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

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.

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 $$|k|$$.