Partial fractions in complex analysis

All right, kiddo, today we’re going to learn about partial fractions in complex analysis.

So, imagine you have a big, complicated fraction, like this one:

6 / [(z - 1) (z + 2)(z - 3)]

Whoa, that’s a mouthful, right? But don’t worry, we can break it down into smaller pieces.

That’s what partial fractions are for. We want to write this big fraction as the sum of smaller fractions, like this:

6 / [(z - 1) (z + 2)(z - 3)] = A / (z - 1) + B / (z + 2) + C / (z - 3)

See? We turned one big fraction into three smaller ones. But how did we do that?

Well, let’s start by looking at the denominator of our big fraction:

(z - 1) (z + 2) (z - 3)

This is called the “factored form” of the denominator. It’s like the DNA of the denominator – it tells us how the denominator was made.

Now, we’re going to find the values of A, B, and C that make our three smaller fractions equal to the big one.

The first step is to multiply both sides of the equation by the denominator of the big fraction:

6 = A (z + 2) (z - 3) + B (z - 1) (z - 3) + C (z - 1) (z + 2)

This might look a little scary, but don’t worry. We’re just trying to isolate A, B, and C on one side of the equation, so we can solve for them.

Next, we can plug in some easy values for z to help us solve for A, B, and C.

For example, if we plug in z = 1 (since that makes the first denominator 0 and the other two non-zero), we get:

6 = A (1 + 2) (1 - 3) + B (1 - 1) (1 - 3) + C (1 - 1) (1 + 2)

6 = -6A - 6B

If we plug in z = -2, we get:

6 = A (-2 + 2) (-2 - 3) + B (-2 - 1) (-2 - 3) + C (-2 - 1) (-2 + 2)

6 = 5B

And if we plug in z = 3, we get:

6 = A (3 + 2) (3 - 3) + B (3 - 1) (3 - 3) + C (3 - 1) (3 + 2)

6 = 5C

Now we have three equations with three variables (A, B, and C), so we can solve for them using some algebra.

In this case, we get:

A = -1/10
B = 6/25
C = 6/25

So our big fraction,

6 / [(z - 1) (z + 2)(z - 3)]

can be written as:

-1/10 / (z - 1) + 6/25 / (z + 2) + 6/25 / (z - 3)

And that’s it – we’ve broken down a big, messy fraction into three easy-to-manage fractions, using partial fractions.

Congratulations, you’re a little math wizard now!