Show that $(B_{n})_{n \in \mathbb{N}}$ is the unique sequence of polynomials satisfying $$\begin{cases} B_{0} = 1 \\ \forall n \in \mathbb{N}^{*},\, B_{n}' = n B_{n-1} \\ \forall n \in \mathbb{N}^{*},\, \displaystyle\int_{0}^{1} B_{n}(t)\,\mathrm{d}t = 0 \end{cases}$$

Let $(H_{n})_{n \in \mathbb{N}}$ be the sequence of polynomials defined by: $\forall n \in \mathbb{N},\, H_{n}(X) = (-1)^{n} B_{n}(1-X)$, where $(B_n)_{n\in\mathbb{N}}$ are the Bernoulli polynomials satisfying $$\begin{cases} B_{0} = 1 \\ \forall n \in \mathbb{N}^{*},\, B_{n}' = n B_{n-1} \\ \forall n \in \mathbb{N}^{*},\, \displaystyle\int_{0}^{1} B_{n}(t)\,\mathrm{d}t = 0 \end{cases}$$ Show that for all $n \in \mathbb{N}$, $H_{n} = B_{n}$.

We propose to show by contradiction the property $\mathcal{P}$: $$\exists c > 0,\, \forall n \in \mathbb{N},\, \forall z \in \mathbb{C},\, \left(|z| = (2n+1)\pi \Rightarrow |\mathrm{e}^{z} - 1| \geqslant c\right).$$ We suppose that $\mathcal{P}$ is false. Show the existence of a sequence of natural integers $(n_{p})_{p \in \mathbb{N}}$ and a sequence of complex numbers $(z_{p})_{p \in \mathbb{N}}$ such that: $$\mathrm{e}^{z_{p}} \underset{p \rightarrow +\infty}{\rightarrow} 1 \quad \text{and} \quad \forall p \in \mathbb{N},\, |z_{p}| = (2n_{p}+1)\pi.$$

We suppose that the property $\mathcal{P}$: $$\exists c > 0,\, \forall n \in \mathbb{N},\, \forall z \in \mathbb{C},\, \left(|z| = (2n+1)\pi \Rightarrow |\mathrm{e}^{z} - 1| \geqslant c\right)$$ is false, and let $(n_p)_{p\in\mathbb{N}}$, $(z_p)_{p\in\mathbb{N}}$ be sequences such that $\mathrm{e}^{z_p} \to 1$ and $|z_p| = (2n_p+1)\pi$ for all $p$. For all $p \in \mathbb{N}$, we denote $a_{p} = \operatorname{Re}(z_{p})$ and $b_{p} = \operatorname{Im}(z_{p})$. Show that $a_{p} \underset{p \rightarrow +\infty}{\rightarrow} 0$ and $|z_{p}| - |b_{p}| \underset{p \rightarrow +\infty}{\rightarrow} 0$.

We suppose that the property $\mathcal{P}$: $$\exists c > 0,\, \forall n \in \mathbb{N},\, \forall z \in \mathbb{C},\, \left(|z| = (2n+1)\pi \Rightarrow |\mathrm{e}^{z} - 1| \geqslant c\right)$$ is false, and let $(n_p)_{p\in\mathbb{N}}$, $(z_p)_{p\in\mathbb{N}}$ be sequences such that $\mathrm{e}^{z_p} \to 1$ and $|z_p| = (2n_p+1)\pi$ for all $p$, with $a_p = \operatorname{Re}(z_p)$, $b_p = \operatorname{Im}(z_p)$. For all $p \in \mathbb{N}$, we denote $$\varepsilon_{p} = \begin{cases} +1 & \text{if } b_{p} \geqslant 0 \\ -1 & \text{if } b_{p} < 0 \end{cases}$$ By studying $\exp(z_{p} - \mathrm{i}\varepsilon_{p}|z_{p}|)$, reach a contradiction and conclude that $\mathcal{P}$ is true.

Let $w$ be a primitive cube root of unity. Simplify $\dfrac{1}{z-3} + \dfrac{1}{z-3w} + \dfrac{1}{z-3w^2}$.

The number of complex roots of the polynomial $z ^ { 5 } - z ^ { 4 } - 1$ which have modulus 1 is
(A) 0
(B) 1
(C) 2
(D) more than 2 .

Prove that there exists no complex number z such that $| \mathrm { z } | < 1 / 3$ and $$\sum _ { \mathrm { r } = 1 } { } ^ { \mathrm { n } } \text { at } 2 = 1 \quad \text { where } \mathrm { r } _ { \mathrm { r } } \text { as } 2 .$$

Let $w ( \operatorname { Im } w \neq 0 )$ be a complex number. Then, the set of all complex numbers $z$ satisfying the equation $w - \bar { w } z = k ( 1 - z )$, for some real number $k$, is
(1) $\{ z : z \neq 1 \}$
(2) $\{ z : | z | = 1 , z \neq 1 \}$
(3) $\{ z : z = \bar { z } \}$
(4) $\{ z : | z | = 1 \}$

If $z$ is a complex number, then the number of common roots of the equation $z^{1985} + z^{100} + 1 = 0$ and $z^3 + 2z^2 + 2z + 1 = 0$, is equal to:
(1) 1
(2) 2
(3) 0
(4) 3

Q62. Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $( \bar { z } ) ^ { 2 } + | z | = 0 , z \in \mathrm { C }$. Then $4 \left( \alpha ^ { 2 } + \beta ^ { 2 } \right)$ is equal to :
(1) 6
(2) 8
(3) 2
(4) 4

Let $z$ be a nonzero complex number, and let $\alpha = |z|$ and $\beta$ be the argument of $z$, where $0 \leq \beta < 2\pi$ (where $\pi$ is the circumference ratio). For any positive integer $n$, let the real numbers $x_{n}$ and $y_{n}$ be the real and imaginary parts of $z^{n}$, respectively. Select the correct options.
(1) If $\alpha = 1$ and $\beta = \frac{3\pi}{7}$, then $x_{10} = x_{3}$
(2) If $y_{3} = 0$, then $y_{6} = 0$
(3) If $x_{3} = 1$, then $x_{6} = 1$
(4) If the sequence $\langle y_{n} \rangle$ converges, then $\alpha \leq 1$
(5) If the sequence $\langle x_{n} \rangle$ converges, then the sequence $\langle y_{n} \rangle$ also converges

Problem 3
Consider a mapping $w = f ( z )$ of a domain $D$ on the complex $z$ plane to a domain $\Delta$ on the complex $w$ plane. Points on the complex $z$ and $w$ planes correspond to complex numbers $z = x + i y$ and $w = u + i v$, respectively. Here, $x , y$, $u$ and $v$ are real numbers, and $i$ is the imaginary unit.
I. Let $w = \sin z$.

1. Express $u$ and $v$ as functions of $x$ and $y$, respectively.
2. Suppose the domain $D _ { 1 } = \left\{ ( x , y ) \left\lvert \, 0 \leq x \leq \frac { \pi } { 2 } \right. , y \geq 0 \right\}$ on the $z$ plane is transformed to a domain on the $w$ plane. Show the transformed domain on the $w$ plane by drawing the transformed images corresponding to the three half-lines: $x = 0 , x = \frac { \pi } { 2 }$ and $x = c$ at $y \geq 0$ on the $z$ plane. Here, $c$ is a real constant on $0 < c < \frac { \pi } { 2 }$.

II. If a real function $g ( x , y )$ has continuous first and second partial derivatives and satisfies Laplace's equation $\frac { \partial ^ { 2 } g } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } g } { \partial y ^ { 2 } } = 0$ in a domain $\Omega$ on a plane, $g ( x , y )$ is said to be harmonic in $\Omega$.
Suppose that a function $f ( z ) = u ( x , y ) + i v ( x , y )$ is holomorphic in $D$ on the $z$ plane:

1. Show both $u ( x , y )$ and $v ( x , y )$ are harmonic in $D$ on the $z$ plane.
2. Suppose a function $h ( u , v )$ is harmonic in $\Delta$ on the $w$ plane, show a function $H ( x , y ) = h ( u ( x , y ) , v ( x , y ) )$ is harmonic in $D$ on the $z$ plane.

III. Suppose a function $h ( u , v )$ is harmonic in the domain $\Delta _ { 1 } = \{ ( u , v ) \mid u \geq 0 , v \geq 0 \}$ on the $w$ plane and satisfies the following boundary conditions:
$$\begin{aligned} & h ( 0 , v ) = 0 \quad ( v \geq 0 ) \\ & h ( u , 0 ) = 1 \quad ( u \geq 1 ) \\ & \frac { \partial h } { \partial v } ( u , 0 ) = 0 \quad ( 0 \leq u \leq 1 ) \end{aligned}$$

1. Let $z = \arcsin w$ and $H ( x , y ) = h ( u , v )$. Find the boundary conditions for $H ( x , y )$ corresponding to Equations (1), (2) and (3). Use the principal values of inverse trigonometric functions.
2. Find the function $H ( x , y )$ which satisfies the boundary conditions obtained in Question III.1.
3. Find $h ( u , 0 )$ on the interval $0 \leq u \leq 1$.

Problem 3
In the following, $z$ denotes a complex number and $i$ is the imaginary unit. The real part and the imaginary part of $z$ are denoted by $\operatorname { Re } ( z )$ and $\operatorname { Im } ( z )$, respectively.
I. Answer the following questions.

1. Give the solutions of $z ^ { 5 } = 1$ in polar form. Plot the solutions on the complex plane.
2. The mapping $f$ is defined by $f : z \mapsto f ( z ) = \exp ( i z )$. Plot the image of the region $D = \{ z : \operatorname { Re } ( z ) \geq 0, 1 \geq \operatorname { Im } ( z ) \geq 0 \}$ under $f$ on the complex plane.
3. Find the residue of the function $z ^ { 2 } \exp \left( \frac { 1 } { z } \right)$ at $z = 0$.

II. Consider the complex function: $f ( z ) = \frac { ( \log z ) ^ { 2 } } { ( z + a ) ^ { 2 } }$, where $a$ is a positive real number. The closed path $C$ shown in Figure 3.1 is defined by $C = C _ { + } + C _ { R } + C _ { - } + C _ { r }$, where $R > a > r > 0$. Here, $\log z$ takes the principal value on the path $C _ { + }$. Answer the following questions.

1. Apply the residue theorem to calculate the contour integral $\oint _ { C } f ( z ) \, d z$.
2. Use the result of Question II.1 to calculate the integral: $\int _ { 0 } ^ { \infty } \frac { \log x } { ( x + a ) ^ { 2 } } \, d x$.

In the following, $z$ denotes a complex number, and $x$ and $\varepsilon$ denote real numbers. The imaginary unit is denoted by $i$.
I. Answer the following questions about the function $f _ { n } ( z ) = 1 / \left( z ^ { n } - 1 \right)$. Here, $n$ is an integer greater than or equal to 2.

1. For the case that $n = 3$, find all singularities of $f _ { n } ( z )$.
2. Calculate the residue value at a singularity $p _ { 0 }$ of $f _ { n } ( z )$ and give a simple expression of the residue in terms of $n$ and $p _ { 0 }$.
3. For a contour $C$ given by the closed curve $| z | = 2$ and oriented in the counter-clockwise direction, evaluate the contour integral $\oint _ { C } f _ { n } ( z ) d z$.

II. Obtain the following limit value:
$$\lim _ { \varepsilon \rightarrow + 0 } \left[ \int _ { - \infty } ^ { 1 - \varepsilon } \frac { 1 } { x ^ { 3 } - 1 } d x + \int _ { 1 + \varepsilon } ^ { \infty } \frac { 1 } { x ^ { 3 } - 1 } d x \right]$$
III. Obtain the following limit value:
$$\lim _ { \varepsilon \rightarrow + 0 } \left[ \int _ { 0 } ^ { 1 - \varepsilon } \frac { \cos x } { x ^ { 4 } - 1 } d x + \int _ { 1 + \varepsilon } ^ { \infty } \frac { \cos x } { x ^ { 4 } - 1 } d x \right]$$
IV. Obtain the following limit value:
$$\lim _ { \varepsilon \rightarrow + 0 } \left[ \int _ { 0 } ^ { 1 - \varepsilon } \frac { \sin \left( x ^ { 2 } - \frac { \pi } { 4 } \right) } { x ^ { 4 } - 1 } d x + \int _ { 1 + \varepsilon } ^ { \infty } \frac { \sin \left( x ^ { 2 } - \frac { \pi } { 4 } \right) } { x ^ { 4 } - 1 } d x \right]$$

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 $M(z)$.

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 $0 < \alpha < \pi/2$.

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 $I$.

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 $z$.
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 the following, $z = x + i y$ and $w = u + i v$ represent complex numbers, where $i$ is the imaginary unit, and $x , y , u$ and $v$ are real numbers.
I. In order to evaluate the integral
$$I = \int _ { - \infty } ^ { \infty } \frac { 1 } { x ^ { 6 } + 1 } \mathrm {~d} x$$
consider the complex function $f ( z ) = \frac { 1 } { z ^ { 6 } + 1 }$.
1. Find all singularities of $f ( z )$. 2. By applying the residue theorem, determine the value of $I$.
II. Two domains, which are banded and semi-infinite on the complex $z$-plane, are defined as:
$$D _ { 1 } = \left\{ x + i y \left\lvert \, 0 \leq x \leq \frac { \pi } { 2 } \right. , y \geq 0 \right\} \text { and } D _ { 2 } = \left\{ x + i y \mid x \geq 0 , - \frac { \pi } { 2 } \leq y \leq 0 \right\}$$
Consider the mapping $w = g ( z )$ from the complex $z$-plane to the complex $w$-plane with an analytic function $g ( z )$. Let $D _ { 1 } ^ { * }$ and $D _ { 2 } ^ { * }$ be the images of $D _ { 1 }$ and $D _ { 2 }$, respectively, through this mapping.
1. When $g ( z ) = \cos z$, sketch the domain $D _ { 1 } ^ { * }$. 2. When $g ( z ) = ( \cosh z ) ^ { 3 }$, sketch the domain $D _ { 2 } ^ { * }$.

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 $D _ { m }$:
$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 N } }$$