Equation with complex variable

1. Apr 12, 2010

sbashrawi

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. The attempt at a solution

2. Apr 12, 2010

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

3. Apr 13, 2010

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.

4. Apr 13, 2010

lanedance

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

5. Apr 13, 2010

lanedance

Last edited: Apr 13, 2010
6. Apr 13, 2010

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

7. Apr 14, 2010

HallsofIvy

Staff Emeritus
Then what was the statement of the question?

8. Apr 14, 2010

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?