Rewrite to and ask, "Just how well-behaved is that exponential for Re(z) < 0?"
Hi.
I'm a little stuck on the following problem:
How many of the roots to the equation:
z - e^z + 5 = 0
lie in the region Re(z) < 0?
I'm just not sure how to approach this. I've solved similar problems through Rouche's theorem, but in all those cases I'm finding the number of roots within a closed circular contour. Here I really don't know what to do! Any help will be greatly appreciated!
|e^z| is equal to e^x. At x = 0, this is equal to 1. As the value of x decreases from 0, e^x will also decrease. Hence, as mentioned in my above post, the expression will lie between 0 and 1 (as x approaches negative infinity, e^x will approach zero). Of course I also know that e^z = e^(x + iy) = e^(x)(cos(y) + isin(y)). This last expression will fluctuate between zero and 1. This we know that e^(z) is also less than or equal to 1. But I don't know how to proceed from here
You're not quite catching the vision.
Expand z + 5 to real and imaginary parts.
Expand e^z to real and imaginary parts.
Equate Real Parts to obtain
Graph left and right sides separately. You should see something about possible intersections.
There are similar lessons to be learned from the Imaginary comparison.