Logarithms of Negative and Imaginary Numbers

By Euler's identity, $ e^{j\pi} = -1$, so that

$\displaystyle \ln(-1) = j\pi

from which it follows that for any $ x<0$, $ \ln(x) = j\pi + \ln(\vert x\vert)$.

Similarly, $ e^{j\pi/2} = j$, so that

$\displaystyle \ln(j) = j\frac{\pi}{2}

and for any imaginary number $ z = jy$, $ \ln(z) = j\pi/2 + \ln(y)$, where $ y$ is real.

Finally, from the polar representation $ z=r e^{j\theta}$ for complex numbers,

$\displaystyle \ln(z) \isdef \ln(r e^{j\theta}) = \ln(r) + j\theta

where $ r>0$ and $ \theta$ are real. Thus, the log of the magnitude of a complex number behaves like the log of any positive real number, while the log of its phase term $ e^{j\theta }$ extracts its phase (times $ j$).

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