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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rate of growth of function with specified zeros

One example of this theorem wanted? Let $ \Lambda =\{\lambda_{n}\}_{n=1}^{\infty} $ be a sequence of distinct complex numbers satisfying the following two properties: (1) $ ...
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83 views

Existence of an entire function with prescribed zeros and asymptotic behavior

One example of this theorem wanted? Let $ \Lambda =\{\lambda_{n}\}_{n=1}^{\infty} $ be a sequence of nonzero complex numbers with the following three properties: There is a constant $ ...
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1answer
55 views

Find 5 odd numbers(1 to 15) to get 30 [duplicate]

---- + ----- +---- + ----- +---- =30 FIll these 5 boxex/dots using 1,3,5,7,9,11,13,15. You can also repeat numbers
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2answers
81 views

An analytic function satisfies $f(1/z)=f(z) $, if $f$ is real on $\{|z|=1\}$, then the coefficients of expansion are real.

An analytic function satisfies $f(1/z)=f(z),\forall z \in \mathbb{C}\backslash\{0\} $, if $f$ has real values on $\{|z|=1\}$, then the coefficients of the Laurent expansion are all real and. Here is ...
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1answer
74 views

Completion of Complex Numbers

In some way, $\mathbb{C}$ completes $\mathbb{R}$, why is there nothing that completes $\mathbb{C}$? Is it just more so that we don't want anything more than $\mathbb{C}$, or is there a property of ...
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1answer
70 views

asymptotical behavior of integral

I'm interest in the asymptotical of $$\int_{-\pi}^{\pi}\exp\Big((\cos z+i\alpha\sin z-1)t\Big)dz\hspace{3mm}\text{as}\hspace{2mm}t\to\infty$$ for $-1<\alpha<1$. Numberical result suggest that ...
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31 views

what does projective line of degree one mean?

I know what is projective line, but I'm confused about degree one. Can someone tell me what is "projective line with degree one"?
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3answers
56 views

Show that the complex closed line integral $\oint\frac{\mathrm{d}z}{p(z)}$ is $0$ ($p$ is polynomial)

Let $p$ be a polynomial of degree $n\geq2$ and has $n$ different roots $z_1,\dots,z_n$. Prove that $\oint\frac{\mathrm{d}z}{p(z)}=0$ where the closed path is large enough so that all roots are in the ...
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1answer
60 views

alternate proof of the fundamental theorem of algebra

I was reading over my notes from complex analysis and saw the fundamental theorem of algebra which states that: A polynomial of positive degree over a field $\mathbb{C}$ of complex numbers has a ...
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122 views

${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$

I have found the following new result connecting two rational log-cosine integrals. Proposition. \begin{align} \displaystyle & {\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos ...
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28 views

If $f\in H(\mathbb{C})$ and $f(e^\frac{2\pi i}{m}z)=f(z)$, then there exists $g\in H(\mathbb{C})$ such that $f(z)=g(z^m)$.

I have a problem I am trying to solve from an old qualifying exam. It is as follows: Suppose that $w=e^\frac{2\pi i}{m}$ where $m$ is a positive integer, and $f\in H(\mathbb{C})$ satisfies ...
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2answers
103 views

Omitting $i$ in calculations

Is it possible in various calculations related to the complex plane which also include analytic geometry , calculating distances etc, to omit $i$ and treat the imaginary axis as simply the cartesian ...
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0answers
16 views

Conformal map two slits to circles

I am trying to find a conformal mapping that maps a double slitted plane onto a plane with two circles. The two slits are both located along the real axis with similar lengths. For a single slit ...
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5answers
226 views

How to find $\sum_{k \in \mathbb{Z}}\frac1{(k+a)(k+b)}$

Let $a,b$ be two unequal integers. I have to find the sum below. $$ \sum_{k \in \mathbb{Z}}\frac1{(k+a)(k+b)} $$ I should use complex analysis, but I have no clue where to start. I only now that I can ...
6
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1answer
53 views

Given two holomorphic functions on a region find two other such that…

Let $\Omega$ be a region in the complex plane and let $f_1$ and $f_2$ be holomorphic functions on $\Omega$ having no common zero. Show that there exist holomorphic functions $g_1$ and $g_2$ on ...
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2answers
69 views

Calculating $\int_0^\infty \frac{\ln x}{(x^2+9)^2} dx$

I try to calculate $$ \int_0^\infty \frac{\ln x}{(x^2+9)^2} dx $$ I use a book that tells me to replace $\ln x \ $ by $ \ \ln(|x|) + i\phi_z$ where $\phi_z$ denotes the argument of $z$, chosen between ...
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1answer
23 views

There is no holomorphic $f$ such that for all $z$ in a neighborhood of $0$, $f^2(z)=\sin z$

Prove doesn't exist function $f(z)$ which is analytic in neighborhood of $0$ (defined to be $S$) such that $\forall z\in S, f^2(z)=\sin z$. I think that the argument principle (or Liouville ...
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1answer
63 views

Prove that A is both open and closed. [closed]

Let $X = \{ z : | z | \leq 1 \} \cup \{ z : | z - 3 | < 1 \} $ be a subset of $\mathbb{C}$. Let the metric be the usual metric $d(x,y) = | x-y |$. Prove that the set A = $\{ z : | z | \leq 1 \}$ ...
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3answers
80 views

How to get a function if you have the Fourier coefficients

So I have $$H(e^{i\omega})=\sum_{n=-\infty}^\infty C_ne^{i\omega n}$$ and I know that: $$C_n = \frac{2}{\pi n}\sin^2\left(\frac{\pi n}{2}\right)$$ How can I work out the function that this makes? I ...
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1answer
49 views

Why do we define the complex exponential as we do?

Why do we define the complex exponential as we do? Defining $e^{z}$ as $e^{x}e^{iy}$ certainly seems to make sense, but I'm not sure the formal reason as to why it's defined like this. Was it from ...
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1answer
33 views

More preliminaries of the Martingale Convergence Theorem

Really struggling with this lemma. Not sure about the general structure of the proof. Why have we chosen g to be orthogonal to all functions of the form 4.3.1? Why should $G(\lambda)=0$, does it ...
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47 views

Contour integration on a rectangle

Another qualifier problem: Suppose $f(z): \mathbb{C} \mapsto \mathbb{C}$ is entire and $\exists M,A>0$ such that $$ |f(x+iy)| \leq \frac{A}{1+x^2} e^{2 \pi M |y|} \: \forall x,y \in \mathbb{R}. ...
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22 views

$\dim \mathcal{S}_k(\Gamma_0(N))$

I'm looking for a formula which gives the dimension of $\mathcal{S}_k(\Gamma_0(N))$ the space of cusp forms of weight $k$ and level $N$. I found the following statement for $k\geq 4$ $$\dim ...
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2answers
42 views

If the real part of analytic function satisfies $u_x=u_y$, then the function is linear

Let f(z) be analytic function and $\forall z=x+iy\in\mathbb C, u_x=u_y$ ($u_x=\frac{\partial f}{\partial x},u_y=\frac{\partial f}{\partial y}$. Prove that $f(z)=az+b$. I thought using Cauchy ...
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25 views

Variation of the argument of a rational function along a circle

Crossposted here on MO. Let $f:\bar{\mathbb C}\to \bar{\mathbb C}$ be a rational function, and take a circle $C$ not crossing the zero- and polar-locus of $f$. The argument principle tells us the ...
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1answer
27 views

Rigorous Definition of Complex Functions, Derivatives and Arithmetic.

I'm a little confused when it comes to the definition of complex functions, for instance the complex logarithm. I know that $i$ can be shown to behaive as a constant, and that it is a constant, but ...
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1answer
40 views

Book Recommendations for Picard Big and Little Theorems

Does anybody have book recommendations for reading about Picard's Little and Big Theorems? Preferably, I am looking for a book that is intended for an undergraduate/first year graduate student who ...
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28 views

Uniqueness of Analytic Continuation

I wasn't very well introduced to Analytic Continuations, but from what I have seen, showing that the analytic continuation is unique is pretty simple. In Real Analysis, from what I can imagine, there ...
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1answer
29 views

Convergence of a Complex Power Series at the radius of convergence

I am currently reviewing some complex analysis, and have come across this question which I absolutely have no idea on how to attempt: Suppose the radius of convergence of the power series $f(z) = ...
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58 views

Proving that two functions involving integrals with Legendre polynomials are equal

I have two functions that I expect to be equal (where $P_{2l}$ are the even Legendre Polynomials): $$F_{2l}(x)=x\, \tanh(\pi x/2)\left|\int_0^1 u^{i x-1}P_{2l}(u)\,du\right|^2$$ ...
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1answer
32 views

Entire function $f$ such that $\lim_{|z| \to \infty} |f(z)|=\infty$

Problem Let $f$ be an entire non constant function such that $\lim_{|z| \to \infty} |f(z)|=\infty$. Prove that $f$ has a positive and finite number of zeros. If $f$ is entire, then $f$ can we ...
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1answer
12 views

Landau Constants precision

Looking at an article in Wikipedia on Landau constants it indicates that the actual values are not known except that they are within a certain interval. This seems surprising to me since for most ...
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31 views

Density of rational functions in open Stein sets

Lately I have been wondering on this problem: if $U \subset \mathbb C^n$ is an open Stein and I denote by $\mathcal R(U)$ the set of rational functions on $\mathbb C^n$ whose restriction to $U$ is ...
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1answer
35 views

Proving that $\lim\limits_{z\to z_0}\frac{f(z)}{g(z)}=\lim\limits_{z\to z_0}\frac{f^\prime(z)}{g^\prime(z)}$

Let $f,g$ both analythic in neighbourhood of $z_0$ and they both have zero of multiplicity $n$ in $z_0$. Prove that $\lim\limits_{z\to z_0}\frac{f(z)}{g(z)}=\lim\limits_{z\to ...
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2answers
69 views

Finding $\int_0^{2\pi}\frac{\sin t + 4}{\cos t + \frac{5}3} dt$

I'd like to ask something about the following integral: $$ \int_0^{2\pi}\frac{\sin t + 4}{\cos t + \frac{5}3} dt $$ I rewrote and took another variable. $$ ...
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0answers
29 views

implicit non-linear equations with complex variables

I am trying to understand a methodology for solving implicit non-linear equations with complex variables. I would like to solve for z1 below where z2 is known. Also both z1 and z2 are complex ...
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68 views

another question about singularity and converge uniformly.

a)Prove if $n\in \mathbb{Z}$ then, $\frac{1}{\sin^2{z}}-\frac{1}{(z-\pi n)^2}$ has a removable singularity at $z=\pi n$ when $n=0$. Actually this is true for all $n$. b) Prove that ...
0
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1answer
28 views

Harmonic function and Poincaré metric

Let $u$ be a harmonic function on the unit disk $\Delta$, taking values in $[0,1]$. Is it true that this implies that $u$ is Lipschitz for the Poincaré metric ? If not, what can be said about a ...
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2answers
23 views

What are the complex solutions of a linear homogenous ODE of order $n$ with constant coefficients?

What are the complex solutions of a linear homogenous ODE of order $n$ with constant coefficients? Where can I read a proof? p.s. I don't even see the answer to the first question with a google ...
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1answer
69 views

Solution of $\exp(z)=z$ in $\Bbb{C}$.

I have posted a related question here. I thinkg this one is more interesting: What about the solution of $\exp(z)=z$ in $\Bbb{C}$? My try : $z \mapsto e^z - z$ is entire non-constant. Perhaps ...
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1answer
35 views

References on estimating capacities (Newton, Martin etc) for sets & alternative formulations.

By G-capacity for capacitable set K I mean: $Cap(K)=[inf\{\int\int G(x,y)d\mu(y)d\mu(x):\mu$ probability measure on K$\}]^{-1}$. where G(x,y) is any kernel eg. the Green kernel. Q1:We've calculated ...
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3answers
112 views

all complex solutions of $z\sin(z)=1$?

A possibly easy question, Can we find all complex solutions of $z\sin(z)=1$ ? My try: Let $$\sin(z) = \frac{e^{iz} - e^{-iz}}{2i}$$ so we have $$ z\frac{e^{iz} - e^{-iz}}{2i}=1 $$ Not sure how ...
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49 views

Choose appropriate contour for a complex integral

I have a problem to solve integral $$ I = \int^{\infty}_0 \frac{\mathrm{d}x}{(x-z)(1+x^2)^{\kappa+2}} $$ I can solve the same integral with borders $-\infty$ to $\infty$ using residue theorem but ...
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0answers
24 views

generalized expression required

suppose i have a set $ {0,1,2.......x-1}$ Now I am generating an i length sequence using the numbers from above set...${a0,a1,....ai}$ where all $ai$$>=0 $ and $ai<=x-1$ Note numbers may ...
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14 views

Solution of definite integral of product of bessel function and exponential

I have an integral $I=\int_{\theta} \int_r J_m(k_1r)e^{-j[P_x r \cos(\theta)+P_y r \sin(\theta)]} r dr d\theta$ $0\leq\theta\leq2\pi; r<\infty$ is there any method to solve this?
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3answers
103 views

find all real valued harmonic functions on the plane that are constant on all vertical lines

find all real valued harmonic functions on the plane that are constant on all vertical lines. Harmonic function is a twice continuously differentiable function that satisfies Laplace's equation. ...
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1answer
22 views

The integral along a circle of the inverse linear function is zero

Assume ${\rm C}$ is a circle and $a,b$ are distinct points in the interior of ${\rm C}$. How can we see that the complex integral $$ \frac{1}{b - a} \int_{\rm C}\left(\frac{1}{z - a} - \frac{1}{z - ...
2
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0answers
36 views

Functional Equation involving derivatives and time-steps [duplicate]

I am attempting to solve the equation $$f(x + 1) = f'(x)$$ for distributions $C \rightarrow C: f(x)$ My first guess to exploit the fact that this seems similar to identity $$\sin\left( ...
5
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2answers
55 views

How to compute $\int_C {e^{3z}-z\over (z+1)^2z^2}$?

I am asked to compute the integral $$ \int_C {e^{3z}-z\over (z+1)^2z^2} $$ where $C$ is a circle with the center at the origin and radius ${1 \over 2}$. My approach was to separate the integral as a ...
3
votes
1answer
70 views

Solving $|z-3| \leq|z-1-i|$

I was trying to solve graphicly: $$|z-3| \leq |z-1-i|$$ I plugged x and y in proper places as real componenets of the comlex number yielding in the end $-4x+2y+7 \leq0$ this might be tackled if ...