Complex Function Grapher

This applet graphs complex functions where the domain is the base plane, the modulus is graphed on the vertical and the color represents the arguments. See the help text for more details.

Help Text

Graphing a complex function is difficult because you need 2 (real) dimensions for the domain and 2 (real) dimensions for the range - a total of 4 dimensions. In this applet, the domain of a complex function is graphed on the base plane. The range is graphed using polar coordinates. The modulus (magnitude) of the complex function is graphed on the vertical axis. The argument (angle) is graphed by using different colors - light blue for positive real, dark blue (shading to purple) for positive imaginary, red for negative real, and yellow-green for negative imaginary. This allows four dimensions to be represented in three spatial dimensions, which are then projected onto a two dimensional screen using a simple orthogonal projection.

Enter complex functions into the f(z) = text box using standard calculator notation. Note that complex functions are usually written as functions of z, and the applet requires you enter the function in terms of a variable z. You can use implied multiplication between numbers and z, but not between an alphabetic constant and z, i.e. it is okay to write 2z but you must enter i*z, not iz. Functions must have parentheses, i.e. sin(z) not sin z. The parser understands the following operations, functions, and constants: +, -, *, /, ^, sqrt(), sin(), cos(), tan(), exp(), ln(), log() (synonymous with ln), sinh(), cosh(), abs(), mod() (synonymous with abs), arg(), conj(), e, pi, i.

To adjust the domain of the graph, right-click (or shift-click if you have a one-button mouse) on the graph. The domain editor only accepts numerical input, i.e. 3.14 not pi.

Assignment

In class we looked at the graphs of the exp(pi*z), cos(pi*z), and cosh(pi*z) and observed that you could identify the periodicity of exp(z) along the imaginary axis and also that the graphs of cos(z) and cosh(z) are the same, just rotated 90°. We will start with this last observation, and then move on to a couple of other ideas about complex functions and their graphs.

Using Euler's formula, e^i*t = cos(t) + i*sin(t), prove that cos(z) = cosh(iz). How does this formula explain the comparison of the graphs of cos(z) and cosh(z) we observed in the lecture hall?
What is the connection between sin(z) and sinh(z)? Can you find a formula similar to the one for cos(z) and cosh(z) that you derived in part 1? Note that the graphs of sin(pi*z) and sinh(pi*z) are linked by two 90° degree rotations, one in the z-plane and one in the w-plane (which appears as spinning the colors in the graphing applet)

In the ordinary two-dimensional graph of y² = x, we don't have y as a function of x since the graph fails the vertical line test, that is a given value of x, say x = 4, will give rise to two different values of y, both y = 2 and y = -2. We deal with this by defining the square root function to always give the positive square root. We can also speak of the two branches of the square root function, the positive square root and the negative square root, which together make up the graph of y² = x. Of course, as a real-valued function, the square root function is only defined for x >= 0. But the point of complex numbers is that you can now take square roots of negative numbers, indeed of any number. In the next two problems we'll try to develop a sense of the two branches of sqrt(z) for the complex plane.

Graph sqrt(z) and -sqrt(z). Observe the "branch cut" along the negative real axis. Why can't we define a branch of square root which is continuous over the whole plane? Hint: The complex numbers e^iθ trace out a circle around the origin of radius 1 starting and ending at -1 as θ varies from -π to π. What are the values of sqrt(e^iθ) at the beginning and end of the circle?
We could build a single graph of the real variables y² = x by stitching the two branches of the square root functions together. How could you stitch the two branches of the square root functions together to get a single graph for the complex case? Hint: The stitching will give you a figure that would have to intersect itself in three dimesnions, but since complex graphs are actually 4 dimensional there is the extra room needed to make stitch things together.

The last topic we will consider are some tricks for finding zeros and poles of complex functions. These tricks can later be used to recognize whether certain systems are stable or unstable. Look at the following functions using the Top View (you can use the Side View in addition if you want).

f(z) = z
f(z) = z^2
f(z) = z^2+1
f(z) = 1/z
f(z) = (z^2 + 1)/z

How can you recognize the zeros and poles of the functions from just looking at the colors you see in the Top View?
How can you distinguish the zeros from the poles by looking at the colors you see in the Top View?