Solutions to $e^z+z-a=0$ with $Re(z)>0$ and $a$ is a real number larger than 1 In an exercise I am required to show that the only solution to $e^z+z-a=0$ in the right half plane is a real one, where $a$ is fixed a real number larger than 1. I am not able to work it out. I tried to used the Argument Principle to compute the number of zeros of this function, but I cannot make my integral converge. Could anybody give me a hand?
Thanks very much!

I tried $a=2$.  Solutions of $e^z+z-2=0$ are $-W_k(e^2)+2$, where $W_k$ are the branches of the Lambert $W$ function.  The whole doubly-infinite sequence of solutions is in the right half-plane.  
Indeed, is there any value of $a>1$ where there is a unique solution in the right half-plane?
