todai-math

Papers (28)
2025
problems1-3 3 todai-engineering-math 6
2024
problem1 1 problem2 1 problem3 1 todai-engineering-math 6
2023
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2022
problem1 1 problem2 1 problem3 1 todai-engineering-math__paper1 2 todai-engineering-math__paper2 2 todai-engineering-math__paper3 7
2021
todai-engineering-math__paper1 6 todai-engineering-math__paper2 3 todai-engineering-math__paper3 3
2020
todai-engineering-math 6
2019
todai-engineering-math 6
2018
ist 3 todai-engineering-math 6
2017
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2016
todai-engineering-math 6 translated 3
2015
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2021 todai-engineering-math__paper3

3 maths questions

QI Complex Numbers Argand & Loci Complex Number Mapping and Image Point Determination View
Consider the complex function $M(z) = \frac{mz}{mz - z + 1}$, where $m$ is a complex number such that $|m| = 1$ and $m \neq 1$.
  1. Find all fixed points of $M(z)$ which satisfy $M(z) = z$.
  2. Express the derivative of $M(z)$ at $z = 0$ by using $m$.
  3. Find $m$ for which the circle $\left| z - \frac{1-i}{2} \right| = \frac{1}{\sqrt{2}}$ on the complex $z$ plane is mapped onto the real axis through $M(z)$.
QII Complex Numbers Argand & Loci Locus Identification from Modulus/Argument Equation View
Deduce the conditions for $z$ and, on the complex $z$ plane, draw the area of $z$ in which the imaginary part of the complex function $J(z) = e^{-i\alpha} z + e^{i\alpha} z^{-1}$ is positive. Here, $\alpha$ is a real number and $0 < \alpha < \pi/2$.
QIII Complex numbers 2 Contour Integration and Residue Calculus View
To calculate the definite integral $I = \int_{0}^{\infty} \frac{x^{\beta}}{(x^{2}+1)^{2}} \mathrm{~d}x$, consider the line integral of the complex function $f(z) = \frac{z^{\beta}}{(z^{2}+1)^{2}}$ on the complex plane. Here, $\beta$ is a real number and $0 < \beta < 1$. The closed integration path $C = C_{1} + C_{R} + C_{2} + C_{r}$ $(0 < r < 1 < R)$ is defined with semicircles and line segments as shown in Figure 3.1.
  1. Using the residue theorem, calculate the line integral $\oint_{C} f(z) \mathrm{d}z$.
  2. Express $\int_{C_{1}} f(z) \mathrm{d}z + \int_{C_{2}} f(z) \mathrm{d}z$ with the definite integral $\int_{r}^{R} \frac{x^{\beta}}{(x^{2}+1)^{2}} \mathrm{~d}x$.
  3. Obtain $\lim_{R \rightarrow \infty} \int_{C_{R}} f(z) \mathrm{d}z$.
  4. Obtain $\lim_{r \rightarrow 0} \int_{C_{r}} f(z) \mathrm{d}z$.
  5. Using the previous results, calculate the definite integral $I$.