Zeros, Poles, and Essential Singularities

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Let be a complex-valued function. Assign a color to each point of the complex plane as a function of , namely the RGB color with four arguments , , , and (red, green, blue, and opacity, all depending on ). If (with chosen by its slider), use black. Otherwise: if , ; if , ; if , .

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For , this colors the four quadrants red, cyan, blue, and yellow.

To illustrate zeros, poles, and essential singularities, choose and three kinds of functions , , and . Note the characteristic -fold symmetry in case of a zero or pole of order .

In the case of a pole, , as .

The following theorem is attributed to Sokhotsky and Weierstrass ([1], p. 116). For any , in any neighborhood of an essential singularity of the function , there will be at least one point at which the value of the function differs from an arbitrary complex number by less than .

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Contributed by: Izidor Hafner (February 2016)
Open content licensed under CC BY-NC-SA


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Reference

[1] A. Sveshnikov and A. Tikhonov, The Theory of Functions of a Complex Variable (G. Yankovsky, trans.), Moscow: Mir Publishers, 1971.



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