Testwiki:Reference desk/Archives/Mathematics/2011 September 24

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September 24

Logarithms of non-positive real numbers

Is there a way to derive the output of the natural logarithm for input values that are not simply integers that are greater than one without using Euler's formula? For instance, I could easily find $\ln (-1)=i\pi +2\pi n\text{, where }n\in \mathbb{Z}$ from the fact that $e^{i\pi +2\pi n}=\cos (\pi +2\pi n)+i\sin (\pi +2\pi n)\text{, where }n\in \mathbb{Z}$. This is the only method I'm familiar with. Is there an alternative method of evaluating the natural logarithm at negative, imaginary, and complex values? — Trevor K. — 17:49, 24 September 2011 (UTC)

Well, it depends on your definition of logarithm. If you use the definition

$\log z=\underset{1}{\overset{z}{\int }}\frac{dw}{w}$

, then you can get the log of −1 by changing to polar coordinates and integrating in a semicircular arc (for the principal value). Other values you would get by letting the integral wrap around more times, or by letting it go clockwise instead of counterclockwise. --Trovatore (talk) 21:18, 24 September 2011 (UTC)

You can find the logarithm of complex or negative values (z) using this formula (correct me if misinterpreted that atan(X,Y) thing):

$\log z=\log |z|+i*\theta$ where $\theta$ is the angle that the number z makes with the real axis.

Source: here. Best, Mattb112885 (talk) 23:43, 24 September 2011 (UTC)