- #1
moo5003
- 207
- 0
Homework Statement
a in Reals, a>1.
f(z) = a + z - exp(z)
a) Show that f has exactly one zero in the left half-plane {z in C : Re(z) < 0}
b) Show that this zero is on the real line
The Attempt at a Solution
Well, I havnt had much progress on the problem as of yet. I'm trying to use Rouches theorem letting:
h(z) = -exp(z)
g(z) = z + a
and then showing that restricting z to the left half-plane
|h(z)|<|g(z)| implying that z+a has as many zeroes as f(z) and therefore showing that f(z) has one zerio in the left half plane (since z+a has one since a>1).
Problem:
I'm unsure how to get the inequality to show |h(z)|<|g(z)| any insight into this would be appreciated.