todai-math 2022 QI.1

todai-math · Japan · todai-engineering-math__paper3 Complex numbers 2 Contour Integration and Residue Calculus

In Questions I.1 and I.2, $z$ denotes a complex number, $i$ the imaginary unit, and $|z|$ the absolute value of $z$.
Calculate the following integral, where $C$ is the closed path on the complex plane as shown in Figure 3.1.
$$I_1 = \oint_C \frac{z}{(z-i)(z-1)} \mathrm{d}z$$
(The contour $C$ is a closed path on the complex plane as depicted in Figure 3.1.)

In Questions I.1 and I.2, $z$ denotes a complex number, $i$ the imaginary unit, and $|z|$ the absolute value of $z$.

Calculate the following integral, where $C$ is the closed path on the complex plane as shown in Figure 3.1.

$$I_1 = \oint_C \frac{z}{(z-i)(z-1)} \mathrm{d}z$$

(The contour $C$ is a closed path on the complex plane as depicted in Figure 3.1.)

Paper Questions

QI.1 QI.2 QII.1 QII.2 QII.3 QII.4 QII.5