Complex Conjugate

**Jason Bourne** · January 4th 2008, 02:57 AM

I have the following question:

Given that

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

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

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

is an analytic function of $z$ . Find $w(z)$ explicitly in terms of $z$ .
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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?

**ThePerfectHacker** · January 4th 2008, 06:11 AM

Originally Posted by Jason Bourne

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

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

Can you solve that?

**Jason Bourne** · January 4th 2008, 09:01 AM

Originally Posted by ThePerfectHacker

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

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.

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