# Complex Analysis

## Homework Statement

For each of the following functions f(z), find f'(z) and identify the maximal region for which f(z) is analytic.

1. $f(z)=1/(z^2+1)$
2. $f(z)=e^{-1/z}$

## The Attempt at a Solution

1. $f'(z)=\frac{-2z}{(z^2+1)^2}$ <--this part is easy. I'm having difficulty being certain of the maximum region for analyticity. Here is my attempt.

f(z) is analytic everywhere but + or - i because f'(z) is undefined there.

Is that a true stament or is the correct statement ... f(z) is analytic everywhere but + or - i because f(z) is undefined there.

2. $f'(z)=\frac{e^{-1/z}}{z^2}$ <--this part is easy. I'm having difficulty being certain of the maximum region for analyticity. Here is my attempt.

f(z) is analytic everywhere but 0 because f'(z) is undefined there. However, f(z) is analytic at infinity.

Is that a true stament or is the correct statement ... f(z) is analytic everywhere but 0 because f(z) is undefined there. However, f(z) is analytic at infinity.

The short answer is that you are correct.

In the future the simplest way to approach these problems is to remember the definition of analytic:

Definition: A function ##f(z)## is analytic at a point ##z_{o}## if ##lim_{z \rightarrow z_{o}} \frac{f(z) - f(z_{o})}{z - z_{o}} = lim_{h \rightarrow 0} \frac{f(z_{o} + h) - f(z_{o})}{h}##.

The maximal region for which ##f(z)## is analytic will be the entire complex plane with any singularities removed (read: with the places it is undefined removed.

For example, for your second function we can write:

The function ##f(z) = e^{\frac{-1}{z}}## is analytic on ℂ - {0}.

vela
Staff Emeritus
Homework Helper

## Homework Statement

For each of the following functions f(z), find f'(z) and identify the maximal region for which f(z) is analytic.

1. $f(z)=1/(z^2+1)$
2. $f(z)=e^{-1/z}$

## The Attempt at a Solution

1. $f'(z)=\frac{-2z}{(z^2+1)^2}$ <--this part is easy. I'm having difficulty being certain of the maximum region for analyticity. Here is my attempt.

f(z) is analytic everywhere but + or - i because f'(z) is undefined there.

Is that a true stament or is the correct statement ... f(z) is analytic everywhere but + or - i because f(z) is undefined there.
The first statement is the one you want. There's a theorem that says if a function is complex differentiable at a point, it's analytic at that point. Just because a function exists at a point doesn't mean it's analytic there.

The first statement is the one you want. There's a theorem that says if a function is complex differentiable at a point, it's analytic at that point.
I don't think so huh vela? Doesn't it have to be complex-differentiable in some disc centered at the point in order for it to be analytic at that point?

vela
Staff Emeritus