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I'm doing the following as an exercise to try and get my head around complex numbers. Specifically, I need to understand what it means to take the natural log of a complex number and what it involves.

Say I wanted to integrate 1/ (1 +x^2) dx

I know this is arcTan(x).

I can also write the integral as dx / [ (1 + ix)(1 - ix) ], which can be solved with partial fractions to arrive at 0.5 Ln [(1 + ix) / (1 - ix) ].

Now it should follow that the 2 results are equivalent. How do I got from Ln to arctan? Any hints?

Mant Thanks for your help,

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# Complex integral

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