Question

evatuate \int_{}^{c} \ f(z)dz, f(z)=\frac{z^{2}}{z+3} , c=\left | z \right |=1

Answer 1

Question 1. Evaluate $\int_{C}f(z)dz$, $f(z) = \frac{z^2}{z + 3}$, $C = \{|z| = 1\}$.

Solution. The function $f(z)$ has the only one singular point in $\mathbb{C}$: $z_0 = -3$. Since $|z_0| = |-3| = 3 > 1$, we conclude that this point does not belong to the domain $D_{\varepsilon} = \{|z| < 1 + \varepsilon\}$ for some small $\varepsilon > 0$. Hence, $f$ is holomorphic in $D_{\varepsilon}$. Since $C \subset D_{\varepsilon}$, by Cauchy theorem the integral of $f$ along $C$ is zero. Thus, $\int_{C} f(z) dz = 0$.

Answer: $\int_{C} f(z) dz = 0$.