10.1.32 Integral of a complex variable

Problem: Using the Cauchy theorem for a multiply connected domain and the Cauchy integral formula, calculate the integral: \[ \int_{L} f(z) d z \text {, where } f(z)=\frac{e^{3 z}}{z^{2}+\frac{\pi^{2}}{9}}, \quad L:|z|=2 . \]