File:Lemniscate sine in the complex plane.svg

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Summary

Description	English: $\operatorname {sl} z$ in the complex plane.^[1] In the picture, it can be seen that the fundamental periods $(1+i)\varpi$ and $(1-i)\varpi$ are "minimal" in the sense that they have the smallest absolute value of all periods whose real part is non-negative.
Date	17 January 2022
Source	Own work
Author	A1E6

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↑ Dark areas represent zeros, and bright areas represent poles. As the argument of $\operatorname {sl} z$ changes from $-\pi$ (excluding $-\pi$ ) to $\pi$ , the colors go through cyan, blue ( $\operatorname {Arg} \approx -\pi /2$ ), magneta, red ( $\operatorname {Arg} \approx 0$ ), orange, yellow ( $\operatorname {Arg} \approx \pi /2$ ), green, and back to cyan ( $\operatorname {Arg} \approx \pi$ ).