Math 240 Home

### Complex Function Grapher

#### 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 as shown at the right.
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.
#### 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°. This week we will use the connections
between complex exponentials and pairs of sines and cosines to solve
homogeneous linear constant coefficient equations. In this particular
studio however, we will look at a couple of other ideas about complex
functions and their graphs, which will be useful to you in the future (I
hope :)
z^3+3*z^2-5*z+2 and the graph of
z^3. At the default domain from -2 to 2, the graphs of these two
polynomials look quite different. But for a domain from -10 to 10 they
look more similar and for a domain from -200 to 200 they look nearly
identical. Because of this, we can conclude that if we take a large
enough window, the number of rainbows around the edge for a polynomial of
degree n is the same as the number of rainbows around the edge for z^n,
that is to say, n rainbows.

Please report any problems with this page to bennett@math.ksu.edu

©2010 Andrew G. Bennett

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. |

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.

In the ordinary two-dimensional graph of
*y*^{2} = *x**y* as a function of *x* since the graph fails the
vertical line test, that is a given value of *x*, say *x*
= 4*y*, both
*y* = 2*y* = -2*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
*y*^{2} = *x**x*
>= 0

- Graph sqrt(z) and -sqrt(z). Observe the "branch cut" along the
negative real axis. Explain 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
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?*y*^{2}=*x**Hint:*The stitching will give you a figure that would have to intersect itself in three dimensions, but since complex graphs are actually 4 dimensional there is the extra room needed to stitch things together. - f(z) = z
- f(z) = z^2
- f(z) = z^2+1
- f(z) = 1/z
- f(z) = 1/z^2
- 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 a single root from a double root or a single pole from a double pole?
- How can you distinguish the zeros from the poles by looking at the colors you see in the Top View?
- Fill in the following table. First use the default window, with both
the Real and Imaginary ranging from -2 to 2. Count how many roots and
poles are shown in the graph. Don't count roots or poles that lie outside
the graphed region, but do count roots and poles with multiplicity. Also
count how many times you cycle through the rainbow as you move around the
edge of the graph window. When counting the rainbows around the edge of
the graph, count rainbows that move counterclockwise as you go from red to
yellow to green to blue to violet as positive and rainbows that go
clockwise as negative. Once you have finished that, right-click on the
graph (or shift-click if you have a Mac with a one-button mouse), and
change the domain to range from -5 to 5 for both the real and imaginary
axes.
Domain from -2 to 2 Domain from -5 to 5 Function # Roots # Poles Rainbows

at the edge# Roots # Poles Rainbows

at the edge(z+3)*(z+1)^2 (z^2+z)/(z-1) (z^2-16)*z^2/(z^2+1) - Look at the values you found in the table above. Can you find a
pattern for the rainbows around the edge in terms of the # of roots and
the # of poles? This pattern is called
*the principle of the argument*

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. They can also lead to a proof of the fundamental theorem of algebra (this is a math class - you should expect us to talk some about theorems and proofs :)

Look at the following functions in the complex function grapher. The Top View will be most useful, but looking at a couple of the examples that have poles using the Side View will help you understand why we use the word "poles" for these points.

Once you learn to recognize zeros and poles, you can look for patterns in how the colors work.

Now from the principle of the argument, that means
the

Please report any problems with this page to bennett@math.ksu.edu

©2010 Andrew G. Bennett