Example 2. The motion of a fluid is described by the complex potential $f(z)=\ln \operatorname{sh} \pi z$. Find the magnitude of the flow $N_{L}$ through the circle $2|z|=3$ and the circulation $\Gamma_{L}$ around it.
Solution. We find the derivative of the complex potential

$$
f^{\prime}(z)=\pi \operatorname{cth} \pi z
$$

Applying formula (2), we get

$$
\Gamma_{L}+i N_{L}=\pi \int_{|z|=3 / 2} \operatorname{cth} \pi z d z=\pi \int_{|z|=3 / 2} \frac{\operatorname{ch} \pi z}{\operatorname{sh} \pi z} d z
$$

The integrand has three simple poles inside the circle $2|z|=3$: $z_{1}=-i, z_{2}=0, z_{3}=i$. The residues of the function at these poles are

$$
\operatorname{res} f^{\prime}\left(z_{k}\right)=\pi \frac{1}{\pi}=1, \quad k=1,2,3
$$

Then

$$
\int_{|z|=3 / 2} \operatorname{cth} \pi z d z=6 \pi i
$$

Therefore,

$$
\Gamma_{L}+i N_{L}=6 \pi^{2} i .
$$

From this, the circulation $\Gamma_{L}=0$, and the flux $N_{L}=6 \pi^{2}$.