How does $\mathrm {e}^z$ and $\log z$ look like as complex functions. I want to visualize complex functions $\mathrm e^z$ and $\log z$ in $C$, here $z\in\Bbb C$. I want to know their behavior and zeros and singularities. Can anyone explain me in an easy way. Thank you in advance. 
 A: $e^z$ is like $e^x$ along the real axis, and like $\cos y + i \sin y$ along the imaginary axis. In other words, constant magnitude, but varying phase. Thus, the real and imaginary parts are wavy. Imagine a corrugated roof that's bent into an exponential along the axis perpendicular to the grooves; that's what the real and imaginary parts look like. (Also, this is possibly the least mathematical description ever!)
The only solution of $e^x = 0$ is negative infinity, and the same goes for the complex version; you won't ordinary find any zeros on it.
$\ln z$ is a bit more fun, since it becomes multi-valued... Since $e^z$ is wavy, you get an infinite set of solutions spaced out by $2\pi$ along the imaginary axis.
A: Exponential function
http://en.wikipedia.org/wiki/Exponential_function#Complex_plane
Logarithmic function
http://en.wikipedia.org/wiki/Complex_logarithm#Plots_of_the_complex_logarithm_function_.28principal_branch.29
A: 
This book has some great color pictures of Complex Functions. It also has a section on Riemann Surfaces on which the functions,"live". Also recommended: Visual Complex Analysis by Tristan Needham.
