<s>[INST] Find the harmonic conjugate Let 

$$u(x,y)=x^3+ax^2y+bxy^2+2y^3$$
  Be a harmonic function and $v(x,y)$ be its harmonic conjugate .If $v(0,0)=1$ then find the 
$$|a+b+v(1,1)|$$

The solution i tried-since $u$ is harmonic so by laplace equation we get
$$6x+2ay+2bx+12y=0$$
By comparing coefficients we get $a=-6 $ and $b=-3$ so we get 
$$u(x,y)=x^3-6x^2y-3xy^2+2y^3$$
Next i am not getting how to proceed please give me a hint
Thankyou.
 [/INST] $$u(x,y) =x^3 -6x^2 y -3xy^2 +2y^3 =(x^3 -3xy^2 ) +2 (y^3 -3x^2 y) =Re( (x+iy)^3 ) - Im(2 (x+iy)^3 )= Re z^3 -2Im z^3 == Re [(1-2i)z^3 ]$$
Hence the conjugate is equal to
$$v(x,y)=Im [(1-2i) z^3 ]$$
$$|a+b +v(1,1) |=|-9 +Im [(1-2i) (1+i)^3]|=|-9 +6|=3$$
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