General method for using Rouche's Theorem to find roots of equations.

Question

I have been presented with the following rather weak version of Rouche's Theorem in lectures:

Let $D\subseteq\mathbb{C}$ be a domain containing a circle $C$ and its interior, and let $f,g:D\rightarrow\mathbb{C}$ be analytic, where $|g(z)|<|f(z)|\:\forall z\in C$ and $f$ has a zero inside $C$. Then $f+g$ has a zero inside $C$.

I have managed to find some solutions to polynomial equations within a disk using this but in some cases I seem unable to find permissible functions $f$ and $g$ for this to work.

For example, proving that there are infinitely many solutions near the imaginary axis for $ze^z=1+z$ .

Any assistance on how to find the right functions to use (for polynomials at least) would be much appreciated.

Answer 1

In the example, you see that for $z$ with a large radius the equation is close to $e^z=1$ with the known solutions at $z=i2\pi k$, $k\in\Bbb Z$.

Then with $|k|$ large, $z=w+i2\pi k$, $w$ presumably small, the equation transforms to $$ f(w)+g(w)=0 ~\text{ where }~ f(w)=e^w-1~~g(w)=-\frac1{w+i2\pi k}. $$ On the circle $|w|=1$ and with $e^w-1=w+\frac12w^2(1+\frac13w+\frac1{12}w^2+...)$ we get with the usual estimate of the remainder of the exponential series by a geometric series $$ |f(w)|=|e^w-1|\ge 1-\frac12\frac1{1-\frac13}=\frac14. $$ Thus whenever $|g(w)|\le\frac1{2\pi|k|-1}<\frac14$, that is, $|k|>1$, the Rouché theorem gives a root in the disk $D(i2\pi k,1)$. One can easily make the circles smaller, in consequence increasing the lower bound for $|k|$.