
A Gallery of Complex Functions
by François Labelle
On these pages, you will see maps of the argument (also called phase) of some complex functions. For clarity and aesthetic reasons, I chose to disregard the magnitude completely. There are other galleries of complex functions on the Internet that will show both information. The color assignment is : 1 > blue, i > magenta, 1 > red, and i > black. When the four colors are touching at a point, it means that the point is a zero or a pole (infinity).
The resolution is of 5 pixels per unit. The range of every image is from 30 to 30 for both the real and the complex axes, except for the band on the left which is the Riemann zeta function from 5 to 5 and from 5*i to 435*i.

Page 1: Miscellaneous functions
f(z) = z. This shows which color gets assigned to each complex number. The center is a good example of what a zero looks like. 
f(z) = 1/z. The center is a good example of what a pole looks like. You can distinguish between a zero and a pole by the order in which colors appear as you go around counterclockwise. 
f(z) = z^2. Recall: 1 or 1 squared give 1 (blue), i or i squared give 1 (red). 
f(z) = sqrt(z). As in the real case, we need to make a choice between two possible square roots. Whatever the choice, there will be at least one discontinuity, which is usually put pointing toward the left. 
f(z) = exp(z). The exponential growth cannot be seen because this is a phase diagram. The phase is cyclic along the imaginary axis with period 2*pi. 
f(z) = log(z). Recall that for 0<x<1, log(x)<0, which is seen as a small red island. 
f(z) = sin(z). Note the series of zeros at each integer multiple of pi. 
f(z) = tan(z). Alternation of zeros and poles along the real axis. 
f(z) = gamma(z). The gamma function extends the factorial function to arbitrary complex numbers: gamma(n) = (n1)!. It has poles at z=0,1,2,... For more information, see Eric Weisstein's page on The Gamma Function. 
f(z) = zeta(z). The Riemann zeta function. It has one pole at z=1 and trivial zeros at z=2,4,6,... . The Riemann Hypothesis asserts that all nontrivial zeros are along a vertical line with real part exactly 1/2. For more information, see Eric Weisstein's page on The Riemann Zeta Function. 
page last updated: March 16, 2002
