Complex Functions

Complex Functions

A complex-valued function $f$ of a complex variable $z$ is a rule that assigns a complex number $f(z)$ to each complex number $z$ in a set $S$.

The set of complex numbers on which a function operates is called the domain of definition.. It can be a domain in the sense described in Regions of the Complex Plane, but it doesn’t have to be. If the domain of definition isn’t explicitly specified, it is assumed to be the entire complex plane.

We often use the format

\[w = f(z)\]

to denote that $f$ maps values of $z$ in the domain to values of $w$ in the range, which is the set of all values the function produces from input values in the domain of definition.

Using this notation, we say that $f$ maps points in the $z$-plane to points in the $w$-plane. Letting $z = x + yi$ and $w = u + vi$, we see that 4 dimensions ($x,y,u,v$) are required to represent the mapping, and thus it is not possible to represent complex-valued functions using conventional graphs.

We can, how ever, represent input sets in the $z$-plane in one graph and output sets in the $w$-plane in another graph to get some notion of how the mapping works.

Some Geometricly Interesting Functions

The function

\[w = z + b\]

where all values are complex, translates any point in the $z$-plane to a new point in the $w$-plane determined by the vector addition of $z$ and $b$.

Next, the function:

\[w = az\]

where all values are complex, rotates $z$ by the argument of $a$, and multiplies the modulus of $z$ by the modulus of $a$ to get a new point in the $w$-plane.

This becomes more obvious if we let $a = pe^{\phi i}$ and $z = re^{\theta i}$. Then we have:

\[w = pre^{(\theta + \phi)i},\]

which we can consider in two parts, a rotation:

\[w^* = e^{\phi i}z\]

and a magnification:

\[w = pw^*\]

Finally, the function:

\[w = f(z) = \frac{1}{z}\]

is called a reciprocation. Again, writing it in the exponential form is more enlightening:

\[w = f(z) = \frac{1}{re^{\theta i}} = \frac{1}{r} e^{-\theta i }.\]

This shows that $f$ inverses the modulus of $z$, and changes the sign of its argument. This has some funky effects; points inside the circle $|z| = 1$ are mapped to points outside the circle $|w| = 1$, and points on the top half of the input circle are mapped to points on the bottom half of the output circle.

Peter Francis has an amazing tool for graphing complex functions.

Bilinear Mappings

Another important class of functions is called bilinear or Mobius mappings. They are of the form

\[w = \frac{az + b}{cz + d}\]

where $a, b, c$ and $d$ are complex numbers for which $ad - bc \neq 0$, which ensures that the mapping is not a constant. Every bilinear transformation is a succession of rotations, magnifications, reciprocations, and/or translations. One property of bilinear transformations is that they map circles and straight lines to circles and straight lines; which one depends on whether the curve to be mapped passes through the point at which the bilinear mapping is undefined.

Real and Imaginary Parts of Complex Functions

Complex functions can be represented as a pair of real-valued functions of two independent variables. For example, if

\[w = z^2 = (x+yi)^2 = (x^2 - y^2) + 2xyi\]

then we can have $ w = u(x,y) + v(x,y)i $ with

\[u(x,y) = x^2 - y^2, \quad v(x,y) = 2xy.\]

Finding the Image of a Curve

The image in the $w$-plane of a curve $C$ in the $z$-plane is denoted $C’$ and can be found by two distinct procedures.

Suppose that $g(x,y) = 0$ is the equation of a curve $C$ in the $z$-plane.

Procedure 1: If it is convenient to invert $w = f(z)$ and find $z$ in terms of $w$, $z = h(w)$, then we effectively have $x$ and $y$ in terms of $u$ and $v$. We can then substitute these values into $g(x,y) = 0$ to give the equation $g[x(u,v), y(u,v)] = 0$ for $C’$.

Procedure 2: If it is inconvenient to find $z$ in terms of $w$, we can represent $C$ parametrically, $x = x(t)$ and $y = y(t)$. If we substitute these into $u = u(x,y)$ and $v = v(x,y)$, we obtain parametric equations $u = u[x(t), y(t)], v = v[x(t), y(t)]$ for $C’$.

The Point at Infinity

There isn’t just one point at infinity. In fact, if you go infinitely far away from the origin in any direction, you will arrive at “the point at infinity”. Depending on the convention of a text, the complex plane can be considered to exclude the point at infinity by default, in which case the complex plane plus the point at infinity is called the extended complex plane, or the complex plane is considerd to include the point at infinity by default, in which case the complex plane without the point at infinity is referred to as the finite complex plane.