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I have been given that $u(x,y)$ is a harmonic function on a star shaped domain $D$. I have to show that it has harmonic conjugate $v(x,y)$ on same domain given up to additive constant by $$v(B)=\int_A^B\left(\frac{\partial u}{\partial x}dy-\frac{\partial u}{\partial y}dx\right)$$

My Solution:

So, if they are harmonic conjugate, they satisfy Cauchy Riemann equation. So, $$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\\\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$$ Simplifying these I get: $$v=\int\left(\frac{\partial u}{\partial x}dy-\frac{\partial u}{\partial y}dx\right)+c$$ without difficulty but the problem is I don't know if I have to show the existence of harmonic conjugate itself. My proof goes along the line of already assuming existence of Harmonic Conjugate.

Also, I don't understand why I need Star Shaped Domain, and what A and B refer to.

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You shouldn't assume that there exists a harmonic conjugate - you should prove it. In order to that you could show that the suggested $v(B)$ has the desired partial derivatives.

Also, you don't necessarily need $D$ to be a star domain, any simply connected domain would have been just as fine.

$A$ is a fixed point in $D$, and $B$ varies. This is similar to the case from calculus: for a fixed point $a$, $F(b)=\int_a^bf(x) \mathrm{d}x$ defines a primitive function of $f(x)$.

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