The theory of functions of one complex variable with an emphasis on the theory of complex analytic (or holomorphic) functions of one complex variable. Typical topics include: Cauchy's integral formula, singularities, poles, meromorphic functions, Laurent and Taylor series, maximum modulus principle, ...

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276 views

Drawing on Argand Diagram

How do I know what $a : \{z \in \mathbb{C}: | z + 2 | + | z - 2 | = 10 \}$ looks like? I tried expanding it with $z = x + iy$, but I end up with a circle radius $1$ (which is clearly wrong!)
4
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1answer
85 views

Interpolation with entire function

Is there any simple way to construct an entiere function $f$ such that : $$\forall p \in {\mathbb N} \quad f(2^p)=(-1)^p$$
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2answers
93 views

Elementary Length of a contour

Parametrize the contour consisting of the perimeter of the square with vertices $-1-i,1-i,1+i,$ and $-1+i$ traversed once in that order. What is the length of this contour? I have already ...
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1answer
163 views

zeros of entire functions

Consider the product $$\displaystyle\prod_{n=1}^{+\infty}(1-e^{-2\pi n}e^{2\pi iz})$$ I've proven that this product converges uniformly on compact subsets of complex plane since the serie ...
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2answers
345 views

Examples of non-Riemann surfaces?

While studying Complex Analysis, I have come across Riemann Surfaces: http://mathworld.wolfram.com/RiemannSurface.html Can anyone please provide some examples of non-Riemannable surfaces? Thanks a ...
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1answer
69 views

proving than an infinite product defines an entire function

Consider the infinite product $$F(z)=\displaystyle\prod_{n=1}^{+\infty}(1-e^{-2\pi n}e^{2\pi iz})$$ How can i prove that $F$ is entire? Can i write $F$ as a Weierstrass product $\prod ...
15
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2answers
819 views

Finding $f$ such that $ \int f = \sum f$

Please see the problem 5 of the given link: http://www.artofproblemsolving.com/Forum/resources.php?c=2&cid=59&year=2005&sid=722231ab4ec5ce280584eb8f24f07656 It asks us to prove that ...
9
votes
3answers
791 views

Automorphisms of the complex plane

How can it be shown that $$Aut(\mathbb{C})=\{f|f(z)=az+b,a\neq 0\},$$ where an automorphism of $\mathbb{C}$ is defined as a bijective entire function with entire inverse? If $f$ is of the form ...
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2answers
369 views

If $f^2$ is an analytic function then so is $f$

I want to show the following: If $f(z) $ is a continuous function on a connected open subset of the complex plane and $f(z)^2$ is an analytic function, then $f(z)$ is analytic. Clearly if $f(z) ...
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1answer
113 views

Why the principal value of complex number argument is taken within $(-\pi,\pi]?$

Why the principal value of complex number argument is taken within $(-\pi,\pi]?$ What is the harm in considering similar other intervals like $[0,2\pi)$ for the purpose?
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0answers
54 views

Internal point transformed in an external one?

Let $f \colon \Omega \to \mathbb{C} $ be an analytic function over a connected open subset $\Omega$ of $\mathbb{C}$ and let $\gamma$ a rectifiable closed curve in $\Omega$. If $a$ is a point which is ...
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1answer
270 views

Smooth path definition

I keep getting confused with the definition of a smooth path. Here is a definition from William T. Shaw's Complex Analysis with Mathematica: A path $\phi$ is a continuous mapping from a segment of ...
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0answers
37 views

Numerical characteristics of a Riemann surface of function

Let $f(x)$ be an analytic function in $\mathbb{C}\setminus A$, where $A$ is a discrete finite set of branch points. I have some questions. Given a set of values of $f(x)$ is some domain $U \subset ...
4
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2answers
176 views

How to show $e^{2 \pi i \theta}$ is not algebraic.

I was wondering if someone could possibly help me figure out how to show $e^{2 \pi i \theta}$ is not algebraic when $\theta$ is irrational. Thanks!
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2answers
162 views

Cauchy Integral Formula

How can I apply the Cauchy Integral Formula if given a contour, two singular points are inside it? for example, how could I evaluate $\int \frac{dz}{z(z-2)}$ given $C: z = 3e^{i\theta}$
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2answers
2k views

Complex Analytic Proof of the Gaussian Integral $\int_{-\infty}^{\infty}e^{-z^2}dz=\sqrt{\pi}$

Prove that $\int_{-\infty}^{\infty}e^{-z^2}dz=\sqrt{\pi}$. Here is my attempted solution: Define $a:=\sqrt{\pi}e^{\frac{\pi i}{4}}$ and let $f(z) = \frac{e^{-z^2}}{1+e^{-2az}}$. Note that ...
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2answers
174 views

The principal value of the following complex exponential

Is the principal value of the following complex exponential: $$(i^i)^i = i^{-1} = -i $$ So principal value is $$ e^{-i\pi/2} = -i?$$
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2answers
272 views

How to prove that the zeros of $z^4+iz^3+1$ are in the disk $D(0,\frac{3}{2})$, and determine how many of them are in the first quadrant?

Prove that the zeros of $z^4+iz^3+1$ are in the disk $D(0,\frac{3}{2})$ and determine how many of them are in the first quadrant.
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1answer
60 views

have to chose correct statement

$f:D\rightarrow\mathbb{C}$ be analytic satisfying $f(1/n)=\frac{2n}{3n+1}$, $D$ is open unit disk,Then 1.$f(0)=2/3$ 2.$f$ has a simple pole at $z=-3$ 3.$f(3)=-3$ 4.No such $f$ exist. well, $1$ ...
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1answer
48 views

Equality of 2 integrals (complex)

let $F(x)=\int_0^x e^{it^k}dt$. Im trying to see why the following holds $$ \lim_{x\to\infty}F(x)=e^{\pi i/2k}\int_0^{\infty}e^{-t^k}dt, $$ and $$\lim_{x\to-\infty}F(x) = -e^{(-1)^k\pi ...
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682 views

Evaluation by methods of complex analysis $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm{dx}$

How would we evaluate the below integral by methods of complex analysis? $$\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm{dx}$$ I asked the question a while ago, but at that time I didn't specify this ...
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2answers
210 views

Calculating $\int_{-\infty}^{\infty}\frac{\sin(ax)}{x}\, dx$ using complex analysis

I am going over my complex analysis lecture notes and there is an example about calculating $$\int_{-\infty}^{\infty}\frac{\sin(ax)}{x}\, dx$$ that I don't understand. The solution in the notes ...
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1answer
99 views

Complex integral with Cauchy integral formula

For $k>1$ integer i want to compute the integral $$\int_0^{\infty}e^{-t^k}dt $$ by using Cauchy's integral formula. Edit: The actual integral im trying to solve is ...
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1answer
146 views

Finding the residue of function with Laurent series $\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{y^n(A+By+Cy^{-1})^k}{\beta (\beta i)^n \ k!}$

I have been trying to find the residue of $f(\omega) = \frac{e^{i \omega a} e^{\frac{-b \omega}{\omega + ib}}}{i \omega}$ at the essential singularity $\omega = -ib$ for a while, but it is giving me ...
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1answer
57 views

Does $ \sum_{k = 0}^{\infty} \sum_{n = 0}^{\infty}\frac{B^k C^{(n+k+1)}}{(ib)^n k! (n+k+1)!}$ converge?

In relation to my question: Finding the residue of function with Laurent series $\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{y^n(A+By+Cy{^-1})^k}{\beta (\beta i)^n \ k!}$ I need to find an expression ...
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1answer
324 views

Arc length and upper bound

Use $||\int_{\lambda} f(z)dz|| \leq max_{z\ on \lambda}|f(z)| \dot\ l(\lambda)$, where $l(\lambda)$ is the arc length, to establish the indicated estimate a.) If $\lambda$ is the vertical ...
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0answers
301 views

Derivation of poisson kernel for disk of radius $R$ from unit disk

Is there a way to derive poisson kernel for disk of radius $R$ from unit disk?
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1answer
42 views

Show that $\sqrt[k]{|z-a_1|\cdots |z-a_k|}$ has a max greater than $R$, and a min less than $R$

This is a homework problem. For $|z| \le R$ and $|a_j| < R$ for $j=1,\ldots, k$, not all zero, show that $\sqrt[k]{|z-a_1|\cdots |z-a_k|}$ has a max greater than $R$, and a min less than $R$. ...
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1answer
105 views

Prove that $\Gamma (-n+x)=\frac{(-1)^n}{n!}\left [ \frac{1}{x}-\gamma +\sum_{k=1}^{n}k^{-1}+O(x) \right ]$

Prove that $\Gamma (-n+x)=\frac{(-1)^n}{n!}\left [ \frac{1}{x}-\gamma +\sum_{k=1}^{n}k^{-1}+O(x) \right ]$ I don't know how to do this ? Note that $\gamma $ is the Euler-Mascheroni constant
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2answers
912 views

Solving for coefficients on a Laurent series

I am having an issue with the following complex analysis problem. I am suppose to find the coefficients of $z^{-1}$, $z^{-2}$ and $z^{-3}$ in the Laurent series for $\displaystyle \frac{1}{\sin z}$ ...
6
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1answer
298 views

Entire function with prescribed values

I am trying to solve the following problem from Ahlfors' Complex Analysis Chapter 5, Section 2.3: Suppose that $\{a_n\}$ is a sequence of distinct complex numbers such that $a_n\to \infty$ and let ...
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3answers
89 views

Show that $\dfrac{x}{e^x-1}$ is non-singular near zero

Show that $\dfrac{x}{e^x-1}$ is non-singular near zero. Does this show boundedness near zero?
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2answers
275 views

Entire function which equals exponential on real axis

I need to find all entire functions $f$ such that $f(x) = e^x$ on $\mathbb{R}$. At first it seems that, since the function $f$ will be real analytic on $\mathbb{R}$ and will have a power series ...
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1answer
176 views

Suppose $|f|$ is constant on $\delta D$. Show that $f$ has at least one zero in $D$.

I'm doing some exercises for an exam and I've come across this one from a past comprehensive that I can't solve, can anyone give me any tips/hints? Suppose that $f$ is analytic in a domain $G$ in the ...
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1answer
247 views

Automorphisms on Punctured Disc

I have to find the automorphism group of the punctured unit disc $D = \{|z| <1\}\setminus \{0\}$. I understand that if $f$ is an automorphism on $D$, then it will have either a (i) removable ...
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1answer
76 views

Simple Laurent series question

I'm trying to find the Laurent series expansion of $f(z)=\frac{1}{z}$ around $z=1+i$ (i.e. the series will be in terms of $z-1-i$), but cannot seem to do it. Can anyone help?
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0answers
62 views

what are the borders of the convergence disks of series?

Let $\mathbb{T}=\{z\in \mathbb{C}\mid |z|=1\}$. For which $S\subseteq \mathbb{T}$, is there a sequence $(a_n)\subseteq \mathbb{C}$ such that the series: $$\sum_{k=1}^\infty{a_kz^k}$$ is convergent on ...
2
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0answers
191 views

Zeros of analytic function and limit points at boundary

Let $S$ be the open ball of center $0$ and radius $1$ with $0$ removed in the complex plane. Is the function $f(z)=\sin(1/z)$ a valid example of analytic function defined in an open subspace whose ...
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2answers
472 views

Finding a branch of complex logarithm $\log(z)$ with parabola branch cut

Find a branch of $\log(z)$ on domain $\mathbb{C}\setminus T$ where $T=\{x+iy:x\ge 0, y=x^2\}$ I know the branch cut will be a parabola, branch cuts are usually rays, and my prof. did explain it ...
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1answer
348 views

Characterizing all entire functions that map the unit circle to itself.

Actually, I'm solving the following problem. there are some steps I can't understand. Can you guys help me to understand? The problem is: Find all entire functions that map the unit circle to itself. ...
2
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1answer
106 views

Explicit Representations of Meromorphic Functions as Quotients of Entire Functions

Any meromorphic function can be expressed as a quotient of two entire functions. However many times meromorphic functions are not given as a quotient. Is there a general algorithm or method, given a ...
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1answer
94 views

Why is this integral zero? (Inner product between two 1-forms on a Riemann surface)

I have a quick question regarding the proof of Proposition II.3.2 in Farkas & Kra (pg. 40). The proposition is that if $\alpha$ is a square-integrable, $C^1$ 1-form, then $\alpha$ lives in a ...
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1answer
66 views

Nice parameterization of linear complex structures on the real plane?

A linear complex structure on a real vector space $V$ is an endomorphism $J$ such that $J \circ J=-\mathrm{id}$. What do all the linear complex structures on $\mathbb{R}^2$ look like? If we let $$ ...
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3answers
224 views

True or false - Function analytic in bounded domain is bounded.

If a function is analytic in a bounded domain then it is bounded. True or false. Why.
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0answers
329 views

Length of a curve in the complex plane

Let $f : \Delta \to \Bbb{C}$ be an injective holomorphic function, where $\Delta$ is the open unit disc in the complex plane. Let $g: \Delta \to \Bbb{C}$ be holomorphic such that $g^2 = f'$ For a ...
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1answer
349 views

Plotting a Function of a Complex Variable

I've been teaching myself complex analysis using "Introductory Complex Analysis" by Richard A. Silverman, and I'm a few chapters in and still have no clue how to plot anything. I know that $$ z = x + ...
6
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1answer
152 views

Solving the sequential equation $\sum_{k=0}^{n-1}\frac{a_{n-k}}{k+1}=\frac{1}{n+1}$

$a_1,a_2,a_3,...$ is a sequence of real numbers such that for $n\in \mathbb{N}$: $$\sum_{k=0}^{n-1}\frac{a_{n-k}}{k+1}=\frac{1}{n+1}$$ How can I prove: ...
3
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1answer
131 views

Showing a holomorphic function is constant

Let $h$ be a holomorphic function on the open unit disk $\Delta$. Assume that there exists $\delta \in (0,1)$ such that for any $r$ with $0 \lt r \lt \delta$, $h$ satisfies: $$|h(0)| = 1/{2\pi} ...
0
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2answers
180 views

Is the alternating sum of exp(-exp(n z)) analytic?

Define $f(z) = \frac{1}{\exp(\exp(z))} - \frac{1}{\exp(\exp(2z))} + \frac{1}{\exp(\exp(3z))} - \frac{1}{\exp(\exp(4z))} + ...$ $f(z) = \sum_{n = 1}^{\infty} (-1)^{n-1} \exp(-\exp(n z))$. Is $f(z)$ ...
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3answers
179 views

Proof of $\displaystyle \lim_{z\to 1-i}[x+i(2x+y)]=1+i$

I am having some difficulty with the epsilon-delta proof of the limit above. I know that $|x+i(2x+y)-(1+i)|<\epsilon$ when $|x+iy-(1-i)|<\delta$. I tried splitting up the expressions above in ...