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

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 $$\lim _ { \varepsilon \rightarrow + 0 } \left[ \int _ { - \infty } ^ { 1 - \varepsilon } g ( x ) d x + \int _ { 1 + \varepsilon } ^ { \infty } g ( x ) d x \right]$$

Let $f ( t )$ be a periodic function of period $T , f ( t + T ) = f ( t ) ( T > 0 )$, and be expanded in the complex Fourier series as follows:
$$f ( t ) = \sum _ { n = - \infty } ^ { \infty } F _ { n } \exp \left( - i \omega _ { n } t \right)$$
Here, $i$ is the imaginary unit, and $t$ is a real number. Answer the following questions.
I. Express $\omega _ { n }$ using $T$ and $n$.
II. Let $M$ be a positive-integer constant and $\delta ( t )$ be the delta function, and define
$$\hat { f } ( t ) = \sum _ { m = 0 } ^ { M - 1 } f ( t ) \delta ( t - m \Delta t ) , \quad \Delta t = \frac { T } { M }$$
Express
$$\lim _ { \varepsilon \rightarrow + 0 } \int _ { - \varepsilon } ^ { T - \varepsilon } \hat { f } ( t ) \exp \left( i \omega _ { k } t \right) d t$$
using $F _ { n } ( n = - \infty , \cdots , - 1,0,1 , \cdots , \infty )$. Here $k$ is an arbitrary integer.
III. $\Delta t$ is given in Question II. Express $F _ { j } ( j = 0,1,2 , \cdots , M - 1 )$ using
$$f ( 0 ) , \quad f ( \Delta t ) , \quad f ( 2 \Delta t ) , \cdots , \quad f ( ( M - 1 ) \Delta t )$$
when $F _ { n } = 0 ( n < 0$ or $n \geq M )$.
IV. Calculate $F _ { j }$ in Question III when $f ( l \Delta t ) = ( - 1 ) ^ { l } \quad ( l = 0,1,2 , \cdots , M - 1 )$.

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

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

In the complex plane, the vertices of the quadrilateral formed by the roots of the equation
$$z ^ { 4 } = 16$$
have what area in square units?
A) 8
B) 12
C) 16
D) $4 \sqrt { 3 }$
E) $6 \sqrt { 2 }$