数学代写|MATH4180 complex analysis
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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）
Here is an attempt [ultimately doomed] at using real methods to expand $\mathrm{H}(\mathrm{x})=$ $1 /\left(1+x^2\right)$ into a power series centred at $x=k$, i.e., into a series of the form $H(x)=\sum_{j=0}^{\infty} c_j X^j$, where $X=(x-k)$. According to Taylor’s Theorem, $c_j=$ $\mathrm{H}^{(j)}(k) / j !$, where $\mathrm{H}^{(j)}(k)$ is the $\mathrm{j}^{\text {th }}$ derivative of $\mathrm{H}$.
(i) Show that $c_0=1 /\left(1+k^2\right)$ and $c_1=-2 k /\left(1+k^2\right)^2$, and find $c_2$. Note how it becomes increasingly difficult to calculate the successive derivatives.
(ii) Recall (or prove) that the $n^{\text {th }}$ derivative of a product $A B$ of two functions $\mathrm{A}(\mathrm{x})$ and $\mathrm{B}(\mathrm{x})$ is given by Leibniz’s rule:
$$
(A B)^{(n)}=\sum_{j=0}^n\left(\begin{array}{l}
n \
j
\end{array}\right) A^{(j)} B^{(n-j)} .
$$
By applying this result to the product $\left(1+x^2\right) \mathrm{H}(x)$, deduce that
$$
\left(1+k^2\right) H^{(n)}(k)+2 n k H^{(n-1)}(k)+n(n-1) H^{(n-2)}(k)=0 .
$$
Because the coefficients in this recurrence relation depend on $n$, we cannot solve it using the technique of Ex. 30 on p. 55.
(iii) Deduce from the previous part that the recurrence relation for the $c_j{ }^{\prime}$ ‘s is
$$
\left(1+k^2\right) c_n+2 k c_{n-1}+c_{n-2}=0
$$
which does have constant coefficients.
(iv) Solve this recurrence relation, and hence recover the result (2.17) on p. 85.
Reconsider the series (2.18) on p. 85 .
(i) Show that we recover the correct series (missing the odd powers of $x$ ) when the centre $k$ of the series is at the origin.
(ii) Find a value of $k$ such that the series is missing all the powers $X^n$, where $\mathrm{n}=2,5,8,11,14, \ldots$. Check your answer using a computer.
Show that each of the following series has the unit circle as its circle of convergence, then investigate the convergence on the unit circle. You can guess the correct answers by “drawing the series” in the manner of [2.18] on p. 89.
(i) $\sum_{n=0}^{\infty} z^n$
(ii) $\sum_{n=1}^{\infty} \frac{z^n}{n}$
(iii) $\sum_{n=1}^{\infty} \frac{z^n}{n^2}$.
[By virtue of (2.29), note that the second series is $-\log (1-z)$.]
We have seen that if we set $P_n(z)=z^n$, then the representation of a complex function $f(z)$ as an infinite series $\sum_{n=0}^{\infty} c_n P_n(z)$ (i.e., a power series) is unique. This is not true, however, if $P_n(z)$ is just any old set of polynomials. The following example is taken from Boas and Boas (2010). Defining
$$
P_0(z)=-1, \quad \text { and } \quad P_n(z)=\frac{z^{n-1}}{(n-1) !}-\frac{z^n}{n !} \quad(n=1,2,3, \ldots),
$$
show that
$$
-2 \mathrm{P}_0-\mathrm{P}_1+\mathrm{P}_3+2 \mathrm{P}_4+3 \mathrm{P}_5+\cdots=e^z=\mathrm{P}_1+2 \mathrm{P}_2+3 \mathrm{P}_3+4 \mathrm{P}_4+\cdots
$$
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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