\mathop{E\/}\nolimits=\mathop{E\/}\nolimits\!\left(k\right)$ denoting the complete elliptic integral of the second kind (§19.2(ii)) and $q\mathop{\exp\/}\nolimits\!\left(2|\imagpart{\zeta}|\right)<1$ ,

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

Symbols:: $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\cos\/}\nolimits z$ : 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

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

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

A related hyperbolic series is

22.11.14 $k^{2}{\mathop{\mathrm{sn}\/}\nolimits^{{2}}}\left(z,k\right)=\frac{\mathop{{E^% {{\prime}}}\/}\nolimits}{\mathop{{K^{{\prime}}}\/}\nolimits}-\left(\frac{\pi}{% 2\!\mathop{{K^{{\prime}}}\/}\nolimits}\right)^{2}\sum_{{n=-\infty}}^{{\infty}}% \left({\mathop{\mathrm{sech}\/}\nolimits^{{2}}}\!\left(\frac{\pi}{2\!\mathop{{% K^{{\prime}}}\/}\nolimits}(z-2n\!\mathop{K\/}\nolimits)\right)\right),$

Symbols:: $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathop{{E^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the second kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\mathrm{sech}\/}\nolimits z$ : hyperbolic secant function, : complex and : modulus
Permalink:: http://dlmf.nist.gov/22.11.E14
Encodings:: TeX, pMML, png

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

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 22.10 Maclaurin Series 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series