Complex Numbers

A complex number is simply a number "z" in which there is a real part and an imaginary part. "z = a +bi" is a complex number with real part "a" and imaginary part "bi."  Note that "a" and "b" are both real numbers!  One useful aspect comes from plotting these complex numbers in what is known as the "complex plane."  The complex plane is exactly the same as the ordinary Cartesian plane except what is plotted isn't x and y, but rather real vs imaginary.  (Real being on the x axis and imaginary being on the y axis)

Plotting the number z on the complex plane, we simply go "a" units on the real axis and "b" units on the imaginary axis.  (Note that you are plotting b, not ib!)  If you connect a line from the origin from this point you now have a line segment (labeled "r") originating from the origin to the point (a,b).  Using trigonometry (taking the angle to be from the horizontal) you will find that..

Where we have used Euler's formula to replace the trigonometric terms with an exponential term.
Therefore, there are two ways to write a complex number, we can write it as a+bi or in its polar exponential form just shown.

A notation that is useful is the Re(z) and Im(z) notation.  Re(z) simply means the real values of the number z, and Im(z) simply means the imaginary values of the number z.  For example, if we had z = 1+2i.  Re(z)=1 and Im(z)=2 (Not 2i!!).

A term that you must know is known as the "modulus" or "norm" of z.  Its notation is that of the absolute value notation and is defined as the length of r in the complex plane.

Where the last term is the square root of z times z bar.  The term z bar is the conjugate of z.  (For those that don't know, the conjugate of any complex number a+bi is a-bi).  A useful relation to the conjugate of z is..

For an example, if we wanted to write the number z = -1-i in polar form, a=-1 and b=-1.  Just like in trigonometry, the angle can take of infinite values. (Because you can simply go around the whole circle and you would be in the same place, yet have a bigger angle value.)  The general formula for the angle in this case is..

Notice that we did not use the value of pi/4 as the answer! This is because the actual point is in the third quadrant and pi/4 is in the first! So we must exclude all angles in the first quadrant in this case because that is not where the point is located.  For simplicity, take n=0.

Complex numbers can also come up in equations.  These are known as (not surprisingly) complex equations.  This is best explained with an example.  Say we want to find the values of x and y in the equation..

All we need to do is expand and equate the real parts on the left with the real parts on the right, and the same for both the imaginary parts.

You may be asking yourself, "Since we have graphed a point using a complex number, does that mean we can have graphs of complex numbers?" The answer is yes!  The way it works is basically exactly how polar graphs work.  The only difference is replacing "r" in the polar graphs with modulus of z, or |z|.  So |z|=3 is a circle of radius 3 in the complex plane.

Next on the agenda is complex infinite series! A complex series in general looks like this..

The conditions for convergence for a complex series is...

So basically, both real parts must converge for the series as a whole to converge.  This makes a lot of sense! Typically you only need to split the series up into its components if the standard ratio test fails to show convergence.  As an example lets test the following series for convergence.

This next example is an example where the ratio test fails and we must write out the first couple of terms to find the real and imaginary part of the summation.  We must then find the convergence of these parts.

As you recall from your Calculus courses, you often needed to find the range in which an infinite series converges.  This was called the radius of convergence.  We will now be looking into the power series..

As an example let us find the convergence of the following series using the ratio test..

So what does this mean?  This means we have a disk of convergence.  Recall that previously I mentioned that |z| acts as r in polar coordinates.  So the disk of convergence is a disk of radius 1.  An interesting thing to note is that the real axis length of the disk is 1, this is the radius (of the disk!) of convergence of the real part of this series! That is the very reason it is called the radius of convergence!

We can use the exponential definition of a complex number to come up with a useful relationship.

This relationship can be used to find the formulas for cos(nt) or sin(nt)! (I am using "t" as theta)  As an example, let us find the equation for finding the cosine of 3 times theta, or cos(3t).

 Very cool!!

The function e^z can be written as a complex infinite series, but another way to write it is as such.

This notation is usually used to find exact values for e^z for a given x and y. (Note that I have replaced a and b with x any y, that shouldn't be a problem! They are just dummy variables!)  If you have e^z and e^(-z)  and set z equal to just its imaginary part "i" times theta, this creates a system of equations in which you can solve for sine and cosine.

These formulas are good to use in integration since products of exponents are far easier to integrate than products of trigonometric functions.  These formulas also work for any value of theta, even complex values!

If we use the above formulas to with a complex value of theta "z" we can get a couple of very useful formulas! Firstly, theta will only have an imaginary part such as "iy".

These come up so often in problems that they actually have their own name, the hyperbolic functions.  Notice that the "i" is not included in the hyperbolic sine function.  These pretty much follow all the rules for the normal trigonometric functions as they should since the hyperbolic functions are simply the imaginary components of the sine and cosine function!

Our next objective is to see if we can find the logarithm of a complex number.  Namely..

So a complex number has infinitely many logarithms! (Since every theta has an infinite amount of other thetas that are equivalent. 

And that's my sch-peal on complex numbers!