Solving z*exp(z)=a: Proving Infinitely Many Roots

[bndnchrs]

This is not a homework problem, it is something that is stumping a group of us right now.

Show that

z*exp(z) = a

Has infinitely many roots in the complex plane.

I would caution against a series approach as we can't guarantee roots of the polynomial
z*exp(z) - a.

Any ideas?

 

[The Bob]

Heya,

What is a? Either way, this is off the top of my head but you could try doing it via norms of a ring which would prehaps reduce it to an equation in the integers. I haven't even stopped to see if it will work but it is an idea :)

The Bob

 

[epenguin]

Write out z as x + iy and don't you soon see there are infinitely many values for which the real part of both sides is equal and infinitely many for which the imaginary part of both sides is equal, proving more than what you are asked?