Summary

DescriptionLemniscate sine in the complex plane.svg	English: ${\displaystyle \operatorname {sl} z}$ in the complex plane.[1] In the picture, it can be seen that the fundamental periods ${\displaystyle (1+i)\varpi }$ and ${\displaystyle (1-i)\varpi }$ 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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1. ↑ Dark areas represent zeros, and bright areas represent poles. As the argument of ${\displaystyle \operatorname {sl} z}$ changes from ${\displaystyle -\pi }$ (excluding ${\displaystyle -\pi }$) to ${\displaystyle \pi }$, the colors go through cyan, blue (${\displaystyle \operatorname {Arg} \approx -\pi /2}$), magneta, red (${\displaystyle \operatorname {Arg} \approx 0}$), orange, yellow (${\displaystyle \operatorname {Arg} \approx \pi /2}$), green, and back to cyan (${\displaystyle \operatorname {Arg} \approx \pi }$).