Why Does Setting m=0 Matter in Calculating Complex Numbers?

[ahmed markhoos]

Hi,I'm facing a problem finding the values of complex numbers, I'll put two examples then I'll explain the issue.

ex1: $$(-e)^{iπ}$$ , my answer is $$(-e)^{π^2±2mπ^2}$$ The book answer is $$(-e)^{π^2}$$

ex2: $$e^{2 arctanh(i)}$$ , my answer is $$e^{[iπ/2±mπ/2]} = ie^{±mπ/2}$$ The book answer is $$i$$

just to mention that the book answer are exactly like wolfram alpha's answers, the problem is that the answers were set as m=0 while they didn't do the same thing for other problems, When can I set m=0 and when I cannot do that?

 

[mfb]

When can I set m=0 and when I cannot do that?

Keep the m if you need some logarithm or inverse trigonometric function somewhere, unless the problem statement specifies the branch you are supposed to use.