- #1

l'Hôpital

- 258

- 0

## Homework Statement

Show that Abel's Limit Theorem holds as the complex number z approaches 1 if instead of taking the requirement [tex] |1 - z| \leq M|1-|z|| [/tex], you restrict z to [tex] |arg(1- z)| \leq \alpha [/tex] where [tex] 0 < \alpha < \frac{\pi}{2} [/tex]

## Homework Equations

## The Attempt at a Solution

The latter requirement should yield us the former requirement.

So, I realize that this requires some geometry, but I was never actually taught geometry well, so I've been trying to do this strictly algebraically i.e. play with inequalities.

This is what I've done: Suppose z = x + iy, x > 1.

[tex]- \alpha < arg(1-z) = \arctan \frac{y}{x-1} < \alpha [/tex]

So, let M be tan(alpha). We obtain the following:

[tex]

y < M(x-1)

[/tex]

Which thus leads to

[tex]

|1-z|^2 = (x-1)^2 + y^2 \leq (1+M^2)(x-1)^2

[/tex]

Letting K = sqrt(1+M^2)...

[tex]

|1-z| = K|x-1| < K|\sqrt{x^2+y^2} - 1| = K| |z| - 1|

[/tex]

This works out well for x > 1, but if x < 1, I can't figure out a way of getting that last inequality.

Help. : (