## MATH3364 complex analysis课程简介

This course covers several topics related to complex analysis, including:

1. The complex number system: This includes a review of complex numbers, their algebraic properties, and geometric representation.
2. Analytic functions: These are functions that can be represented as power series expansions, and they have a number of important properties such as being differentiable and having a unique analytic continuation.
3. The Cauchy integral theorem: This theorem states that the value of a complex integral around a closed curve is determined by the behavior of the function inside the curve. It is a powerful tool for evaluating complex integrals and has many applications in physics and engineering.
4. Series representation: Complex functions can be represented as power series expansions, Laurent series, and Fourier series. These representations are useful for understanding the behavior of functions and for numerical computations.
5. Residue theory: This theory is concerned with the behavior of singularities of complex functions and their residues. It provides a powerful tool for evaluating complex integrals and has many applications in physics and engineering.
6. Conformal mapping: This is a technique for mapping one region of the complex plane onto another region while preserving angles. It has important applications in fluid dynamics, electrostatics, and other areas of physics and engineering.

## PREREQUISITES

Chapter 1: Complex Numbers
1.1 The Algebra of Complex Numbers
1.2 Point Representation of Complex Numbers
1.3 Vectors and Polar Forms
1.4 The Complex Exponential
1.5 Powers and Roots
1.6 Planar Sets
1.7 The Riemann Sphere and Stereographic Projection
Chapter 2: Analytic Functions
2.1 Functions of a Complex Variable
2.2 Limits and Continuity
2.3 Analyticity
2.4 The Cauchy-Riemann Equations
2.5 Harmonic Functions
Chapter 3: Elementary Functions
3.1 Polynomials and Rational Functions
3.2 The Exponential, Trigonometric and Hyperbolic Functions
3.3 The Logarithmic Function
3.4 Washers, Wedges, and Walls
3.5 Complex Powers and Inverse Trigonometric Functions

## MATH3364 complex analysis HELP（EXAM HELP， ONLINE TUTOR）

Problem: Chapter 1: #1: Describe geometrically the sets of points $z$ in the complex plane defined by the following relations: (a) $\left|z-z_1\right|=\left|z-z_2\right|$ where $z_1, z_2 \in \mathbb{C}$; (b) $1 / z=\bar{z}$; (c) $\operatorname{Re}(z)=3$; (d) $\operatorname{Re}(z)>c$ (resp., $\geq c$ ) where $c \in \mathbb{R}$.

(a) The set of points $z$ in the complex plane that satisfy the equation $\left|z-z_1\right|=\left|z-z_2\right|$ is the perpendicular bisector of the line segment connecting $z_1$ and $z_2$. This is because any point $z$ that is equidistant from $z_1$ and $z_2$ must lie on the perpendicular bisector of the line segment connecting them.

(b) The set of points $z$ in the complex plane that satisfy the equation $1/z=\bar{z}$ is the unit circle centered at the origin. To see this, note that $\bar{z}$ is the reflection of $z$ across the real axis, and $1/z$ is the inversion of $z$ about the unit circle centered at the origin. Thus, $1/z=\bar{z}$ if and only if $z$ lies on the unit circle and is reflected across the real axis.

(c) The set of points $z$ in the complex plane that satisfy the equation $\operatorname{Re}(z)=3$ is the vertical line through $3$ on the real axis.

(d) The set of points $z$ in the complex plane that satisfy the inequality $\operatorname{Re}(z)>c$ is the open half-plane to the right of the vertical line through $c$ on the real axis. The set of points $z$ that satisfy $\operatorname{Re}(z)\geq c$ is the closed half-plane to the right of the same vertical line.

Problem: Chapter 1: #3: With $\omega=s e^{i \varphi}$, where $s \geq 0$ and $\varphi \in \mathbb{R}$, solve the equation $z^n=\omega$ in $\mathbb{C}$ where $n$ is a natural number. How many solutions are there?

We can write $\omega$ in polar form as $\omega = s e^{i \varphi}$. Now, we want to solve the equation $z^n=\omega$ in $\mathbb{C}$.

Let $z=r e^{i\theta}$ be a solution to this equation. Then, we have $z^n=r^n e^{in\theta}$ and $\omega=s e^{i \varphi}$. Equating these two expressions, we get:

$$r^n e^{in\theta} = s e^{i\varphi}$$

Taking the modulus of both sides, we get:

$$r^n = s$$

Taking the argument of both sides, we get:

$$n\theta = \varphi + 2k\pi$$

where $k$ is an integer. Solving for $\theta$, we get:

$$\theta = \frac{\varphi + 2k\pi}{n}$$

Thus, we have $n$ solutions for $z$, given by:

$$z_k = s^{1/n} e^{i(\varphi + 2k\pi)/n}$$

where $k=0,1,\ldots,n-1$. Note that these solutions are equally spaced on a circle centered at the origin with radius $s^{1/n}$, and the angle between consecutive solutions is $2\pi/n$.

Therefore, there are exactly $n$ solutions to the equation $z^n=\omega$ in $\mathbb{C}$, given by the formula above.

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

Statistics-lab™可以为您提供rochester.edu MATH208 Operations Research运筹学的代写代考和辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。

## MATH3364 complex analysis课程简介

This course covers several topics related to complex analysis, including:

1. The complex number system: This includes a review of complex numbers, their algebraic properties, and geometric representation.
2. Analytic functions: These are functions that can be represented as power series expansions, and they have a number of important properties such as being differentiable and having a unique analytic continuation.
3. The Cauchy integral theorem: This theorem states that the value of a complex integral around a closed curve is determined by the behavior of the function inside the curve. It is a powerful tool for evaluating complex integrals and has many applications in physics and engineering.
4. Series representation: Complex functions can be represented as power series expansions, Laurent series, and Fourier series. These representations are useful for understanding the behavior of functions and for numerical computations.
5. Residue theory: This theory is concerned with the behavior of singularities of complex functions and their residues. It provides a powerful tool for evaluating complex integrals and has many applications in physics and engineering.
6. Conformal mapping: This is a technique for mapping one region of the complex plane onto another region while preserving angles. It has important applications in fluid dynamics, electrostatics, and other areas of physics and engineering.

## PREREQUISITES

Chapter 1: Complex Numbers
1.1 The Algebra of Complex Numbers
1.2 Point Representation of Complex Numbers
1.3 Vectors and Polar Forms
1.4 The Complex Exponential
1.5 Powers and Roots
1.6 Planar Sets
1.7 The Riemann Sphere and Stereographic Projection
Chapter 2: Analytic Functions
2.1 Functions of a Complex Variable
2.2 Limits and Continuity
2.3 Analyticity
2.4 The Cauchy-Riemann Equations
2.5 Harmonic Functions
Chapter 3: Elementary Functions
3.1 Polynomials and Rational Functions
3.2 The Exponential, Trigonometric and Hyperbolic Functions
3.3 The Logarithmic Function
3.4 Washers, Wedges, and Walls
3.5 Complex Powers and Inverse Trigonometric Functions

## MATH3364 complex analysis HELP（EXAM HELP， ONLINE TUTOR）

Problem 3.1. Let $E \subset \mathbb{C}$ be an open set and $f: E \rightarrow \mathbb{C}$ be a function. If $f$ is differentiable at a point $z \in E$, show that $f$ is also continuous at $z$.

To show that $f$ is continuous at $z$, we need to show that for any $\epsilon > 0$, there exists a $\delta > 0$ such that $|f(w) – f(z)| < \epsilon$ for all $w \in E$ with $|w – z| < \delta$.

Since $f$ is differentiable at $z$, we know that the limit $$\lim_{w \rightarrow z} \frac{f(w) – f(z)}{w – z}$$ exists. Let $L$ be the value of this limit.

Now, let $\epsilon > 0$ be given. We want to find a $\delta > 0$ such that $|f(w) – f(z)| < \epsilon$ for all $w \in E$ with $|w – z| < \delta$.

Since the limit above exists, we know that there exists a $\delta_1 > 0$ such that if $0 < |w – z| < \delta_1$, then $$\left|\frac{f(w) – f(z)}{w – z} – L\right| < \frac{\epsilon}{2}.$$ This implies that $$|f(w) – f(z) – L(w – z)| < \frac{\epsilon}{2} |w – z|$$ for $0 < |w – z| < \delta_1$.

Now, let $\delta = \min{\delta_1, \epsilon/(2|L|+2)}$. Then, if $0 < |w – z| < \delta$, we have \begin{align*} |f(w) – f(z)| &= |f(w) – f(z) – L(w – z) + L(w – z)| \ &\leq |f(w) – f(z) – L(w – z)| + |L(w – z)| \ &< \frac{\epsilon}{2} |w – z| + |L| |w – z| \ &\leq \frac{\epsilon}{2} \delta + |L| \delta \ &\leq \frac{\epsilon}{2} \cdot \frac{\epsilon}{2(|L|+1)} + |L| \cdot \frac{\epsilon}{2(|L|+1)} \ &= \frac{\epsilon}{2(|L|+1)}(|L|+1) \ &= \frac{\epsilon}{2} \ \end{align*} Therefore, we have shown that for any $\epsilon > 0$, there exists a $\delta > 0$ such that $|f(w) – f(z)| < \epsilon$ for all $w \in E$ with $|w – z| < \delta$. This proves that $f$ is continuous at $z$.

Problem 3.2. Prove the product formula: if $f$ and $g$ are complex functions that are differentiable at $z$, then $f g$ is differentiable at $z$ with derivative $(f g)^{\prime}(z)=$ $f^{\prime}(z) g(z)+f(z) g^{\prime}(z)$

To prove the product formula, we need to show that the limit $$\lim_{w \rightarrow z} \frac{(fg)(w) – (fg)(z)}{w – z}$$ exists, and to find its value.

We have \begin{align*} \frac{(fg)(w) – (fg)(z)}{w – z} &= \frac{f(w)g(w) – f(z)g(z)}{w – z} \ &= \frac{f(w)g(w) – f(w)g(z) + f(w)g(z) – f(z)g(z)}{w – z} \ &= f(w)\frac{g(w) – g(z)}{w – z} + g(z)\frac{f(w) – f(z)}{w – z} \end{align*}

Now, since $f$ and $g$ are differentiable at $z$, we know that the limits $$\lim_{w \rightarrow z} \frac{f(w) – f(z)}{w – z} = f'(z)$$ and $$\lim_{w \rightarrow z} \frac{g(w) – g(z)}{w – z} = g'(z)$$ exist.

Therefore, we have \begin{align*} \lim_{w \rightarrow z} \frac{(fg)(w) – (fg)(z)}{w – z} &= \lim_{w \rightarrow z} \left[f(w)\frac{g(w) – g(z)}{w – z} + g(z)\frac{f(w) – f(z)}{w – z}\right] \ &= f(z)g'(z) + g(z)f'(z) \end{align*} This proves the product formula.

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

Statistics-lab™可以为您提供rochester.edu MATH208 Operations Research运筹学的代写代考和辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。