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

§22.10 Maclaurin Series

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

§22.10(i) Maclaurin Series in

Notes:: See Lawden (1989, pp. 36–37).
Keywords:: Jacobian elliptic functions
Referenced by:: §22.17(i)
Permalink:: http://dlmf.nist.gov/22.10.i

Initial terms are given by

22.10.1 $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)=z-\left(1+k^{2}\right)\frac{z^% {3}}{3!}+\left(1+14k^{2}+k^{4}\right)\frac{z^{5}}{5!}-\left(1+135k^{2}+135k^{4% }+k^{6}\right)\frac{z^{7}}{7!}+\mathop{O\/}\nolimits\!\left(z^{9}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, : : factorial, : complex and : modulus
A&S Ref:: 16.22.1
Permalink:: http://dlmf.nist.gov/22.10.E1
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22.10.2 $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)=1-\frac{z^{2}}{2!}+\left(1+4k^% {2}\right)\frac{z^{4}}{4!}-\left(1+44k^{2}+16k^{4}\right)\frac{z^{6}}{6!}+% \mathop{O\/}\nolimits\!\left(z^{8}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, : : factorial, : complex and : modulus
A&S Ref:: 16.22.2
Permalink:: http://dlmf.nist.gov/22.10.E2
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22.10.3 $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)=1-k^{2}\frac{z^{2}}{2!}+k^{2}% \left(4+k^{2}\right)\frac{z^{4}}{4!}-k^{2}\left(16+44k^{2}+k^{4}\right)\frac{z% ^{6}}{6!}+\mathop{O\/}\nolimits\!\left(z^{8}\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, : : factorial, : complex and : modulus
A&S Ref:: 16.22.3
Permalink:: http://dlmf.nist.gov/22.10.E3
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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 $|z|<\min\left(\mathop{K\/}\nolimits\!\left(k\right),\mathop{{K^{{\prime}}}\/}% \nolimits\!\left(k\right)\right)$ .

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

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

Initial terms are given by

22.10.4 $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)=\mathop{\sin\/}\nolimits z-% \frac{k^{2}}{4}(z-\mathop{\sin\/}\nolimits z\mathop{\cos\/}\nolimits z)\mathop% {\cos\/}\nolimits z+\mathop{O\/}\nolimits\!\left(k^{4}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, : complex and : modulus
A&S Ref:: 16.13.1
Referenced by:: §22.10(ii), ¶ ‣ §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.10.E4
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22.10.5 $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)=\mathop{\cos\/}\nolimits z+% \frac{k^{2}}{4}(z-\mathop{\sin\/}\nolimits z\mathop{\cos\/}\nolimits z)\mathop% {\sin\/}\nolimits z+\mathop{O\/}\nolimits\!\left(k^{4}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, : complex and : modulus
A&S Ref:: 16.13.2
Referenced by:: ¶ ‣ §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.10.E5
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22.10.6 $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)=1-\frac{k^{2}}{2}{\mathop{\sin% \/}\nolimits^{{2}}}z+\mathop{O\/}\nolimits\!\left(k^{4}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\sin\/}\nolimits z$ : sine function, : complex and : modulus
A&S Ref:: 16.13.3
Referenced by:: §22.10(ii), ¶ ‣ §22.20(iii)
Permalink:: http://dlmf.nist.gov/22.10.E6
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22.10.7 $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)=\mathop{\tanh\/}\nolimits z-% \frac{{k^{{\prime}}}^{2}}{4}(z-\mathop{\sinh\/}\nolimits z\mathop{\cosh\/}% \nolimits z){\mathop{\mathrm{sech}\/}\nolimits^{{2}}}z+\mathop{O\/}\nolimits\!% \left({k^{{\prime}}}^{4}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\mathrm{sech}\/}\nolimits z$ : hyperbolic secant function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function, : complex, : 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
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22.10.8 $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)=\mathop{\mathrm{sech}\/}% \nolimits z+\frac{{k^{{\prime}}}^{2}}{4}(z-\mathop{\sinh\/}\nolimits z\mathop{% \cosh\/}\nolimits z)\mathop{\tanh\/}\nolimits z\mathop{\mathrm{sech}\/}% \nolimits z+\mathop{O\/}\nolimits\!\left({k^{{\prime}}}^{4}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\mathrm{sech}\/}\nolimits z$ : hyperbolic secant function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function, : complex, : modulus and $k^{{\prime}}$ : complementary modulus
A&S Ref:: 16.15.2
Permalink:: http://dlmf.nist.gov/22.10.E8
Encodings:: TeX, pMML, png

22.10.9 $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)=\mathop{\mathrm{sech}\/}% \nolimits z+\frac{{k^{{\prime}}}^{2}}{4}(z+\mathop{\sinh\/}\nolimits z\mathop{% \cosh\/}\nolimits z)\mathop{\tanh\/}\nolimits z\mathop{\mathrm{sech}\/}% \nolimits z+\mathop{O\/}\nolimits\!\left({k^{{\prime}}}^{4}\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\mathrm{sech}\/}\nolimits z$ : hyperbolic secant function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function, : complex, : 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
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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 -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.

© 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. 22.9 Cyclic Identities 22.11 Fourier and Hyperbolic Series

§22.10 Maclaurin Series

Contents

§22.10(i) Maclaurin Series in

§22.10(ii) Maclaurin Series in and

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