22 Jacobian Elliptic Functions Properties 22.9 Cyclic Identities 22.11 Fourier and Hyperbolic Series

§22.10 Maclaurin Series

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Permalink:: http://dlmf.nist.gov/22.10
See also:: Annotations for Ch.22

§22.10(i) Maclaurin Series in $z$

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Keywords:: Jacobian elliptic functions, Maclaurin series, in $z$
Notes:: See Lawden (1989, pp. 36–37).
Referenced by:: §22.17(i)
Permalink:: http://dlmf.nist.gov/22.10.i
See also:: Annotations for §22.10 and Ch.22

Initial terms are given by

22.10.1		$\operatorname{sn}\left(z,k\right)=z-\left(1+k^{2}\right)\frac{z^{3}}{3!}+\left% (1+14k^{2}+k^{4}\right)\frac{z^{5}}{5!}-\left(1+135k^{2}+135k^{4}+k^{6}\right)% \frac{z^{7}}{7!}+O\left(z^{9}\right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $!$ : factorial (as in $n!$ ), $z$ : complex and $k$ : modulus A&S Ref: 16.22.1 Permalink: http://dlmf.nist.gov/22.10.E1 Encodings: TeX, pMML, png See also: Annotations for §22.10(i), §22.10 and Ch.22

22.10.2		$\operatorname{cn}\left(z,k\right)=1-\frac{z^{2}}{2!}+\left(1+4k^{2}\right)% \frac{z^{4}}{4!}-\left(1+44k^{2}+16k^{4}\right)\frac{z^{6}}{6!}+O\left(z^{8}% \right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $!$ : factorial (as in $n!$ ), $z$ : complex and $k$ : modulus A&S Ref: 16.22.2 Permalink: http://dlmf.nist.gov/22.10.E2 Encodings: TeX, pMML, png See also: Annotations for §22.10(i), §22.10 and Ch.22

22.10.3		$\operatorname{dn}\left(z,k\right)=1-k^{2}\frac{z^{2}}{2!}+k^{2}\left(4+k^{2}% \right)\frac{z^{4}}{4!}-k^{2}\left(16+44k^{2}+k^{4}\right)\frac{z^{6}}{6!}+O% \left(z^{8}\right).$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $!$ : factorial (as in $n!$ ), $z$ : complex and $k$ : modulus A&S Ref: 16.22.3 Permalink: http://dlmf.nist.gov/22.10.E3 Encodings: TeX, pMML, png See also: Annotations for §22.10(i), §22.10 and Ch.22

Further terms may be derived by substituting in the differential equations (22.13.13), (22.13.14), (22.13.15). The full expansions converge when $|z|<\min\left(K\left(k\right),{K^{\prime}}\left(k\right)\right)$ .

§22.10(ii) Maclaurin Series in $k$ and $k^{\prime}$

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Keywords:: Jacobian elliptic functions, Maclaurin series, in $k,k^{\prime}$
Notes:: See Lawden (1989, pp. 50–52). The expansions in powers of $k^{\prime}$ follow from those in powers of $k$ by means of Table 22.6.1 and (4.28.8)–(4.28.10), and division of series (§1.9(vi)).
Referenced by:: §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.10.ii
See also:: Annotations for §22.10 and Ch.22

Initial terms are given by

22.10.4		$\operatorname{sn}\left(z,k\right)=\sin z-\frac{k^{2}}{4}(z-\sin z\cos z)\cos z% +O\left(k^{4}\right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $z$ : complex and $k$ : modulus A&S Ref: 16.13.1 Referenced by: §22.10(ii), §22.20(iii) Permalink: http://dlmf.nist.gov/22.10.E4 Encodings: TeX, pMML, png See also: Annotations for §22.10(ii), §22.10 and Ch.22

22.10.5		$\operatorname{cn}\left(z,k\right)=\cos z+\frac{k^{2}}{4}(z-\sin z\cos z)\sin z% +O\left(k^{4}\right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $z$ : complex and $k$ : modulus A&S Ref: 16.13.2 Referenced by: §22.20(iii) Permalink: http://dlmf.nist.gov/22.10.E5 Encodings: TeX, pMML, png See also: Annotations for §22.10(ii), §22.10 and Ch.22

22.10.6		$\operatorname{dn}\left(z,k\right)=1-\frac{k^{2}}{2}{\sin}^{2}z+O\left(k^{4}% \right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\sin\NVar{z}$ : sine function, $z$ : complex and $k$ : modulus A&S Ref: 16.13.3 Referenced by: §22.10(ii), §22.20(iii) Permalink: http://dlmf.nist.gov/22.10.E6 Encodings: TeX, pMML, png See also: Annotations for §22.10(ii), §22.10 and Ch.22

22.10.7		$\operatorname{sn}\left(z,k\right)=\tanh z-\frac{{k^{\prime}}^{2}}{4}(z-\sinh z% \cosh z){\operatorname{sech}}^{2}z+O\left({k^{\prime}}^{4}\right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\sinh\NVar{z}$ : hyperbolic sine function, $\tanh\NVar{z}$ : hyperbolic tangent function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 16.15.1 Referenced by: §22.10(ii) Permalink: http://dlmf.nist.gov/22.10.E7 Encodings: TeX, pMML, png See also: Annotations for §22.10(ii), §22.10 and Ch.22

22.10.8		$\operatorname{cn}\left(z,k\right)=\operatorname{sech}z+\frac{{k^{\prime}}^{2}}% {4}(z-\sinh z\cosh z)\tanh z\operatorname{sech}z+O\left({k^{\prime}}^{4}\right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\sinh\NVar{z}$ : hyperbolic sine function, $\tanh\NVar{z}$ : hyperbolic tangent function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 16.15.2 Permalink: http://dlmf.nist.gov/22.10.E8 Encodings: TeX, pMML, png See also: Annotations for §22.10(ii), §22.10 and Ch.22

22.10.9		$\operatorname{dn}\left(z,k\right)=\operatorname{sech}z+\frac{{k^{\prime}}^{2}}% {4}(z+\sinh z\cosh z)\tanh z\operatorname{sech}z+O\left({k^{\prime}}^{4}\right).$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\sinh\NVar{z}$ : hyperbolic sine function, $\tanh\NVar{z}$ : hyperbolic tangent function, $z$ : complex, $k$ : modulus and $k^{\prime}$ : complementary modulus A&S Ref: 16.15.3 Referenced by: §22.10(ii) Permalink: http://dlmf.nist.gov/22.10.E9 Encodings: TeX, pMML, png See also: Annotations for §22.10(ii), §22.10 and Ch.22

Further terms may be derived from the differential equations (22.13.13), (22.13.14), (22.13.15), or from the integral representations of the inverse functions in §22.15(ii). The radius of convergence is the distance to the origin from the nearest pole in the complex $k$ -plane in the case of (22.10.4)–(22.10.6), or complex $k^{\prime}$ -plane in the case of (22.10.7)–(22.10.9); see §22.17.

22.9 Cyclic Identities 22.11 Fourier and Hyperbolic Series

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§22.10 Maclaurin Series

Contents

§22.10(i) Maclaurin Series in z

§22.10(ii) Maclaurin Series in k and k′

§22.10(i) Maclaurin Series in $z$

§22.10(ii) Maclaurin Series in $k$ and $k^{\prime}$