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what is the application of log(x) where x is negative number?

Anyone knows real usecase?

enter image description here

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More generally, your question can be rephrased as "why is complex analysis useful?" Virtually anywhere the subject applies, you're probably going to see the need for the complex exponential and logarithmic functions. Are you asking for a mathematical use case or physical/engineering? – dls Mar 3 '12 at 16:30
What is this graph? – Did Mar 3 '12 at 16:50
@Didier i plotted this graph in octave, so i guess it plots only real part of log(x) for negative numbers. – P K Mar 3 '12 at 16:54
Seems pretty useless. – Did Mar 3 '12 at 17:02
@Didier - I agree. – P K Mar 3 '12 at 17:15
up vote 1 down vote accepted

Logarithm is normally defined only on positive reals. Logarithm of $0$ seems to make little sense, and there is no natural choice for the negative numbers. Note that the minimal requirement for $\log$ is that $\exp(\log(x)) = x$, so for $x < 0$ you need to take $$\log(x) = \log(|x|) + \pi i + 2k \pi i,\qquad k \in \mathbb{N}$$ where the terms correspond (consecutively) to the right absolute value, the minus sign, and the fact that $\exp$ is periodic with period $2 \pi i$ if you onsider it as a function on the complex plane (if you wanted to stick to reals, you have no way of making $\exp(y)$ negative). You can, by convention, take some fixed value of $k$ (say $k = 0,-1,...$), but no such choice is canonical.

A fact that might interest you is that if you removed a halfline starting from $0$ in the complex place (say, different than $\mathbb{R}_+$), then on the remainder there would be exactly one possible choice of logarithm so that $\exp \circ \log = \mathrm{id}$.

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I think in the last paragraph you mean exactly one continuous choice? – Ben Blum-Smith Mar 3 '12 at 16:11
Yes, sorry, I should have mentioned it. This unique continuous function actually turns out to be holomorphic. – Jakub Konieczny Mar 3 '12 at 17:37

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