Problem 1

I. Find the general solutions $y ( x )$ for the following differential equations:

1. $\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { y } { x } = \left( \frac { y } { x } \right) ^ { 3 }$,
2. $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 6 [ y + \cos ( 3 x ) ] = 0$.

II. Consider the curve $C$ given by the following polar equation in the polar coordinate system $( r , \theta )$ with the origin $O$ on the $x y$-orthogonal coordinate plane as the pole, and the positive part of the $x$-axis as the starting line:
$$r = 2 + \cos \theta \quad ( 0 \leq \theta < 2 \pi )$$

1. Calculate the area of the region enclosed by the curve $C$.
2. Consider the tangent line at the point $( r , \theta ) = \left( \frac { 4 + \sqrt { 2 } } { 2 } , \frac { \pi } { 4 } \right)$ on the curve $C$. Find the slope of this tangent line in the $x y$-orthogonal coordinate system.

Problem 2

For a square matrix $\boldsymbol { A } , e ^ { \boldsymbol { A } }$ is defined as:
$$e ^ { A } = \boldsymbol { E } + \sum _ { k = 1 } ^ { \infty } \frac { 1 } { k ! } \boldsymbol { A } ^ { k } ,$$
where $\boldsymbol { E }$ is the identity matrix and $e$ is the base of natural logarithm.
I. Let $\boldsymbol { A }$ be a $3 \times 3$ square matrix which can be diagonalized by a regular matrix $\boldsymbol { P }$, i.e., $\boldsymbol { A } = \boldsymbol { P } \boldsymbol { D } \boldsymbol { P } ^ { - 1 }$, where $\boldsymbol { D }$ is a diagonal matrix:
$$\boldsymbol { D } = \left( \begin{array} { c c c } \lambda _ { 1 } & 0 & 0 \\ 0 & \lambda _ { 2 } & 0 \\ 0 & 0 & \lambda _ { 3 } \end{array} \right)$$
Here, $\lambda _ { 1 } , \lambda _ { 2 }$, and $\lambda _ { 3 }$ are complex numbers. Prove the following equation:
$$e ^ { \boldsymbol { A } } = \boldsymbol { P } \left( \begin{array} { c c c } e ^ { \lambda _ { 1 } } & 0 & 0 \\ 0 & e ^ { \lambda _ { 2 } } & 0 \\ 0 & 0 & e ^ { \lambda _ { 3 } } \end{array} \right) \boldsymbol { P } ^ { - 1 }$$
II. Let $\boldsymbol { A } = \left( \begin{array} { c c c } - 1 & 4 & 4 \\ - 5 & 8 & 10 \\ 3 & - 3 & - 5 \end{array} \right)$.

1. Find the regular matrix $\boldsymbol { P }$ and the diagonal matrix $\boldsymbol { D }$ such that $\boldsymbol { A } = \boldsymbol { P } \boldsymbol { D } \boldsymbol { P } ^ { - 1 }$.
2. Calculate $e ^ { \boldsymbol { A } }$.

III. Consider $\boldsymbol { A } = \left( \begin{array} { c c c } 0 & - x & 0 \\ x & 0 & 0 \\ 0 & 0 & 1 \end{array} \right) , \boldsymbol { B } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 1 \end{array} \right)$ and $\boldsymbol { a } = \left( \begin{array} { l } 1 \\ 1 \\ e \end{array} \right)$, where $x$ is a real number. In the following, the transpose of a vector $\boldsymbol { v }$ is denoted by $\boldsymbol { v } ^ { T }$.

1. Express the sum of the eigenvalues of $e ^ { \boldsymbol { A } }$ using $e$ and $x$.
2. Let $\boldsymbol { C } = \boldsymbol { B } e ^ { \boldsymbol { A } }$. Find the minimum and maximum values of $\frac { \boldsymbol { y } ^ { T } \boldsymbol { C } \boldsymbol { y } } { \boldsymbol { y } ^ { T } \boldsymbol { y } }$ for a real three-dimensional vector $\boldsymbol { y } ( \boldsymbol { y } \neq \mathbf { 0 } )$.
3. Let $f ( \boldsymbol { z } ) = \frac { 1 } { 2 } \boldsymbol { z } ^ { T } \boldsymbol { C } \boldsymbol { z } - \boldsymbol { a } ^ { T } \boldsymbol { z }$ for a real three-dimensional vector $\boldsymbol { z } = \left( \begin{array} { c } z _ { 1 } \\ z _ { 2 } \\ z _ { 3 } \end{array} \right)$ and $\boldsymbol { C }$ in III.2. Find $\sqrt { z _ { 1 } ^ { 2 } + z _ { 2 } ^ { 2 } + z _ { 3 } ^ { 2 } }$ for $\boldsymbol { z }$ such that $\frac { \partial f ( \boldsymbol { z } ) } { \partial z _ { 1 } } = \frac { \partial f ( \boldsymbol { z } ) } { \partial z _ { 2 } } = \frac { \partial f ( \boldsymbol { z } ) } { \partial z _ { 3 } } = 0$.

Problem 3

In the following, $z$ is a complex number and $i$ is the imaginary unit. Answer the following questions on the complex function
$$f ( z ) = \frac { \cot z } { z ^ { 2 } }$$
Here, $\cot z = \frac { 1 } { \tan z }$. For a positive integer $m$, we define
$$D _ { m } = \lim _ { z \rightarrow 0 } \frac { \mathrm {~d} ^ { m } } { \mathrm {~d} z ^ { m } } ( z \cot z )$$
If necessary, you may use $D _ { 2 } = - \frac { 2 } { 3 }$ and
$$\lim _ { z \rightarrow n \pi } \frac { z - n \pi } { \sin z } = ( - 1 ) ^ { n }$$
for an integer $n$.
I. Find all poles of $f ( z )$. Also, find the order of each pole.
II. Find the residue of each pole found in I.
III. Let $M$ be a positive integer and set $R = \pi ( 2 M + 1 )$. As shown in Fig.1, for a parameter $t$ that ranges in the interval $- \frac { R } { 2 } \leq t \leq \frac { R } { 2 }$, consider the following four line segments $C _ { k } ( k = 1,2,3,4 )$:
$$\begin{aligned} & C _ { 1 } : z ( t ) = \frac { R } { 2 } + i t \\ & C _ { 2 } : z ( t ) = - t + i \frac { R } { 2 } \\ & C _ { 3 } : z ( t ) = - \frac { R } { 2 } - i t \\ & C _ { 4 } : z ( t ) = t - i \frac { R } { 2 } \end{aligned}$$
The initial point of each line segment corresponds to $t = - \frac { R } { 2 }$ and the terminal point corresponds to $t = \frac { R } { 2 }$. For each complex integral $I _ { k } = \int _ { C _ { k } } f ( z ) \mathrm { d } z$ along $C _ { k } ( k = 1,2,3,4 )$, find $\lim _ { M \rightarrow \infty } I _ { k }$.
IV. Let $C$ be the closed loop composed of the four line segments $C _ { 1 } , C _ { 2 } , C _ { 3 }$, and $C _ { 4 }$ in III. By applying the residue theorem to the complex integral $I = \oint _ { C } f ( z ) \mathrm { d } z$ along the closed loop $C$, find the value of the following infinite series:
$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } }$$
V. $f ( z )$ is now replaced with the complex function
$$g ( z ) = \frac { \cot z } { z ^ { 2 N } }$$
where $N$ is a positive integer. By following the steps in I-IV, express the following infinite series in terms of $D _ { m }$:
$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 N } }$$

Problem 4

In the three-dimensional orthogonal coordinate system $x y z$, consider the surface $S$ defined by the following equation:
$$\left( \begin{array} { c } x ( \theta , \phi ) \\ y ( \theta , \phi ) \\ z ( \theta , \phi ) \end{array} \right) = \left( \begin{array} { c c c } \cos \theta & - \sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{array} \right) \left( \begin{array} { c } \cos \phi + 2 \\ 0 \\ \sin \phi \end{array} \right)$$
where $\theta$ and $\phi$ are parameters of the surface $S$, and $0 \leq \theta < 2 \pi , 0 \leq \phi < 2 \pi$. Let $V$ be the region surrounded by the surface $S$, and let $W$ be the region satisfying the inequality $x ^ { 2 } + y ^ { 2 } \leq 4$. Answer the following questions for the surface $S$.
I. Find the unit normal vector oriented inward the region $V$ at the point $P \left( \begin{array} { c } \frac { 1 } { \sqrt { 2 } } \\ \frac { 1 } { \sqrt { 2 } } \\ 0 \end{array} \right)$ on the surface $S$.
II. Find the area of the surface $S$ included in the region $W$.
III. Find the overlapping volume created by the two regions $V$ and $W$.
IV. Consider the three-dimensional curve $C$ on the surface $S$, which is defined by setting $\theta = \phi$. Find the curvature of the curve $C$ at the point $Q \left( \begin{array} { l } 0 \\ 2 \\ 1 \end{array} \right)$ on the curve $C$. Note that, in general, given a three-dimensional curve defined by $c ( t ) = \left( \begin{array} { c } x ( t ) \\ y ( t ) \\ z ( t ) \end{array} \right)$ represented by a parameter $t$, the curvature $\kappa ( t )$ of the curve at the point $c ( t )$ on the curve is given by the following equation:
$$\kappa ( t ) = \frac { \left| \frac { \mathrm { d } c ( t ) } { \mathrm { d } t } \times \frac { \mathrm { d } ^ { 2 } c ( t ) } { \mathrm { d } t ^ { 2 } } \right| } { \left| \frac { \mathrm { d } c ( t ) } { \mathrm { d } t } \right| ^ { 3 } }$$

Problem 5

Consider a function $f ( t )$ of a real number $t$, where $| f ( t ) |$ and $| f ( t ) | ^ { 2 }$ are integrable. Let $F ( \omega ) = \mathcal { F } [ f ( t ) ]$ denote the Fourier transform of $f ( t )$. It is defined as
$$F ( \omega ) = \mathcal { F } [ f ( t ) ] = \int _ { - \infty } ^ { \infty } f ( t ) \exp ( - i \omega t ) \mathrm { d } t$$
where $\omega$ is a real number and $i$ is the imaginary unit. Then, the following equation is satisfied:
$$\int _ { - \infty } ^ { \infty } | F ( \omega ) | ^ { 2 } \mathrm {~d} \omega = 2 \pi \int _ { - \infty } ^ { \infty } | f ( t ) | ^ { 2 } \mathrm {~d} t$$
Also, let $R _ { f } ( \tau )$ denote the autocorrelation function of $f ( t )$. It is defined as
$$R _ { f } ( \tau ) = \int _ { - \infty } ^ { \infty } f ( t ) f ( t - \tau ) \mathrm { d } t$$
where $\tau$ is a real number.
I. Consider a case where $f ( t )$ is defined as follows:
$$f ( t ) = \begin{cases} \cos ( a t ) & \left( | t | \leq \frac { \pi } { 2 a } \right) \\ 0 & \left( | t | > \frac { \pi } { 2 a } \right) \end{cases}$$
Here, $a$ is a positive real constant. Find the followings:

1. $F ( \omega )$,
2. $R _ { f } ( \tau )$,
3. $\mathcal { F } \left[ R _ { f } ( \tau ) \right]$.

II. Find the values of the following integrals. Here, you may use the results of I.

1. $\int _ { - \infty } ^ { \infty } \frac { \cos ^ { 2 } \frac { \pi x } { 2 } } { \left( x ^ { 2 } - 1 \right) ^ { 2 } } \mathrm {~d} x$,
2. $\int _ { - \infty } ^ { \infty } \frac { \cos ^ { 4 } \frac { \pi x } { 2 } } { \left( x ^ { 2 } - 1 \right) ^ { 4 } } \mathrm {~d} x$.

Problem 6

Consider the following procedure that generates a sequence of random variables that take the value 0 or 1. For an integer $n \geq 1$, we denote the $n$-th random variable of a sequence generated by the procedure as $X _ { n }$.

• $X _ { 1 }$ becomes $X _ { 1 } = 0$ with a probability $\frac { 2 } { 3 }$ and $X _ { 1 } = 1$ with a probability $\frac { 1 } { 3 }$.
• For an integer $n = 1,2 , \ldots$ in order, the following is repeated until the procedure terminates:
• The procedure terminates with a probability $p ( 0 < p < 1 )$ if $X _ { n } = 0$ and with a probability $q ( 0 < q < 1 )$ if $X _ { n } = 1$. Here, $p$ and $q$ are constants.
• If the procedure does not terminate as above, $X _ { n + 1 }$ becomes $X _ { n + 1 } = 0$ with a probability $\frac { 2 } { 3 }$ and becomes $X _ { n + 1 } = 1$ with a probability $\frac { 1 } { 3 }$.

When the procedure terminates at $n = \ell$, a sequence of length $\ell$, composed of random variables $\left( X _ { 1 } , \ldots , X _ { \ell } \right)$, is generated, and no further random variables are generated. Answer the following questions.
I. For an integer $k \geq 1$, consider the following matrix:
$$\boldsymbol { P } _ { k } = \left( \begin{array} { c c } \operatorname { Pr } \left( X _ { n + k } = 0 \mid X _ { n } = 0 \right) & \operatorname { Pr } \left( X _ { n + k } = 1 \mid X _ { n } = 0 \right) \\ \operatorname { Pr } \left( X _ { n + k } = 0 \mid X _ { n } = 1 \right) & \operatorname { Pr } \left( X _ { n + k } = 1 \mid X _ { n } = 1 \right) \end{array} \right)$$
Here, $\operatorname { Pr } ( A \mid B )$ is the conditional probability of an event $A$ given that an event $B$ has occurred.

1. Express $\boldsymbol { P } _ { 1 }$ and $\boldsymbol { P } _ { 2 }$ using $p$ and $q$.
2. Express $\boldsymbol { P } _ { 3 }$ using $\boldsymbol { P } _ { 1 }$.
3. $\boldsymbol { P } _ { k }$ can be expressed as $\boldsymbol { P } _ { k } = \gamma _ { k } \boldsymbol { P } _ { 1 }$ using a real number $\gamma _ { k }$. Find $\gamma _ { k }$.

II. For an integer $m \geq 2$, find the respective probabilities that $X _ { m } = 0$ and $X _ { m } = 1$, given that the procedure does not terminate before $n = m$.
III. Find the expected value and the variance of the length of the sequence, $\ell$, generated by the procedure. If necessary, you may use $\sum _ { m = 1 } ^ { \infty } m r ^ { m - 1 } = \frac { 1 } { ( 1 - r ) ^ { 2 } }$ and $\sum _ { m = 1 } ^ { \infty } m ^ { 2 } r ^ { m - 1 } = \frac { 1 + r } { ( 1 - r ) ^ { 3 } }$ for a real number $r$ whose absolute value is smaller than 1.
IV. For an integer $k \geq 1$, find the probability $\operatorname { Pr } \left( X _ { n } = 0 \mid X _ { n + k } = 1 \right)$.