§22.11 Fourier and Hyperbolic Series

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

If , then

22.11.1		$\mathrm{sn}(z,k)$	$={\displaystyle \frac{2\pi }{Kk}}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{q^{n+\frac{1}{2}}\sin \left((2n+1)\zeta \right)}{1-q^{2n+1}}},$
ⓘ 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: TeX, pMML, png See also: Annotations for §22.11 and Ch.22
22.11.2		$\mathrm{cn}(z,k)$	$={\displaystyle \frac{2\pi }{Kk}}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{q^{n+\frac{1}{2}}\cos \left((2n+1)\zeta \right)}{1+q^{2n+1}}},$
ⓘ 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: TeX, pMML, png See also: Annotations for §22.11 and Ch.22
22.11.3		$\mathrm{dn}(z,k)$	$={\displaystyle \frac{\pi }{2K}}+{\displaystyle \frac{2\pi }{K}}{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{q^n\cos \left(2n\zeta \right)}{1+q^{2n}}}.$
ⓘ 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: TeX, pMML, png See also: Annotations for §22.11 and Ch.22

22.11.4		$\mathrm{cd}(z,k)$	$={\displaystyle \frac{2\pi }{Kk}}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^n{q}^{n+\frac{1}{2}}\cos \left((2n+1)\zeta \right)}{1-q^{2n+1}}},$
ⓘ 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: TeX, pMML, png See also: Annotations for §22.11 and Ch.22
22.11.5		$\mathrm{sd}(z,k)$	$={\displaystyle \frac{2\pi }{Kk{k}^{\prime }}}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^n{q}^{n+\frac{1}{2}}\sin \left((2n+1)\zeta \right)}{1+q^{2n+1}}},$
ⓘ 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: TeX, pMML, png See also: Annotations for §22.11 and Ch.22
22.11.6		$\mathrm{nd}(z,k)$	$={\displaystyle \frac{\pi }{2K{k}^{\prime }}}+{\displaystyle \frac{2\pi }{K{k}^{\prime }}}{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^n{q}^n\cos \left(2n\zeta \right)}{1+q^{2n}}}.$
ⓘ Symbols: $\mathrm{nd}(z,k)$: 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, $k^{\prime }$: 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 , then

22.11.7		$\mathrm{ns}(z,k)-{\displaystyle \frac{\pi }{2K}}\csc \zeta$	$={\displaystyle \frac{2\pi }{K}}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{q^{2n+1}\sin \left((2n+1)\zeta \right)}{1-q^{2n+1}}},$
ⓘ Symbols: $\mathrm{ns}(z,k)$: 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, $\csc z$: cosecant function, $q$: nome, $\sin z$: sine function, $z$: complex, $k$: 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		$\mathrm{ds}(z,k)-{\displaystyle \frac{\pi }{2K}}\csc \zeta$	$=-{\displaystyle \frac{2\pi }{K}}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{q^{2n+1}\sin \left((2n+1)\zeta \right)}{1+q^{2n+1}}},$
ⓘ Symbols: $\mathrm{ds}(z,k)$: 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, $\csc z$: cosecant function, $q$: nome, $\sin z$: sine function, $z$: complex, $k$: 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		$\mathrm{cs}(z,k)-{\displaystyle \frac{\pi }{2K}}\cot \zeta$	$=-{\displaystyle \frac{2\pi }{K}}{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{q^{2n}\sin \left(2n\zeta \right)}{1+q^{2n}}},$
ⓘ Symbols: $\mathrm{cs}(z,k)$: 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, $\cot z$: cotangent function, $q$: nome, $\sin z$: sine function, $z$: complex, $k$: 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		$$\mathrm{dc}(z,k)-\frac{\pi }{2K}\sec \zeta =\frac{2\pi }{K}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{2n+1}\cos \left((2n+1)\zeta \right)}{1-q^{2n+1}},$$
ⓘ Symbols: $\mathrm{dc}(z,k)$: 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, $\sec z$: secant function, $z$: complex, $k$: 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		$$\mathrm{nc}(z,k)-\frac{\pi }{2K{k}^{\prime }}\sec \zeta =-\frac{2\pi }{K{k}^{\prime }}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{2n+1}\cos \left((2n+1)\zeta \right)}{1+q^{2n+1}},$$
ⓘ Symbols: $\mathrm{nc}(z,k)$: 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, $\sec z$: secant function, $z$: complex, $k$: 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		$$\mathrm{sc}(z,k)-\frac{\pi }{2K{k}^{\prime }}\tan \zeta =\frac{2\pi }{K{k}^{\prime }}\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{2n}\sin \left(2n\zeta \right)}{1+q^{2n}}.$$
ⓘ Symbols: $\mathrm{sc}(z,k)$: 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, $\tan z$: tangent function, $z$: complex, $k$: 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 ,

22.11.13		$${\mathrm{sn}}^2(z,k)=\frac{1}{k^2}\left(1-\frac{E}{K}\right)-\frac{2{\pi }^2}{k^2{K}^2}\sum _{n=1}^{\mathrm{\infty }}\frac{n{q}^n}{1-q^{2n}}\cos \left(2n\zeta \right).$$
ⓘ 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, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\cos z$: cosine function, $q$: nome, $z$: complex, $k$: 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 ${\mathrm{cn}}^2(z,k)$ and ${\mathrm{dn}}^2(z,k)$ follow immediately from (22.6.1).

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

A related hyperbolic series is

22.11.14		$$k^2{\mathrm{sn}}^2(z,k)=\frac{E^{\prime }}{K^{\prime }}-{\left(\frac{\pi }{2K^{\prime }}\right)}^2\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left({\mathrm{sech}}^2\left(\frac{\pi }{2K^{\prime }}(z-2nK)\right)\right),$$
ⓘ Symbols: $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $E^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the second kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{sech}z$: hyperbolic secant function, $z$: complex and $k$: 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 ${\mathrm{cn}}^2(z,k)$ and ${\mathrm{dn}}^2(z,k)$ may be derived via (22.6.1). See Dunne and Rao (2000).