Equation with complex variable

[sbashrawi]

Homework Statement

Do you know how to find the solution of the equation:

a - z - exp(-z) = 0 , where a > 1 and z is a complex variable

Homework Equations

The Attempt at a Solution

 

[Gregg]

$$Xe^X=Y \iff W(Y)=X$$

For example

$$e^{-z}=a-z$$

try to show that

$$C=f(z)e^{f(z)}$$

where C is a constant and you will get the right answer

W(y) is Lambert W function

 

[sbashrawi]

I couldn't write it in the form :

C = f(x) exp (fx))

Also I don't see how this will give me the right answer.

 

[lanedance]

how about this
$$e^{-z}=a-z$$
$$-1= e^{z}(z-a)$$
$$-e^{-a}= e^{z-a}(z-a)$$

 

[lanedance]

not too sure about the W function bit though wiki has a bit on it
http://en.wikipedia.org/wiki/Lambert_W_function

do you have any more context/constraints on the question?

 

[sbashrawi]

So the solution is:

-e^(-a)=(z-a) e^(z-a)
W(-e^(-a) )=z-a
z=W(-e^(-a) )+a=-a+a=0

am I right or not?

If so, the answer according ot the question should be in the have plane Re z >= 0

and must be real.

What happen to the solution if a goes to 1.

from the statement of the question I can guess that my answer is not on the right way

 

[HallsofIvy]

Then what was the statement of the question?

 

[sbashrawi]

The full statement:

Let a >= 1 then show that the given equation has exactly one solution in the half plane Rez>= 0, and that solution is real. What happen to the solution if a goes to 1?