Complex Variables

Complex Variables

Complex numbers are really two numbers packaged into one entity (much like matrices). The two “numbers” are the real and imaginary portions of the complex number:

We may plot complex numbers in a complex plane: the horizontal axis corresponds to the real part and the vertical axis corresponds to the imaginary part. Im{z} z = x + jy y Re{z} x

Often, we wish to use polar coordinates to specify the complex number. Instead of horizontal x and vertical y, we have radius r and angle q. Im{z} z = x + jy y r Re{z} q x

The best way to express a complex number in polar coordinates is to use Euler’s identity: So, and

We also have A summary of the complex relationships is on the following slide.

Im{z} r y = r sin q q Re{z} x = r cos q

The magnitude of a complex number is the square-root of the sum of the squares of the real and imaginary parts: If we set the magnitude of a complex number equal to a constant, we have

or, This is the equation of a circle, centered at the origin, of radius c.

Im{z} c2 = |z|2 = x2 + y2 z = x + jy y c Re{z} x

Suppose we wish to find the region corresponding to This would be a disk, centered at the origin, of radius c.

Im{z} x2 + y2 = |z|2 < c2 y c Re{z} x

Suppose we wish to find the region corresponding to This would be a disk, centered at z0, of radius c.

Im{z} (x-x0)2 + (y-y0)2 = |z-z0|2 < c2 = |z-z0|2 c y0 z0 Re{z} x0

Functions of Complex Variables Suppose we had a function of a complex variable, say Since z is a complex number, w will be a complex number. Since z has real and imaginary parts, w will have real and imaginary parts.

The standard notation for the real and imaginary parts of z are x and y respectively. The standard notation for the real and imaginary parts of w are u and v respectively.

where Both u and v are functions of x and y.

So a complex function of one complex variable is really two real functions of two real variables.

Exercise: Find u(x,y) and v(x,y) for each of the following complex functions:

Continuity of Complex Functions In order to perform operations such as differentiation and integration of complex functions, we must be able to verify of the complex function is continuous. A complex function is said to be continuous at a point z0 if as z approaches z0 (from any direction) then f(z) can be made arbitrarily close to f(z0).

A more mathematical definition of continuity would be for any e, we can make for some d such that Since we are dealing with complex numbers, the geometric interpretation of this statement is different from that of real numbers.

The region |z-z0| < d defines a disk in the complex plane of radius d centered about z0. d Im{z} z0 Re{z}

So, if we wish |f(z)-f(z0)| < e we must find a d to make this so. e Im{w} f(z0) Re{w}

Example: Suppose Find d such that for

Solution:

All we need to do is to find a value of d such that if then

We can do some calculations on a spreadsheet (continuity.xls). A value of d< 0.1 seems to do it.

A MATLAB plot (by continuity.m) of the previous example is shown on the following slide.

Differentiation of Complex Functions How do we take derivatives of complex functions with respect to complex variables? If what is

The differential dz can vary in one of two ways: along the real axis (dx) or along the imaginary axis (dy). Im{z} y+dy dy y dx Re{z} x x+dx

As z varies in either direction, the derivative must be the same. x direction y direction So, we must have

These last two conditions are called the Cauchy-Riemann equations. These equations are the criteria for a complex function to be differentiable (with respect to z = x + jy).

Example: Show that the function is differentiable Solution: We have shown that

Now that we have determined that this function is differentiable, the derivative can be found using or

If we apply these formulas to where

we have or

We see that the derivative in both cases is The answer is what we would expect to get if z were treated as a real variable. As it turns out, for most well-behaved complex functions, the derivative can be found by treating z as if it were a real variable.

Example: Show that the function is not differentiable Solution: We have shown that

Exercise: Is differentiable?

Definition: A function is said to be analytic if it is differentiable throughout a region in the complex plane.

Integration of Complex Functions What happens when we try to take the integral of a complex function along some path in the complex plane?

A complex integral is like a line integral in two dimensions. The real and the imaginary parts of the integral are nearly identical to classic line integrals.

Example: Integrate over the real interval z = 0 + j0 to z = 2 + j0. Solution: We have shown that

Since we are integrating along the real (x) axis, all integrals with respect to dy are zero. In addition y=0. So,

The result is exactly what we would expect to get if we simply integrated a real variable from 0 to 2.

Example: Integrate over the imaginary interval z = 0 + j0 to z = 0 + j2. Solution: The integral becomes

Complex Variables