Circumventing discontinuity of complex function fails with different parameters

Questions
1

Hello Wolfram Community!

The attached notebook shows the discontinuity of the complex function as it should. However, one introduces A Log function using IF- statement to 2 Pi I when arg [z] < 0 to circumvent the discontinuity. While this works well in some cases (as shown in the attached notebook), it does not work when using a different set of parameters (as shown in the attached notebook).

I was wondering if there is a broader and more robust way that encompasses the second plot shown at the end of the attached notebook.

Thank you in advance!

Open in Wolfram Cloud ↗

Wolfram_Community.nb (130 KB)

2

Your myLog is discontinuous on the positive real axis:

myLog[z_] := If[Arg[z] < 0, Log[z] + I 2 \[Pi], Log[z]]
FunctionDiscontinuities[myLog[z], z]
ComplexPlot[myLog[z], {z, -1 - I, 1 + I}]

You cannot redefine Log in the complex plane without discontinuities. You may have studied that the vector field (x,-y)/(x^2+y^2) has locally a potential but not globally. It is basically the same thing.

3

Sorry, I meant (-y, x)/(x^2+y^2).

4

Hi!

Thank you for your note. MyLog still works physically for all intents and purposes; in fact it renders accurate data. If any discontinuities left, they are negligibly insignificant.

Thanks!

HH