September 24 Information

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 ${\displaystyle \ln(-1)=i\pi +2\pi n{\text{, where }}n\in \mathbb {Z} }$ from the fact that ${\displaystyle e^{i\pi +2\pi n}=\cos(\pi +2\pi n)+i\sin(\pi +2\pi n){\text{, where }}n\in \mathbb {Z} }$. 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

${\displaystyle \log z=\int _{1}^{z}{\frac {dw}{w}}}$

, 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) reply

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

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

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