Functions of a Complex Variable

Traditionally, in mathematics, complex analysis is known as the theory of functions of a complex variable. It is the branch of mathematical analysis that analyses functions of complex numbers. It is helpful in multiple branches of mathematics, including number theory, algebraic geometry, analytic combinatorics, and applied mathematics. In this article, you will learn what the functions of a complex variable are, and how to solve the functions of a complex variable.

Learn: Complex numbers

Definition of Functions of a Complex Variable

Complex variable functions or complex functions are functions that assign complex numbers for complex numbers. Let C be the set of complex numbers.

A function f : C → C is a rule which associates with z ∈ C, a unique w ∈ C, written as w = f(z).

Here,

z = x + iy

w = u + iv, where u = u(x, y) and v = v(x, y)

Thus, u and v are functions of x and y.

w = f(z) = u(x, y) + i v(x, y)

From this we can say that the standard notations for real and imaginary parts of z are x and y. Also, the standard notations for real and imaginary parts of w are u and v, where u and v are functions of x and y.

A function of a complex variable, i.e., w = f(z) is generally expressed as a mapping or transformation from the complex plane z to the complex plane w. This can be shown as:

The graph of the function often reveals properties of a real-valued function of a real variable. However, no suitable graphical representation is available for a function, w = f(z), where z and w are complex since the numbers z and w are located in a plane rather than a line.

Solved Examples

Example 1:

Consider f(z) = z² + iz, and express it in terms of real and imaginary parts.

Solution:

We know that z = x + iy

Now, f(z) = z² + iz

= (x + iy)² + i(x + iy)

= x² + i²y² + 2ixy + ix + i²y

= x² – y² + 2ixy + ix – y {since i² = -1}

= x² – y² – y + i(x + 2xy)

This is of the form w = u + iv such that u = x² – y² – y and v = x + 2xy.

Or

u = u(x, y) = x² – y² – y

v = v(x, y) = x + 2xy

Example 2:

Find u(x, y) and v(x, y) for the complex function f(z) = z³.

Solution:

Given, f(z) = z³

As we know, z = x + iy

So, f(z) = (x + iy)³

= x³ + i³y³ + 3ixy(x + iy)

= x³ – iy³ + 3ix²y + 3i²xy² {since i³ = -i}

= x³ – iy³+ 3ix²y – 3xy² {since i² = -1}

= (x³ – 3xy²) + i(3x²y – y³)

This is of the form w = f(z) = u + iv.

So, u = u(x, y) = x³ – 3xy²

v = v(x, y) = 3x²y – y³

Example 3:

For what values of z is the complex function w = 1/z defined? Also, find u, v and w when z = 3 – 2i.

Solution:

We know that, z = x + iy

Given, w = 1/z

Thus, w is defined for all values of z ≠ 0.

Now,

w = 1/z

= 1/(x + iy)

By rationalizing the denominator, we get;

w = [1/(x + iy)] . [(x – iy)/(x – iy)]

= (x – iy)/ (x² – i²y²)

= (x – iy)/(x² + y²) {since i² = -1}

This can be written as:

w = x/(x² + y²) – iy/(x² + y²)

Comparing with w = u + iv, we have

u = u(x, y) = x/(x² + y²)

v = v(x, y) = -y/(x² + y²)

Consider z = 3 – 2i

By comparing with z = x + iy, we get x = 3 and y = -2

And x² + y² = (3)² + (-2)² = 9 + 4 = 13

So, u = 3/13 and v = -(-2)/13 = 2/13

Therefore, w = (3/13) + i(2/13)

Practice Problems

1. Find the real and imaginary parts of w = f(z) = z² – z.
2. Solve: f(z) = sin z
3. Find the functions of u and v for w = az² + bz + c.