How can we define $\Gamma$ (circulation) around a point vortex?

Question

I'm having trouble understanding something in a book that I'm reading (Chorin & Marsden intro to fluid mechanics):

Next we shall examine a model of incompressible, inviscid flow. We imagine the vorticity is concentrated in $N$ point vortices, (i.e. points where the vorticity field is singular), centred at $z_1,z_2,...z_N$ in the plane. The stream function of the j$^{\text{th}}$ vortex, ignoring other vortices for a moment, is

$$ \psi_j(z) = -\frac{\Gamma_j}{2\pi}\log(|z-z_j|).$$

My question is, how do we evaluate $\Gamma_j$? The only definition given for circulation that I've seen is one in terms of a line integral $$\Gamma = \oint_C \vec{u} \cdot d\vec{s}$$ around a closed curve $C$. How can we evaluate this if our "$C$" is really a single point?

I believe that the answer has to do with path deformation being valid, and so perhaps it doesn't even matter which closed curve we choose, but I'm not sure why that would be the case.

Follow-up optional question: Why does it follow that $\psi_j$ is defined in this way?

Answer 1

Your definition of $\Gamma(C)$ is correct and as you see it depends on C. Of course, you can deform as usual the curve C without changing circulation as soon as you are not crossing singular points. If you now start with your $\psi$ and integrate over a C around just one of the points $z_j$ you will get as a result $\Gamma_j$. This is because you can shrink C to the point, say take a small circumference around it with radius $\epsilon\to 0$, and then the flow will be tangential with modulus equal to $\Gamma_j/(2\pi \epsilon)$. In this limit the integral $\Gamma(C_\epsilon)$ will tend to $\Gamma_j$.