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### Complex Function Grapher

Launch the Complex Function Grapher

#### Introduction

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 the Complex Function Grapher app, 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 either from the side or the top, depending on the view selected.In class we discussed connections between exp(z), sin(z), and cos(z) that could be deduced from their Taylor series. This week we will use the connections between complex exponentials and pairs of sines and cosines to find real solutions to linear constant coefficient differential equations. In this studio we will look at the connections between these geometrically. We will also look at some other ideas about complex functions and their graphs, which will be helpful in some future courses (particularly if you study control theory).

#### Assignment

- Graph exp(pi z). According to Euler's formula, we should expect
exp(pi z) = cos(pi z/i) + i sin(pi z/i) to be periodic, since cos(z) and sin(z) are periodic. How does the periodicity show up in the graph? What is the period of exp(pi z)? - Graph cos(pi z) and cosh(pi z). How are the two graphs related?
- Graph i*sin(pi z) and sinh(i*pi z). How are the two graphs related? What is the relationship between i*sin(pi z) and sinh(i*pi z)?
- 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 live in 4 dimensions 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. Count roots and poles of the following
functions with multiplicity (i.e. a double root counts as two). 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.
Function # Roots # Poles Rainbows

at the edge(z+1)(z-1)^2 (z^2+z)/(z-1) (z^2-2)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 hyperbolic cosine and hyperbolic sine are defined as

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

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

©2014 Andrew G. Bennett

If you have any problems with this page, please contact bennett@math.ksu.edu.

©2010, 2014 Andrew G. Bennett