1. ## Finding complex roots

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!

2. Rewrite to $z + 5 = e^{z}$ and ask, "Just how well-behaved is that exponential for Re(z) < 0?"

3. Hi.

Well on the real line, where we have z = x + 0i, the exponential will lie in the interval [0,1]. Am I onto something?

4. Maybe a hint, but e^z is really well-behaved for y = 0. What happens as x and y head negative? What happens to |e^z|?

5. Originally Posted by TKHunny
Maybe a hint, but e^z is really well-behaved for y = 0. What happens as x and y head negative? What happens to |e^z|?
|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

6. 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 $\frac{x+5}{e^{x}} = \cos(y)$

Graph left and right sides separately. You should see something about possible intersections.

There are similar lessons to be learned from the Imaginary comparison.