# analytic complex functions

• May 15th 2007, 03:31 PM
Hollysti
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???
• May 15th 2007, 03:41 PM
ThePerfectHacker
Hier.
• May 15th 2007, 05:43 PM
Hollysti
ok, now I am supposed to find the derivative of the function. Do I just add the two partial derivatives?
• May 15th 2007, 07:42 PM
ThePerfectHacker
It is equal to: u_x + i*v_x
• May 15th 2007, 07:47 PM
Hollysti
The derivative of e^(-x)e^(-iy) equals u_x + i*v_x? Ok, I hadn't realized that.
• May 15th 2007, 08:15 PM
ThePerfectHacker
Quote:

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?)
• May 15th 2007, 08:45 PM
Hollysti
Eulers formula?
• May 15th 2007, 08:52 PM
ThePerfectHacker
Quote:

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.
• May 15th 2007, 08:55 PM
Hollysti
Oh, right. That makes sense. Thank you for taking time to explain that to me!