# 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?