# Math Help - analytic complex functions

1. ## analytic complex functions

I am supposed to prove that f(z) = e^(-x)e^(-iy) is analytic everywhere. I will be able to do it, I think, but I do not know how to break f(z) into the real and imaginary parts (u(x.y) + iv(x,y)). Can someone please help???

2. Hier.

3. ok, now I am supposed to find the derivative of the function. Do I just add the two partial derivatives?

4. It is equal to: u_x + i*v_x

5. The derivative of e^(-x)e^(-iy) equals u_x + i*v_x? Ok, I hadn't realized that.

6. Originally Posted by Hollysti
The derivative of e^(-x)e^(-iy) equals u_x + i*v_x? Ok, I hadn't realized that.
But remember to write it in standard form:
u(x,y)+iv(x,y)

In fact, instead we can write,
u_y-i*v_y
(Why?)

7. Eulers formula?

8. Originally Posted by Hollysti
Eulers formula?
No! The Cauchy-Riemann equations.

If f(z) is analytic then u_x = u_y and u_y = -v_x.

9. Oh, right. That makes sense. Thank you for taking time to explain that to me!