What are the real and complex roots of z = exp(-z)?

[gmans]

of the equation
z= exp(-z)

could someone possibly point me in the right direction to start this problem?
this area of math is still new to me so please go easy
thanks

 

[berkeman]

I moved this thread to the homework section. gmans, what do you mean by the equation that you have written? What is "exp" representing?

 

[arildno]

Probably, he means:
$$z=e^{-z}$$

 

[quasar987]

That's a fun problem. I say start by defining z=a+ib where a and b are real numbers and also note z=|z|exp(i$\Phi$) the polar form of z, so that you're looking for the solutions to

$$|z|e^{i\Phi}=e^{-a-ib}$$

And use the fact that two complex numbers are equal iff their modulus are the same and their polar angle are the same up to a difference of $2n\pi$, $n\in\mathbb{Z}$.