# [SOLVED] Complex Problem

• Feb 4th 2009, 03:08 PM
ronaldo_07
[SOLVED] Complex Problem
Consider the complex function

f(z) = $\displaystyle e^{-iz}$.

(a) Express f(z) in the form u(x, y) + iv(x, y), give u and v.

(b) Use the Cauchy-Riemann equations and the existence and continuity of the partial derivatives of u and v to show that f '(z) exists for all z.

(c) Using your results from b), find an expression for f '(z) in terms of z.

Help with this question please, Thanks.
• Feb 4th 2009, 03:18 PM
Plato
$\displaystyle e^{iz} = e^{\left( { - y + ix} \right)} = \left[ {e^{ - y} \cos (x)} \right] + i\left[ {e^{ - y} \sin (x)} \right]$
Now, you finish the other parts.
• Feb 5th 2009, 04:36 AM
ronaldo_07
Sorry I made a mistake its ie^{-iz}
• Feb 5th 2009, 09:05 AM
ThePerfectHacker
Quote:

Originally Posted by ronaldo_07
Sorry I made a mistake its ie^{-iz}

So? If you know what $\displaystyle e^{-iz}$ is (what Plato did) then it is easy to find $\displaystyle ie^{-iz}$.
• Feb 5th 2009, 09:47 AM
ronaldo_07
$\displaystyle ie^{iz} = e^{\left( { - y + ix} \right)} = \left[ {e^{ - iy} \cos (x)} \right] + i\left[ {e^{ - iy} \sin (x)} \right]$

is this correct?
• Feb 5th 2009, 12:40 PM
ronaldo_07
Quote:

Originally Posted by Plato
$\displaystyle e^{iz} = e^{\left( { - y + ix} \right)} = \left[ {e^{ - y} \cos (x)} \right] + i\left[ {e^{ - y} \sin (x)} \right]$
Now, you finish the other parts.

should it not be y-xi?
• Feb 5th 2009, 01:19 PM
Plato
Quote:

Originally Posted by ronaldo_07
$\displaystyle ie^{iz} = {\color{red}\large i}e^{\left( { - y + ix} \right)} = \left[ {{\color{red}\large i}e^{ - iy} \cos (x)} \right] {\color{red}\large -} \left[ {e^{ - iy} \sin (x)} \right]$

You missed a sign and two i's. See the red above.

Quote:

Originally Posted by ronaldo_07
should it not be y-xi?

No indeed!
$\displaystyle i(x + yi) = ix - y$
• Feb 5th 2009, 01:26 PM
ronaldo_07
Quote:

Originally Posted by ronaldo_07
$\displaystyle ie^{-iz} = e^{\left( { - y + ix} \right)} = \left[ {e^{ - iy} \cos (x)} \right] + i\left[ {e^{ - iy} \sin (x)} \right]$

is this correct?

The - sign was there when i wrote the question origionally.