# Complex Conjugate

• Jan 4th 2008, 02:57 AM
Jason Bourne
Complex Conjugate
I have the following question:

Given that

$\displaystyle u(x,y) = x~cosx~coshy + y~sinx~sinhy$

Find a funvtion $\displaystyle v(x,y)$ such that

$\displaystyle w(z) = u(x,y) + iv(x,y)$

is an analytic function of $\displaystyle z$. Find $\displaystyle w(z)$ explicitly in terms of $\displaystyle z$.
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I know how to do these type of questions by using Cauchy-Riemann equations but I was wondering if there was a way of making this one simpler as it becomes very tedious. Is there an identity I can use to make the process much simpler?
• Jan 4th 2008, 06:11 AM
ThePerfectHacker
Quote:

Originally Posted by Jason Bourne
$\displaystyle u(x,y) = x~cosx~coshy + y~sinx~sinhy$

You need to find $\displaystyle u(x,y)+iv(x,y)$ so that $\displaystyle \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\displaystyle \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}$.

Can you solve that?
• Jan 4th 2008, 09:01 AM
Jason Bourne
Quote:

Originally Posted by ThePerfectHacker
You need to find $\displaystyle u(x,y)+iv(x,y)$ so that $\displaystyle \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\displaystyle \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}$.

Can you solve that?

Yes I know how to solve it, but the problem involved lots of calculus and it got all messed up and way too long, so I was thinking that there must be an identity so that I could simplify the problem and make it more bearable so that im not insane by the end of the question.