(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

For [tex]u(x,y)=e^{-y}(x\sin(x)+y\cos(x))[/tex] find a harmonic conjugate [tex]v(x,y)[/tex] and express the analytic function [tex]f=u +iv[/tex] as a function of z alone (where [tex]z=x+iy[/tex]0

2. Relevant equations

The Cauchy Riemann equations [tex]u_x=v_y[/tex] and [tex]u_y=-v_x[/tex]

and possibly:

[tex]sin(x) = \frac{e^{ix}-e^{-ix}}{2i}[/tex]

[tex]cos(x) = \frac{e^{ix}+e^{-ix}}{2}[/tex]

3. The attempt at a solution

I've been toying around with algebra and I've been able to reduce up to [tex]u=\frac {e^{-y+ix}(y-ix)+e^{-y-ix}(y+ix)}{2}[/tex] but I can't get past here. I have also noted that if I divide by i, u(x,y) becomes u(y,-x) so I can replace these variables and I have [tex] u/i = e^{x}(y\sin(y)-x\cos(y)[/tex] which is, as someone on another forum pointed out, the real part of [tex]-ze^{z}[/tex]. So I suppose I could write [tex]f=Re(-ze^{iz})i + iv[/tex] no?

But how should I go about finding v? I am familiar with the straightforward procedure of finding u_x, integrating w.r.t v, then finding v_x and setting it equal to u_y... but I can't seem to simplify u to the point of easy differentiation. I don't know if I should just try to brute force it, or if that's even possible. It seems like u should simplify more.

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# Homework Help: Complex Analysis: Harmonic Conjugates

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