22 Jacobian Elliptic Functions Properties 22.2 Definitions 22.4 Periods, Poles, and Zeros

§22.3 Graphics

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Permalink:: http://dlmf.nist.gov/22.3
See also:: Annotations for Ch.22

§22.3(i) Real Variables: Line Graphs
§22.3(ii) Real Variables: Surfaces
§22.3(iii) Complex $z$ ; Real $k$
§22.3(iv) Complex $k$

§22.3(i) Real Variables: Line Graphs

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Keywords:: Jacobian elliptic functions, graphics, real variable
Notes:: Figures 22.3.1–22.3.12 were produced at NIST and by the authors.
Referenced by:: Figure 22.16.2, Figure 22.16.2, §23.4(i)
Permalink:: http://dlmf.nist.gov/22.3.i
See also:: Annotations for §22.3 and Ch.22

Line graphs of the functions $\operatorname{sn}\left(x,k\right)$ , $\operatorname{cn}\left(x,k\right)$ , $\operatorname{dn}\left(x,k\right)$ , $\operatorname{cd}\left(x,k\right)$ , $\operatorname{sd}\left(x,k\right)$ , $\operatorname{nd}\left(x,k\right)$ , $\operatorname{dc}\left(x,k\right)$ , $\operatorname{nc}\left(x,k\right)$ , $\operatorname{sc}\left(x,k\right)$ , $\operatorname{ns}\left(x,k\right)$ , $\operatorname{ds}\left(x,k\right)$ , and $\operatorname{cs}\left(x,k\right)$ for representative values of real $x$ and real $k$ illustrating the near trigonometric ( $k=0$ ), and near hyperbolic ( $k=1$ ) limits.

See accompanying text — Figure 22.3.1: $k=0.4$ , $-3K\leq x\leq 3K$ , $K=1.6399\dots$ . Magnify

§22.3(ii) Real Variables: Surfaces

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Keywords:: Jacobian elliptic functions, graphics, real variable
Notes:: These graphics were produced at NIST.
Permalink:: http://dlmf.nist.gov/22.3.ii
See also:: Annotations for §22.3 and Ch.22

$\operatorname{sn}\left(x,k\right)$ , $\operatorname{cn}\left(x,k\right)$ , and $\operatorname{dn}\left(x,k\right)$ as functions of real arguments $x$ and $k$ . The period diverges logarithmically as $k\to 1-$ ; see §19.12.

§22.3(iii) Complex $z$ ; Real $k$

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Keywords:: Jacobian elliptic functions, complex variable, graphics
Notes:: These graphics were produced at NIST and by the authors.
Referenced by:: §23.4(ii)
Permalink:: http://dlmf.nist.gov/22.3.iii
See also:: Annotations for §22.3 and Ch.22

In the graphics shown in this subsection height corresponds to the absolute value of the function and color to the phase. See About Color Map.

§22.3(iv) Complex $k$

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Keywords:: Jacobian elliptic functions, complex modulus, graphics
Notes:: Figures 22.3.22–22.3.23 and the graphics 22.3.24–22.3.25 were produced at NIST and by the authors. Figures 22.3.26–22.3.29 were produced by the authors.
Referenced by:: §22.1
Permalink:: http://dlmf.nist.gov/22.3.iv
See also:: Annotations for §22.3 and Ch.22