# 复分析|MATH0013 Analysis 3: Complex Analysis代写2023

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## Instructions:

Complex analysis is a branch of mathematics that studies complex functions, which are functions that have complex numbers as their inputs and outputs. It is an important area of study because complex functions have many interesting and useful properties that are not shared by real-valued functions.

Starting from complex numbers is a good way to introduce the subject, as it allows you to develop the algebraic and geometric properties of complex numbers that are essential for understanding complex functions. Cauchy’s theorem and Cauchy’s integral formulae are fundamental results in complex analysis that relate the values of a complex function inside a closed contour to its values on the boundary of the contour. These theorems have many important applications, such as evaluating definite integrals and finding the number of zeros of a complex polynomial in a region.

The theorem of residues is another important result in complex analysis that allows you to compute certain types of integrals using the residues of a function. This theorem is particularly useful for evaluating integrals involving trigonometric and logarithmic functions.

Laurent’s theorem is a generalization of Taylor’s theorem for complex functions, which allows you to express a complex function as a power series with both positive and negative powers of the complex variable. This theorem is useful for studying the singularities of a complex function, which are points where the function is not defined or behaves in an unusual way.

Overall, the course on complex functions seems to cover a lot of important topics in complex analysis that have many applications in mathematics and other fields.

Let $G$ be an undirected graph with $n$ vertices. Then a Hamilton path is a simple path in $G$ that visits each vertex once (i.e., has $n$ vertices and $n-1$ edges), while a Hamilton cycle is a simple cycle in $G$ that visits each vertex once (i.e., has $n$ vertices and $n$ edges). Let HAMPATH and HAMCYCLE be the problems of deciding whether $G$ has a Hamilton path and Hamilton cycle respectively, given $G$ as input. (a) Show that if $G$ has a Hamilton cycle, then $G$ also has a Hamilton path.

(a) If $G$ has a Hamilton cycle, then we can simply remove one edge from the cycle to obtain a Hamilton path. Since a Hamilton cycle has $n$ edges, removing one edge from it leaves a path with $n-1$ edges that visits every vertex exactly once. Therefore, if $G$ has a Hamilton cycle, it also has a Hamilton path.

(b) Give an example of a graph $G$ that has a Hamilton path but no Hamilton cycle.

(b) Consider the graph $G$ consisting of three vertices $a,b,c$ and three edges $(a,b),(b,c),(c,a)$. This graph has a Hamilton path $a\rightarrow b\rightarrow c$, but no Hamilton cycle. To see this, note that any cycle in $G$ must contain all three vertices, and therefore must have length at least 3. But the only cycle in $G$ is of length 3, and it visits each vertex twice.

(c) Give a polynomial-time reduction from $H A M C Y C L E$ to $H A M P A T H$.

(c) To reduce $HAMCYCLE$ to $HAMPATH$, we can simply remove one vertex $v$ from $G$ and ask whether the resulting graph $G’$ has a Hamilton path. If $G$ has a Hamilton cycle, then $G’$ has a Hamilton path by part (a). Conversely, if $G’$ has a Hamilton path, we can add the vertex $v$ back into the path to obtain a Hamilton cycle in $G$. This reduction takes polynomial time since we only need to remove one vertex and its adjacent edges, which can be done in constant time per vertex.