Find the conjugate harmonic function

Question

1) Find the conjugate harmonic function of:

$$u(x,y) = e^{4x}(x\cdot\cos (4y) - y\cdot\sin (4y)) $$

2) Express $w = u + iv$ as an analytic function of $z$ only ($z = x + iy$)

Solution:

I found $v$ to be the expression below $$v = e^{4x}(x\cdot\sin (4y) + y\cdot\cos (4y))$$ But I am not sure if this is correct.

Answer 1

![check m solution ][https://i.stack.imgur.com/Bq7tE.jpg]1[my solution is]

Answer 2

$u+iy=ze^{4z}$ by some simple algebraic manipulation. Since $ze^{4z}$ i analytic your answer computation of v is correct. Notw, however, that v is only determined up to a constant.