Complec potential and streamfunction

[MaxManus]

Homework Statement

The complex potential is:
W = Uz + (m/2pi) [ln(z + a) − ln(z − a)]
where z = x +yi
Find the streamfunction

The Attempt at a Solution

The streamfunction is the complex part of the complex potential

Psi = uy + (m/2pi)(ln(y+a) - ln(y-a))
= uy + (m/2pi)ln((y+a)/(y-a))

,but the solution says:
psi = uy - $$\frac{m}{2pi}$$arctan((2ay)/(x^2+y^2-a^2))

What am I doing wrong?

I have tried comparing them with a = 0.5, x = 1, y =1 and do not get the same answer.

 

[KataKoniK]

Hello! Your attempt at finding the streamfunction is incorrect. The correct expression for the streamfunction is:

Psi = uy + (m/2pi)arctan((2ay)/(x^2+y^2-a^2))

To understand why, let's first recall the definition of the streamfunction:

Psi = -Im(W)

where W is the complex potential. In your attempt, you have only considered the imaginary part of the second term in the complex potential. However, you have ignored the imaginary part of the first term, which is Uz = U(x+iy) = Ux - iUy. Thus, the correct expression for the streamfunction should be:

Psi = Uy + (m/2pi)arctan((2ay)/(x^2+y^2-a^2))

which matches the solution given. I hope this helps clarify any confusion.