Let $f$ be the function defined by $f ( x ) = e ^ { 2 x }$. Which of the following is the Maclaurin series for $f ^ { \prime }$, the derivative of $f$?
(A) $1 + x + \frac { x ^ { 2 } } { 2 ! } + \frac { x ^ { 3 } } { 3 ! } + \cdots + \frac { x ^ { n } } { n ! } + \cdots$
(B) $2 + 2 x + \frac { 2 x ^ { 2 } } { 2 ! } + \frac { 2 x ^ { 3 } } { 3 ! } + \cdots + \frac { 2 x ^ { n } } { n ! } + \cdots$
(C) $1 + 2 x + \frac { ( 2 x ) ^ { 2 } } { 2 ! } + \frac { ( 2 x ) ^ { 3 } } { 3 ! } + \cdots + \frac { ( 2 x ) ^ { n } } { n ! } + \cdots$
(D) $2 + 2 ( 2 x ) + \frac { 2 ( 2 x ) ^ { 2 } } { 2 ! } + \frac { 2 ( 2 x ) ^ { 3 } } { 3 ! } + \cdots + \frac { 2 ( 2 x ) ^ { n } } { n ! } + \cdots$

Show that for all $a \in \mathbb{R}$, $$\int_0^a \sin(x^2) \mathrm{d}x = \sum_{n=0}^{+\infty} (-1)^n \frac{a^{4n+3}}{(2n+1)!(4n+3)}$$

Show that for all $a \in \mathbb { R }$, $$\int _ { 0 } ^ { a } \sin \left( x ^ { 2 } \right) \mathrm { d } x = \sum _ { n = 0 } ^ { + \infty } ( - 1 ) ^ { n } \frac { a ^ { 4 n + 3 } } { ( 2 n + 1 ) ! ( 4 n + 3 ) }$$

Let $\sum _ { k \geqslant 0 } c _ { k } x ^ { k }$ be a power series with radius of convergence $R > 0$. We set $$\forall x \in ] - 1,1 [ , \quad g ( x ) = \sum _ { k = 0 } ^ { + \infty } x ^ { k }.$$ Show that $g$ is of class $\mathcal { C } ^ { \infty }$ on $] - 1,1 [$ and that $$\forall j \in \mathbb { N } , \quad \forall x \in ] - 1,1 [ , \quad g ^ { ( j ) } ( x ) = \frac { j ! } { ( 1 - x ) ^ { j + 1 } }.$$

We denote by $\mathcal{E}$ the set of functions $f : \mathbb{C} \rightarrow \mathbb{C}$ expandable as a power series with radius of convergence infinity, and $\omega(t) = e^{2i\pi t}$ for $t \in [0,1]$. Let $f \in \mathcal{E}$ whose power series expansion we denote $\sum a_{n} z^{n}$. Show that, for all $k \in \mathbb{Z}$: $$\int_{0}^{1} f(\omega(t)) \omega(t)^{-k} \,\mathrm{d}t = \begin{cases} a_{k} & \text{if } k \in \mathbb{N} \\ 0 & \text{otherwise} \end{cases}$$

Recall that $x$ is a fixed element of $]0;1[$. Show that:
$$\int _ { 0 } ^ { 1 } \frac { t ^ { x - 1 } } { 1 + t } \mathrm {~d} t = \sum _ { k = 0 } ^ { + \infty } \frac { ( - 1 ) ^ { k } } { k + x }$$

Give an example of a power series expansion of a rational function whose antiderivative is not the expansion of a rational function.
(The antiderivative of a power series $f(x) = \sum_{n=0}^{\infty} c_n x^n$ is defined as $\int_0^x f(t)\,dt \stackrel{\text{def}}{=} \sum_{n=0}^{\infty} \frac{c_n}{n+1} x^{n+1}$.)

Answer all the following questions.
I. Let a periodic function $f ( x )$ satisfy the condition $f ( x + \pi ) = f ( x - \pi )$. Find the Fourier series expansion of $f ( x )$ for each case, where $f ( x )$ is expressed as follows for the interval $- \pi \leq x \leq \pi$.
1. $f ( x ) = x \quad ( - \pi < x < \pi ) , \quad f ( - \pi ) = f ( \pi ) = 0$ 2. $f ( x ) = x ^ { 2 }$
For the Fourier series expansion, the following equations should be used.
$$\begin{aligned} & f ( x ) = \frac { a _ { 0 } } { 2 } + \sum _ { n = 1 } ^ { \infty } \left( a _ { n } \cos n x + b _ { n } \sin n x \right) \\ & a _ { 0 } = \frac { 1 } { \pi } \int _ { - \pi } ^ { \pi } f ( x ) \mathrm { d } x \\ & a _ { n } = \frac { 1 } { \pi } \int _ { - \pi } ^ { \pi } f ( x ) \cos n x \mathrm {~d} x \\ & b _ { n } = \frac { 1 } { \pi } \int _ { - \pi } ^ { \pi } f ( x ) \sin n x \mathrm {~d} x \end{aligned}$$
II. Consider the function shown in Figure 5.1 as
$$V ( t ) = A \left| \sin \left( \frac { \omega t } { 2 } \right) \right| \quad \left( \omega = \frac { 2 \pi } { T } \right) .$$
Note that $A$ and $T$ are positive real numbers. The complex Fourier series expansion of $V ( t )$ is given as
$$V ( t ) = - \frac { 2 A } { \pi } \sum _ { n = - \infty } ^ { \infty } \frac { 1 } { 4 n ^ { 2 } - 1 } e ^ { i n \omega t }$$
Let $I ( t )$ be the periodic solution that satisfies the ordinary differential equation
$$L \frac { \mathrm {~d} I ( t ) } { \mathrm { d } t } + R I ( t ) = V ( t )$$
Note that $L$ and $R$ are positive real numbers. Find the coefficient $C _ { n }$, when the complex Fourier series expansion of $I ( t )$ is expressed as
$$I ( t ) = \sum _ { n = - \infty } ^ { \infty } C _ { n } e ^ { i n \omega t }$$

Problem 5

I. We consider a continuous and absolutely integrable function $f ( t )$ of a real variable $t$ and denote the Fourier transform of the function $f ( t )$ as $\mathcal { F } \{ f ( t ) \}$. We define a function $F ( \omega )$ by the following formula:
$$F ( \omega ) = \mathcal { F } \{ f ( t ) \} = \int _ { - \infty } ^ { \infty } f ( t ) \exp ( - i \omega t ) \mathrm { d } t$$
where $\omega$ is a real variable and $i$ is the imaginary unit.

1. We define $g ( t ) = f ( a t )$ for a constant $a$ satisfying $a > 0$ and $$G ( \omega ) = \mathcal { F } \{ g ( t ) \} = \mathcal { F } \{ f ( a t ) \}$$ Express $G ( \omega )$ using the function $F$.
2. When $f ( t ) = \exp \left( - t ^ { 2 } \right)$ and $a = 2$, sketch the graph of the Fourier transformed functions $F ( \omega )$ and $G ( \omega )$ defined by Equation (2) as a function of $\omega$ to show the difference between them.
3. We define $h ( t ) = f ( t ) \exp ( - i b t )$ for a constant $b$ satisfying $b > 0$ and $$H ( \omega ) = \mathcal { F } \{ h ( t ) \} = \mathcal { F } \{ f ( t ) \exp ( - i b t ) \}$$ Express $H ( \omega )$ using the function $F$.
4. When $f ( t ) = \exp \left( - t ^ { 2 } \right)$ and $b = 2$, sketch the graph of the Fourier transformed functions $F ( \omega )$ and $H ( \omega )$ defined by Equation (3) as a function of $\omega$ to show the difference between them.

II. Let $N$ be a positive integer. We define a discrete Fourier transform $D _ { 1 } , \cdots , D _ { N }$ by the following formula:
$$D _ { m } = \frac { 1 } { \sqrt { N } } \sum _ { n = 1 } ^ { N } c _ { n } \exp \left( - i \frac { 2 \pi } { N } n m \right)$$
for a complex sequence $c _ { 1 } , \cdots , c _ { N }$. Here, $m$ is an integer satisfying $1 \leq m \leq N$.

1. Calculate $S \left( n , n ^ { \prime } \right)$ : $$S \left( n , n ^ { \prime } \right) = \frac { 1 } { N } \sum _ { m = 1 } ^ { N } \exp \left\{ i \frac { 2 \pi } { N } \left( n - n ^ { \prime } \right) m \right\}$$ Here, $n$ is an integer satisfying $1 \leq n \leq N$, and $n ^ { \prime }$ is an integer satisfying $1 \leq n ^ { \prime } \leq N$.
2. Let $U _ { m n }$ be a complex number satisfying $$D _ { m } = \sum _ { n = 1 } ^ { N } U _ { m n } c _ { n }$$ Show that the matrix $\mathbf { U } = \left[ U _ { m n } \right] _ { 1 \leq m \leq N , 1 \leq n \leq N }$ is a unitary matrix.
3. Derive an equation for the inverse discrete Fourier transform $c _ { n }$ from $D _ { 1 } , \cdots , D _ { N }$. Here, $n$ is an integer satisfying $1 \leq n \leq N$.
4. For any complex value $z , \bar { z }$ is the complex conjugate of $z$. We define $Q$ by $$Q = \sum _ { n = 1 } ^ { N } \left( \overline { c _ { n } } c _ { n + 1 } + \overline { c _ { n + 1 } } c _ { n } \right)$$ Express $Q$ in terms of $D _ { m }$ and $\overline { D _ { m } }$. Here, we impose the condition $c _ { N + 1 } = c _ { 1 }$.

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