# Complex variables

1. Oct 19, 2009

### sara_87

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

let:
f(z)=u(x,y)+iv(x,y)
I want to express the following function like the one above:
$$f(z)=cos(z)\equiv=\frac{1}{2}(e^{iz}+e^{-iz})$$

2. Relevant equations

(i=sqrt(-1))
f(z) is a complex function

3. The attempt at a solution

$$f(z)=cos(z)\equiv=\frac{1}{2}(e^{iz}+e^{-iz})$$
$$=\frac{1}{2}(e^{i(x+iy)}+e^{-i(x+iy)})=\frac{1}{2}(e^{x-iy}+e^{-xi+y})$$

this is where i stopped because i got stuck. any help please?

2. Oct 19, 2009

### Staff: Mentor

Check your work in the expression before the last =.
i(x + iy) = ix + i2y = -y + ix. The other one is OK.

So one of your expressions will be e-y + ix = e-yeix = e-y(cos x + i sin x) = e-y cos x + i e-y sin x. Do about the same thing to the other expression and add the two together.