todai-math 2022 QI.2

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 $z$.
Consider the definite integral $I_2$ expressed as
$$I_2 = \int_0^{2\pi} \frac{d\theta}{10 + 8\cos\theta}$$
2.1. Find a complex function $G(z)$ when $I_2$ is rewritten as an integral of a complex function as $$I_2 = \oint_{|z|=1} G(z) \mathrm{d}z$$ Note that the integration path is a unit circle centered at the origin on the complex plane oriented counterclockwise. Show the derivation process with your answer.
2.2. Find all singularities of $G(z)$.
2.3. Using the residue theorem, obtain $I_2$.

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

Consider the definite integral $I_2$ expressed as

$$I_2 = \int_0^{2\pi} \frac{d\theta}{10 + 8\cos\theta}$$

2.1. Find a complex function $G(z)$ when $I_2$ is rewritten as an integral of a complex function as
$$I_2 = \oint_{|z|=1} G(z) \mathrm{d}z$$
Note that the integration path is a unit circle centered at the origin on the complex plane oriented counterclockwise. Show the derivation process with your answer.

2.2. Find all singularities of $G(z)$.

2.3. Using the residue theorem, obtain $I_2$.

Paper Questions

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