# §22.3 Graphics

## §22.3(i) Real Variables: Line Graphs

Line graphs of the functions , , , , , , , , , , , and for representative values of real and real illustrating the near trigonometric (), and near hyperbolic () limits.

 Figure 22.3.1: , , . Symbols: : Legendre’s complete elliptic integral of the first kind, : real and : modulus Referenced by: §22.3(i) Permalink: http://dlmf.nist.gov/22.3.F1 Encodings: pdf, png Figure 22.3.2: , , . For the curve for is a boundary between the curves that have an inflection point in the interval , and its translates, and those that do not; see Walker (1996, p. 146). Symbols: : Jacobian elliptic function, : Legendre’s complete elliptic integral of the first kind, : real and : modulus Permalink: http://dlmf.nist.gov/22.3.F2 Encodings: pdf, png
 Figure 22.3.3: , , . Symbols: : Legendre’s complete elliptic integral of the first kind, : real and : modulus Permalink: http://dlmf.nist.gov/22.3.F3 Encodings: pdf, png Figure 22.3.4: , , . Symbols: : Legendre’s complete elliptic integral of the first kind, : real and : modulus Permalink: http://dlmf.nist.gov/22.3.F4 Encodings: pdf, png
 Figure 22.3.5: , , . Symbols: : Legendre’s complete elliptic integral of the first kind, : real and : modulus Permalink: http://dlmf.nist.gov/22.3.F5 Encodings: pdf, png Figure 22.3.6: , , . Symbols: : Legendre’s complete elliptic integral of the first kind, : real and : modulus Permalink: http://dlmf.nist.gov/22.3.F6 Encodings: pdf, png
 Figure 22.3.7: , , . Symbols: : Legendre’s complete elliptic integral of the first kind, : real and : modulus Permalink: http://dlmf.nist.gov/22.3.F7 Encodings: pdf, png Figure 22.3.8: , , . Symbols: : Legendre’s complete elliptic integral of the first kind, : real and : modulus Permalink: http://dlmf.nist.gov/22.3.F8 Encodings: pdf, png
 Figure 22.3.9: , , . Symbols: : Legendre’s complete elliptic integral of the first kind, : real and : modulus Permalink: http://dlmf.nist.gov/22.3.F9 Encodings: pdf, png Figure 22.3.10: , , . Symbols: : Legendre’s complete elliptic integral of the first kind, : real and : modulus Permalink: http://dlmf.nist.gov/22.3.F10 Encodings: pdf, png
 Figure 22.3.11: , , . Symbols: : Legendre’s complete elliptic integral of the first kind, : real and : modulus Permalink: http://dlmf.nist.gov/22.3.F11 Encodings: pdf, png Figure 22.3.12: , , . Symbols: : Legendre’s complete elliptic integral of the first kind, : real and : modulus Referenced by: §22.3(i) Permalink: http://dlmf.nist.gov/22.3.F12 Encodings: pdf, png

## §22.3(ii) Real Variables: Surfaces

, , and as functions of real arguments and . The period diverges logarithmically as ; see §19.12.

Figure 22.3.13: for , to 20, .
 Figure 22.3.14: for , to 20, . Symbols: : Jacobian elliptic function, : base of exponential function, : real, : modulus and : integer Permalink: http://dlmf.nist.gov/22.3.F14 Encodings: VRML, X3D, pdf, png Figure 22.3.15: for , to 20, . Symbols: : Jacobian elliptic function, : base of exponential function, : real, : modulus and : integer Permalink: http://dlmf.nist.gov/22.3.F15 Encodings: VRML, X3D, pdf, png

## §22.3(iii) Complex ; Real

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.

 Figure 22.3.18: for , , . , . Symbols: : Jacobian elliptic function, : Legendre’s complementary complete elliptic integral of the first kind, : Legendre’s complete elliptic integral of the first kind, : real, : real and : modulus Permalink: http://dlmf.nist.gov/22.3.F18 Encodings: VRML, X3D, pdf, png Figure 22.3.19: for , , . , . Symbols: : Jacobian elliptic function, : Legendre’s complementary complete elliptic integral of the first kind, : Legendre’s complete elliptic integral of the first kind, : real, : real and : modulus Permalink: http://dlmf.nist.gov/22.3.F19 Encodings: VRML, X3D, pdf, png
 Figure 22.3.20: for , , . , . Symbols: : Jacobian elliptic function, : Legendre’s complementary complete elliptic integral of the first kind, : Legendre’s complete elliptic integral of the first kind, : real, : real and : modulus Permalink: http://dlmf.nist.gov/22.3.F20 Encodings: VRML, X3D, pdf, png Figure 22.3.21: for , , . , . Symbols: : Jacobian elliptic function, : Legendre’s complementary complete elliptic integral of the first kind, : Legendre’s complete elliptic integral of the first kind, : real, : real and : modulus Permalink: http://dlmf.nist.gov/22.3.F21 Encodings: VRML, X3D, pdf, png

## §22.3(iv) Complex

Figure 22.3.22: , , as a function of , .
Figure 22.3.23: , , as a function of , .

In Figures 22.3.24 and 22.3.25, height corresponds to the absolute value of the function and color to the phase. See p. About Color Map.

 Figure 22.3.24: for , , . , . Symbols: : Jacobian elliptic function, : Legendre’s complementary complete elliptic integral of the first kind, : Legendre’s complete elliptic integral of the first kind, : real, : real and : modulus Keywords: Jacobian elliptic functions Referenced by: §22.3(iv), §22.3(iv), § ‣ DLMF Update; Version 1.0.3 Permalink: http://dlmf.nist.gov/22.3.F24 Encodings: VRML, X3D, pdf, png Figure 22.3.25: as a function of complex , , . Compare §22.17(ii). Symbols: : Jacobian elliptic function, : imaginary part, : real part and : modulus Keywords: Jacobian elliptic functions Referenced by: §22.17(ii), §22.3(iv), §22.3(iv) Permalink: http://dlmf.nist.gov/22.3.F25 Encodings: VRML, X3D, pdf, png
 Figure 22.3.26: Density plot of as a function of complex , , . Grayscale, running from 0 (black) to 10 (white), with truncated to 10. White spots correspond to poles. Symbols: : Jacobian elliptic function, : imaginary part, : real part and : modulus Keywords: Jacobian elliptic functions Referenced by: §22.3(iv) Permalink: http://dlmf.nist.gov/22.3.F26 Encodings: pdf, png Figure 22.3.27: Density plot of as a function of complex , , . Grayscale, running from 0 (black) to 10 (white), with truncated to 10. White spots correspond to poles. Symbols: : Jacobian elliptic function, : imaginary part, : real part and : modulus Keywords: Jacobian elliptic functions Permalink: http://dlmf.nist.gov/22.3.F27 Encodings: pdf, png
 Figure 22.3.28: Density plot of as a function of complex , , . Grayscale, running from 0 (black) to 10 (white), with truncated to 10. White spots correspond to poles. Symbols: : Jacobian elliptic function, : imaginary part, : real part and : modulus Keywords: Jacobian elliptic functions Permalink: http://dlmf.nist.gov/22.3.F28 Encodings: pdf, png Figure 22.3.29: Density plot of as a function of complex , , . Grayscale, running from 0 (black) to 10 (white), with truncated to 10. White spots correspond to poles. Symbols: : Jacobian elliptic function, : imaginary part, : real part and : modulus Keywords: Jacobian elliptic functions Referenced by: §22.17(ii), §22.3(iv) Permalink: http://dlmf.nist.gov/22.3.F29 Encodings: pdf, png
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