Statistics-lab™可以为您提供cornell.edu MATH4180 complex analysis复分析的代写代考和辅导服务！
MATH4180 complex analysis课程简介
This course covers several topics related to complex analysis, including:
- The complex number system: This includes a review of complex numbers, their algebraic properties, and geometric representation.
- 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.
- 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.
- 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.
- 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.
- 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
MATH4180 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 $c \in \mathbb{R}$.
Solution: (a) When $z_1 \neq z_2$, this is the line that perpendicularly bisects the line segment from $z_1$ to $z_2$. When $z_1=z_2$, this is the entire complex plane.
(b)
$$
\frac{1}{z}=\frac{\bar{z}}{z \bar{z}}=\frac{\bar{z}}{|z|^2} .
$$
So
$$
\frac{1}{z}=\bar{z} \Leftrightarrow \frac{\bar{z}}{|z|^2}=\bar{z} \Leftrightarrow|z|=1 .
$$
This is the unit circle in $\mathbb{C}$.
(c) This is the vertical line $x=3$.
(d) This is the open half-plane to the right of the vertical line $x=c$ (or the closed half-plane if it is $\geq$ ).
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?
Solution: Notice that
$$
\omega=s e^{i \varphi}=s e^{i(\varphi+2 \pi m)}, m \in \mathbb{Z}
$$
It’s worth spending a moment or two thinking what is the best choice for our generic integer. Clearly $n$ is a bad choice as it is already used in the problem; as we often use $t$ for the imaginary part, that is out too. The most natural is to use $m$ (though $k$ would be another fine choice); at all costs do not use $i$ !
Based on this relationship, we have
$$
z^n=s e^{i(\varphi+2 \pi m)}
$$
So,
$$
z=s^{1 / n} e^{\frac{i(c+2 \pi m)}{m}}
$$
Thus, we will have $n$ unique solutions since each choice of $m \in{0,1, \ldots, n-1}$ yields a different solution so long as $s \neq 0$. Note that $m=n$ yields the same solution as $m=0$, in general, if two choices of $m$ differ by $n$ then they yield the same solution, and thus it suffices to look at the $n$ specified values of $m$. If $s=0$, then we have only 1 solution.
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.
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