# Math Help - given u, find v, the corresponding harmonic function of u.

1. ## given u, find v, the corresponding harmonic function of u.

Can someone check my working here? thanks

form the corresponding analytic function f(z)= u+iv
$u=4xy^3 - 4x^3y+x$
$u_x=4y^3-12x^2y+1=v_y$

$v=\int4y^3-12x^2y+1\cdot dx$
$v=y^4-6y^2x^2+y+f(x)$

$v_x=-12xy^2 + f'(x) = -u_y= -12xy^2 + 4x^3$
$f'(x)= 4x^3$
$f(x)=x^4 + c$
$v=y^4 - 6y^2x^2 + y + x^4 +c$

$f(z) = 4xy^3 - 4x^3y+x + i(y^4 - 6y^2x^2 + y + x^4 +c)$

2. You answer is correct. If you want to check it just verify the Cauchy Riemann equations hold. (They do)

3. You might just want to conclude by saying that $f(z) = iz^4+z+ic$.