Question

Show that u= 12 log⁡(x2+y2) is Harmonic and find its Harmonic conjugate function​

Answer 1

Answer: What you got in the first step is correct. That is, when we integrate

∂v∂y=xx2+y2=1x1+(yx)2=∂∂y(yx)1+(yx)2=∂∂y(tan−1(yx)),

we obtain v=tan−1(yx). Note that in your second step:

∂v∂x=−yx2+y2=−yx21+(yx)2=∂∂x(yx)1+(yx)2=∂∂x(tan−1(yx)).

Integrate it, again we get v=tan−1(yx).

Explanation: