Although I seem to understand the Proof of Rouche's Theorem and how to apply it in the case of Complex variable general polynomials, I am having great difficulty with the following problem:
A.
Prove that $ z e^{ z} = a $ , where $a$ is non-zero real number has infinitely many roots.
B.
Prove that $ \tan z=a\, z, a>0, $ has
(1) infinitely many real roots,
(2)only two pure imaginary roots if $0<a<1$ , and,
(3) all real roots if $a>1 $ or if $a=1$.
C.
Prove that $ z\, \tan z =a , a>0 $ has infinitely many real roots but no imaginary roots.....