Question

Obtain the harmonic conjugate v of the function u=2x(1-y)

Answer 1

Answer on Question #85332 – Math – Complex Analysis

Question

Obtain the harmonic conjugate $\mathbf{v}$ of the function $u = 2x(1 - y)$

Solution

We have

$$u_{x}(x, y) = \frac{\partial u(x, y)}{\partial x} = 2(1 - y) = \frac{\partial v(x, y)}{\partial y}$$

So $v(x, y) = 2y - y^2 + c(x)$, where $c(x)$ is an arbitrary function of $x$.

After that

$$\begin{array}{l} - \frac{\partial v(x, y)}{\partial x} = -c'(x) = \frac{\partial u(x, y)}{\partial y} = -2x, \\ c(x) = x^2 + C_1 \\ \end{array}$$

So $v(x) = 2y - y^2 + x^2 + C_1$.

Answer: $v(x, y) = 2y - y^2 + x^2 + C_1$.

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