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

= 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
 * $$\log z=\int_1^z \frac{dw}{w}$$
 * , then you can get the log of &minus;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)