- #1

usn7564

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## Homework Statement

[tex]u(x,y) = sin(x^2-y^2)cosh(2xy)[/tex]

Find a function f(x+iy) = u(x,y) + iv(x,y), where v(x,y) is a real function, such that f is analytical in all of the complex plane. Find all such f.

**The attempt at a solution**

I expanded using Euler's for sin and cosh which gave me

[tex]u(x,y) = \frac{1}{4i} \Bigg( e^{i(x^2-y^2) +2xy} + e^{i(x^2-y^2)-2xy} - e^{-i(x^2-y^2)+2xy} - e^{-i(x^2-y^2)-2xy}\Bigg)[/tex]

Which I can simplify to

[tex]\frac{1}{4i}\Bigg(e^{i\bar{z}^2} - e^{-i\bar{z}^2} + e^{iz^2}-e^{-iz^2}\Bigg)[/tex]

I don't think the simplification will be necessary to solve it but it shows that u depends on the conjugate of z. This makes me assume v has to have the same terms depending on the conjugate terms, but negative to cancel them out (else f cannot be analytic).

And that's as far as I've come. Simply looking at u(x,y) makes me pretty sure I shouldn't solve it 'conventionally' using the Cauchy-Riemann equations, got to be some trick to it. Though I don't know how to proceed.

The answer should be

[tex]v(x,y) = sinh(2xy)cos(x^2-y^2) + C \implies f = sin(z^2) + iC[/tex]