I want to decide the domain and range of $e^z + z$ and find some properties of its inverse function such as multivalued function. And I guess the domain and range should be whole complex plane
Question 1: for the range I want to show it can reach any complex number. so $$e^z + z = e^{x+iy} + x + iy = (e^x \cos y + x) + i (e^x \sin y + y) = a + ib$$ for any $a$ and $b$, then $e^x \cos y + x = a$ and $e^x \sin y + y = b$, but then I dont know how to get the $x, y$ from this system of equations.
Question 2: How to get the inverse function of this function ? And is the inverse function multivalued function? Since if it is, then we should have some $e^z + z = e^w + w$, also means there exists different x and y such that $$e^{x_1} \cos y_1 + x_1 = e^{x_2} \cos y_2 + x_2 \space \text{and} \space e^{x_1} \sin y_1 + y_1 = e^{x_2} \sin y_2 + y_2$$ but I cannot find the way to do that. Thank you for any help.
Let us denote $V(w)$ the inverse of $z+e^z$, i.e. $V(w)$ is the (multivalued) function that gives solutions to $z+e^z=w$. Its range is the domain of $z+e^z$, which is the entire complex plane. Its domain is the range of $z+e^z$, but it is not obvious what it is. By the Little Picard theorem the range of an entire function is either the whole complex plane or the plane without a single point. For example, for $e^z$ that single point is $0$, it never takes that value.
For the inverse, define $f(s):=se^s$ then $e^{z+e^z}=f(e^z)$. So $z+e^z=w$ when $e^{w}=f(e^z)$, and finding $V$ is equivalent to inverting $f$. But the inverse of $f$ is well-known, it is a multivalued function called Lambert $W$ function, so $V(w)=\ln W(e^w)$. We can also get an alternative expression for $V$ in terms of $W$ that does not involve $\text{Ln}$. By definition of $W$ as the inverse, we have $e^w=W(e^w)e^{W(e^w)}$ or $e^{w-W(e^w)}=W(e^w)$. Taking the logarithms, $$V(w)=\ln\, W(e^w)=w-W(e^w).$$
Now we can resolve the domain question. Just like complex logarithm $\ln$ Lambert $W$ function has countably many branches. The principal branch is defined for all complex values and other branches only for non-zero values (also like $\ln$). But $e^w$ is never zero, so $W(e^w)$ is defined for all $w$ for any branch. In particular, this means that the range of $z+e^z$ is the whole complex plane, there is no exceptional point.