Let $z$ be a complex variable, and write $x = \Re(z)$ and $y = \Im(z)$ for the real and the imaginary parts, respectively. Let $f(z)$ be a complex polynomial. Let $R > 0$ be a real number and $\gamma$ the circle in $\mathbb{C}$ of radius $R$ and centre at 0, oriented in the counterclockwise direction. What is the value of $$\frac{1}{2\pi \imath R} \int_{\gamma} \left( \Re(f(z))\,\mathrm{d}x + \Im(f(z))\,\mathrm{d}y \right)$$

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, set $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z)\, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Show that, for any complex number $z$ such that $|z| < 1$, $\frac{1}{2\pi} \int_0^{2\pi} \mathcal{P}(t,z)\, \mathrm{d}t = 1$.

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, set $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z)\, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Show that, for all $r \in [0,1[$ and all real $t$ and $\theta$, $$\mathcal{P}\left(t, r\mathrm{e}^{\mathrm{i}\theta}\right) = \frac{1 - r^2}{1 - 2r\cos(t-\theta) + r^2}$$

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Show that, for any complex number $z$ such that $|z| < 1$, $\frac{1}{2\pi} \int_0^{2\pi} \mathcal{P}(t,z) \, \mathrm{d}t = 1$.

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Show that, for all $r \in [0,1[$ and all real $t$ and $\theta$, $$\mathcal{P}\left(t, r\mathrm{e}^{\mathrm{i}\theta}\right) = \frac{1 - r^2}{1 - 2r\cos(t-\theta) + r^2}$$

We are given a polynomial $P \in \mathbb{C}[X]$ of degree $d$. Show that, for all $z \in \mathbb{C}$ and all $r \in \mathbb{R}$, we have: $$P(z) = \frac{1}{2\pi} \int_0^{2\pi} P\left(z + re^{i\theta}\right) d\theta$$

Let $M \in \mathcal{B}_{n}$. We define $$b^{\prime}(M) = \sup\left\{\varphi_{M}(z); z \in \mathbf{C} \backslash \mathbb{D}\right\}$$ where $\varphi_{M}: z \in \mathbf{C} \backslash \mathbb{D} \longmapsto (|z|-1)\left\|R_{z}(M)\right\|_{\mathrm{op}}$, and $b^{\prime}(M) \leq b(M)$.
We fix $r \in ]1, +\infty[$ and $(X,Y) \in \Sigma_{n}^{2}$. For $\rho \in \mathbf{R}^{+*}$, we denote $\mathbb{D}_{\rho} = \{z \in \mathbf{C}; |z| \leq \rho\}$.
Show that there exists an element $F_{r}$ of $\mathcal{R}_{n}$ whose poles are all in $\mathbb{D}_{1/r}$ and such that the following two properties are satisfied: $$\begin{gathered} \forall z \in \mathbf{C} \backslash \mathbb{D}_{1/r}, \quad \left|F_{r}(z)\right| \leq \frac{b^{\prime}(M)}{r|z|-1} \\ \forall k \in \mathbf{N}, \quad X^{T}M^{k}Y = \frac{r^{k+1}}{2\pi} \int_{-\pi}^{\pi} F_{r}\left(e^{it}\right) e^{i(k+1)t} \mathrm{~d}t \end{gathered}$$

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. For all $n \in \mathbb{N}$, we define $$\gamma_{n} : \begin{cases} [0,1] \rightarrow \mathbb{C} \\ t \mapsto (2n+1)\pi\, \mathrm{e}^{2\mathrm{i}\pi t} \end{cases}$$ and for all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, $$Q_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\gamma_{n}(t)}}{(\mathrm{e}^{\gamma_{n}(t)} - 1)\gamma_{n}(t)^{n-1}} \,\mathrm{d}t.$$ Show that, for all $n \in \mathbb{N}$, $Q_{n} \in \mathcal{E}$.

For all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, let $$Q_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\gamma_{n}(t)}}{(\mathrm{e}^{\gamma_{n}(t)} - 1)\gamma_{n}(t)^{n-1}} \,\mathrm{d}t$$ where $\gamma_{n}(t) = (2n+1)\pi\,\mathrm{e}^{2\mathrm{i}\pi t}$. Show that $$\forall n \in \mathbb{N}^{*},\, \forall z \in \mathbb{C}, \quad Q_{n}(z+1) - Q_{n}(z) = n z^{n-1}.$$

Problem 3

Answer the following questions. Here, $i , e$, and $\log$ denote the imaginary unit, the base of the natural logarithm, and the natural logarithm, respectively. I. Consider the definite integral $I$ expressed as
$$I = \int _ { 0 } ^ { 2 \pi } \frac { \cos \theta d \theta } { ( 2 + \cos \theta ) ^ { 2 } }$$

1. Find a complex function $G ( z )$ of a complex variable $z$ when we rewrite $I$ as an integral of a complex function as

$$\oint _ { | z | = 1 } G ( z ) d z$$
where the integration path is a unit circle in the counter clockwise direction.
2. Find all poles and the respective orders and residues.
3. Evaluate the integral $I$. II. Let a function of a real variable $\theta$ with real parameters $\alpha$ and $\beta$ be
$$f ( \theta ; \alpha , \beta ) = 1 + e ^ { 2 i \beta } + \alpha e ^ { i ( \theta + \beta ) }$$
Consider the definite integral
$$F ( \alpha , \beta ) = \int _ { 0 } ^ { 2 \pi } d \theta \frac { d } { d \theta } [ \log f ( \theta ; \alpha , \beta ) ]$$

1. Find a complex function $G ( z )$ of a complex variable $z$ when we rewrite $F ( \alpha , \beta )$ as an integral of a complex function as

$$\oint _ { | z | = 1 } G ( z ) d z$$
where the integration path is a unit circle in the counter clockwise direction.
2. Find all poles and the respective orders and residues.
3. Evaluate $F ( \alpha , \beta )$ by classifying cases with respect to $\alpha$ and $\beta$. Ignore the case in which the integration path passes through any poles.

Answer the following questions concerning complex functions defined over the $z$-plane ( $z = x + i y$ ), where $i$ denotes the imaginary unit.
I. For the function $f ( z ) = \frac { z } { \left( z ^ { 2 } + 1 \right) ( z - 1 - i a ) }$, where $a$ is a positive real number:

1. Find all the poles and respective residues of $f ( z )$.
2. Using the residue theorem, calculate the definite integral $$\int _ { - \infty } ^ { \infty } \frac { x } { \left( x ^ { 2 } + 1 \right) ( x - 1 - i a ) } d x$$

II. Consider the function $g ( z ) = \frac { z } { \left( z ^ { 2 } + 1 \right) ( z - 1 ) }$ and the closed counter-clockwise path of integration $C$, which consists of the upper half circle $C _ { 1 }$ with radius $R \left( z = R e ^ { i \theta } , 0 \leq \theta \leq \pi \right)$, the line segment $C _ { 2 }$ on the real axis $( z = x , - R \leq x \leq 1 - r )$, the lower half circle $C _ { 3 }$ with its center at $z = 1 \left( z = 1 - r e ^ { i \theta } , 0 \leq \theta \leq \pi \right)$, and the line segment $C _ { 4 }$ on the real axis $( z = x , 1 + r \leq x \leq R )$. Here, $e$ denotes the base of the natural logarithm, and let $r > 0 , r \neq \sqrt { 2 }$ and $R > 1 + r$.
Answer the following questions.

1. Calculate the integral $\int _ { C } g ( z ) d z$.
2. Using the result from Question II.1, calculate the following value $$\lim _ { \varepsilon \rightarrow + 0 } \left[ \int _ { - \infty } ^ { 1 - \varepsilon } g ( x ) d x + \int _ { 1 + \varepsilon } ^ { \infty } g ( x ) d x \right]$$

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:
$$\lim _ { \varepsilon \rightarrow + 0 } \left[ \int _ { 0 } ^ { 1 - \varepsilon } \frac { \sin \left( x ^ { 2 } - \frac { \pi } { 4 } \right) } { x ^ { 4 } - 1 } d x + \int _ { 1 + \varepsilon } ^ { \infty } \frac { \sin \left( x ^ { 2 } - \frac { \pi } { 4 } \right) } { x ^ { 4 } - 1 } d x \right]$$

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

Answer the following questions. Here, for any complex value $z , \bar { z }$ is the complex conjugate of $z$, arg $z$ is the argument of $z , | z |$ is the absolute value of $z$, and $i$ is the imaginary unit.
I. Sketch the region of $z$ on the complex plane that satisfies the following:
$$z \bar { z } + \sqrt { 2 } ( z + \bar { z } ) + 3 i ( z - \bar { z } ) + 2 \leq 0$$
II. Answer the following questions on the complex valued function $f ( z )$ below.
$$f ( z ) = \frac { z ^ { 2 } - 2 } { \left( z ^ { 2 } + 2 i \right) z ^ { 2 } }$$

1. Find all the poles of $f ( z )$ as well as the orders and residues at the poles.
2. By applying the residue theorem, find the value of the following integral $I _ { 1 }$. Here, the integration path $C$ is the circle on the complex plane in the counterclockwise direction which satisfies $| z + 1 | = 2$. $$I _ { 1 } = \oint _ { C } f ( z ) \mathrm { d } z$$

III. Answer the following questions.

1. Let $g ( z )$ be a complex valued function, which satisfies $$\lim _ { | z | \rightarrow \infty } g ( z ) = 0$$ for $0 \leq \arg z \leq \pi$. Let $C _ { R }$ be the semicircle, with radius $R$, in the upper half of the complex plane with the center at the origin. Show $$\lim _ { R \rightarrow \infty } \int _ { C _ { R } } e ^ { i a z } g ( z ) \mathrm { d } z = 0$$ where $a$ is a positive real number.
2. Find the value of the following integral, $I _ { 2 }$ : $$I _ { 2 } = \int _ { 0 } ^ { \infty } \frac { \sin x } { x } \mathrm {~d} x$$

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