Wednesday, February 22, 2012

An Approach to Plotting Functions in the Complex Plane using Parametric Mode

Graphing functions on the complex plane is not easily handled on a graphing calculator. Few options exist: (1) build a program to build an x-list and a y-list and build a Scatterplot, and (2) use the parametric graphing mode using a numerous amount of paths.

All screen shots are taken with an iPad 2 and a TI nSpire CX CAS calculator.

The complex plane is shown below:

There are pros and cons to each method. The pro of method (1) is that you would get a complete picture of any desired mesh. However, you are also faced with high execution time, even on the best graphing calculators, a lot of RAM usage, and you can usually only use a small area to graph on: for example the interval [-2,2] with 25 point intervals.

Method (2) involves parametric graphing that splits the function into two parts: the real component x(t), and its imaginary component y(t). For any complex function f(z):

x(t) = real(f(z))
y(t) = imag(f(z))

You will either have to separate the real and imaginary components, or if you graphing calculator can handle complex operations, use the real and imag functions.

Caution: If you calcualtor can handle complex numbers, some may not have a built in routine for the trigonometric or advanced operations. This applies to the TI-84 family, and most (if not all) Casio graphing calculators. I list some advanced operations for convenience.

Source: HP-41C Math Pac, Hewlett Packard, 1980?

Let z = x + yi, r = abs(z), θ = arg(z) = angle(z), i = √ -1

ln z = ln r + θ i
a ^ z = e ^ (z ln a)
sin z = sin x cosh y + i cos x sinh y
cos z = cos x cosh y - i sin x sinh y
z ^ n = r ^ n * e ^ ( i n θ )
z ^ ( 1/n ) = r ^ ( 1/n ) * e ^ ( i ( θ / n + ( 2 π k ) / n ) ) , k = 0, 1,...,n - 1
e ^ z = e ^ x * cos y + i * e ^ x * sin y

There is one big obstacle with using the parametric mode. There is only one independent variable, t. So using a set of parametric equations describe only one path.

For example, in graphing sin z set the pair of parametric equations as:
x(t) = real(sin(t + t i))
y(t) = imag(sin(t + t i))

graphs sin z along the path y = x. (x=t, y=t)

In order to get a more complete picture (or really, just more detail), you will have to repeat the pair, but using different paths.

For path y = x, use t + t i (sub x = t, y = t)
For path y = -x, use t - t i (sub x = t, y = -t)
For path y = 2x, use 2t + t i (sub x = t, y = 2t)
For path y = e^x, use e^t + t i (sub x = t, y = e^t)
For path y = ln x, use t + e^t * i (sub x = e^t, y = t)
For path y = x^n, use t + t^n * i (sub x = t, y = t^n)
For path y^n = x, use t^n + t * i (sub x = t^n, y = t)

This method is good if you don't want to program, just want to get a quicker picture (method (2) still takes a lot of effort), or you are working with a monochromatic screen (most graphing calculators) where too much detail can just "paint the screen black".

Below are examples of graphs of complex functions from the use of method (2). Keep in mind that these are not complete graphs but enough to get some idea of what is going on.

Hope you enjoyed this blog. Eddie

This blog is property of Edward Shore. © 2012

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