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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10 views

Automorphisms of the upper half plane

STATEMENT: Suppose $(x_1,x_2,x_3)$ and $(y_1,y_2,y_3)$ are two pairs of three distinct points on the real axis with$$x_1<x_2<x_3 \;\;\;\;\text{and} \;\;\;\;\;y_1<y_2<y_3$$ Prove that ...
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15 views

Maximum modulus principle on open strip

If $\Omega= \{ z\in \mathbb{C} : |\operatorname{Re} z|<\alpha\}$ and $f\in{H}(\Omega)\cap C(\bar{\Omega})$ is such that $f(z)\leq \exp\big(\gamma(|z|) e^{\frac{\pi}{2\alpha} |z|}$ for all $z\in ...
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0answers
9 views

Continuity of Fourier-Stieltjes transform

I read in Kolmogorov-Fomin's (p. 419 here) that, if $F$ is a function having bounded variation on $\mathbb{R}$ then the Fourier-Stieltjes transform$$g(\lambda):=\int_{-\infty}^\infty e^{-i\lambda ...
4
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2answers
74 views

a holomorphic function is uniform limit of polynomials

I was reviewing my complex analysis, and found this problem in a problem set. It says "prove that every holomorphic function on the disc $D=\{|z|<1\}$ is a uniform limit of polynomials". I'm ...
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0answers
31 views

Zeroes of a complex valued function

How to find the zeroes of $f_r(z)=ze^{r-z}-1$ for any $r>0$ and real in say the unit disc? Or in any other domain? I understand we have to use Rouche's theorem somehow but I am not sure what is ...
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1answer
18 views

Finding the poles of a complex function

I am asked to find the poles of: $$f(z)=\sinh \frac{z}{z^2-1}$$ My answer is $\pm 1$. To check I was not missing anything I asked Wolfram (see here) who tells me it cannot find any poles for this ...
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15 views

Proof of convergence of Dirichlet's Eta Function

I'd like to check directly the convergence of Dirichlet's Eta Function, also known as the Alteranting Zeta Function or even Alternating Euler's Zeta Function: ...
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1answer
29 views

Geometric Interpretation of Liouville's Theorem?

The only bounded entire functions in $\mathbb{C}$ are constants. Could someone please give me a geometric interpretation of the theorem above?
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13 views

what is the next term in the Kronecker Limit formula?

The Kronecker Limit formula gives the constant term in the Laurent expansion about s=1 of the Eisenstein series E(s,\tau). What is the next term? I.e., the coefficient of the first power of (s-1)? I ...
3
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20 views

Reversing an “inverse Fourier transform”

Let $g$ be the Fourier transform of an unknown function $y\in L_1(-\infty,\infty)$:$$g(\lambda)=\int_{\mathbb{R}}y(x)e^{-i\lambda x}d\mu_x$$Let $f$ be defined as ...
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15 views

Answer to Plane Trigonometry Ex XLIX Q16?

If $\alpha, \beta, \gamma, \cdots$ be the roots of the equation $sin(mx) - nx cos(mx) = 0$ prove that $\tan^{-1}\frac{x}{\alpha} + \tan^{-1}\frac{x}{\beta} + \cdots + \tan^{-1}\frac{x}{v} = 0$. The ...
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3answers
26 views

Express $f (z) = u + iv$ as a function of $ z = x + iy$?

I have $u(x,y)=\sin(x)\cosh(y)$ and I got conjugate harmonic of $u(x,y)$ as $v(x,y) = \cos(x)\sinh(y)+c$. So, $f(z) = \sin(x)\cosh(y)+\mathrm i(\cos(x)\sinh(y)+c)$. How to express $f(z) = u + iv$ ...
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0answers
13 views

Integrability condition for a complex valued function of a real variable [on hold]

What is the integrability condition for a function from the set of positive real numbers to $\Bbb C$
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26 views

Removable singularity of a harmonic function

Assume that $h$ is harmonic in the punctured unit disk $\mathbb D\backslash\{0\}$ such that $$ \lim_{r\to0}h(re^{it})=0 $$ for all $t\in\mathbb R$. Can $h$ be extended to a function harmonic in ...
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58 views

Is the closed function for zeta(3) equal to this answer? [on hold]

To whom it may concern: Is the closed function for $$\zeta(3) = \sum_{n=1}^\infty \frac{(-1)^ne^{inx}}{n^3}$$ with $x=pi$?. I got this function in the following way: 1.Consider the power series ...
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1answer
40 views

Does this type of function exist?

I am struggling with this question: Determine whether there exists a holomorphic function $f: \mathbb{C}^*\rightarrow\mathbb{C}$, such that $$\lim_{r \rightarrow ...
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2answers
18 views

Find the conditions under which the given equation, in one complex unknown, has exactly one solution, and compute that solution

The equation in question: $az + b\bar{z}+c= 0$ The best i've managed so far, is to assume a and b are real numbers, but c is also a complex number. Then, solving for the real and imaginary parts of ...
3
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0answers
50 views

Problems on Hardy's Introduction to Divergent Series

In the introduction to Hardy's last work, "Divergent Series", he writes that $$\frac{1}{1-e^{ix}}$$ generates this sort of power series: $$1+ e^{ix} + e^{2ix} + \ldots $$ According to (1.2.2) of the ...
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1answer
23 views

How to compute contour integral?

Use Residue theorem to compute contour integral $$\int_C \frac{4e^z}{\sin z} dz$$ I need help figuring out singularities that are within the circle $|z|= 4$. I am stuck at that part. Thanks in ...
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0answers
21 views

Inversion formula for $\int_{\mathbb{R}}f(x)e^{-izx}dx$

Let $f:\mathbb{R}\to\mathbb{C}$ be a measurable function such that$$\forall x\ge 0\quad|f(x)|<Ce^{\gamma_0 x}$$$$\forall x<0\quad f(x)=0$$I must specify that all the integrals I am going to ...
2
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1answer
29 views

Exponent of convergence $\limsup_{n\to \infty}\frac{\log n}{\log |a_n|}$

Let $(a_n)_{n\ge1}$ be the sequence of zeros on an entire function $f$. We define the convergence exponent of $(a_n)_{n\ge1}$ as $$\lambda=\inf\left\{\mu>0\ :\ ...
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10 views

Find the schwarz-christoffel transformation of the upper half-plane U onto the region

Find the schwarz-christoffel transformation of the upper half-plane U onto the region what I have done is considering the other region: let $x_1=-1,\ x_2=0,\ w_1=ia,\ w_2=-R$, then we have ...
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1answer
22 views

Conformal mappings to polygons: why is my integral conformal?

I'm learning about conformal mappings into polygons in a class,(undergrad complex analysis) and am having trouble understanding one of the examples given in my book. (Stein & Shakarchi) Here it ...
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2answers
41 views

$f$ is either univalent or constant

Let $\mathrm{\Omega}$ be an open connected set in $\mathbb{C}$. Let ${f_n}$ be a sequence of univalent analytic functions on $\mathrm{\Omega}$ and assume that ${f_n}$ converges locally uniformly to a ...
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1answer
16 views

Why is $z^{1/k}=|z|^{1/k}e^{iarg(z)/k}$ for $k\in \mathbb{Z}$ true?

I'm reading a short paper here Which attempts to prove/discuss all the properties of the complex logarithm, exponential, and power functions. In the paper, the following statement is made on pg 9, ...
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1answer
38 views

analytic function with a fixed point

let $\omega \subset \mathbb C$ be a simple connected set and $f:\omega \to A$ is an analytic function where $A \subset \omega$ is compact. prove that $f$ has an unique fixed point. i think we can ...
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1answer
33 views

If $f$ is ananlytic in $D_r(z_0)$\ {$z_0$} and $Ref(z)>0$ for all $z\in D_r(z_0)$\ {$z_0$} then $z_0$ is a removable singularity

If $f$ is ananlytic in $D_r(z_0)$\ {$z_0$} and $Ref(z)>0$ for all $z\in D_r(z_0)$\ {$z_0$} then $z_0$ is a removable singularity. I want to prove this statement but I just cannot seem to find a ...
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0answers
19 views

Euler Characteristic $\mathbb{C} \backslash\{0,1\}$ and non-compact surfaces

I am looking at various proofs of the Little and Big Picard Theorems. I am interested in the following question: Without the Uniformization Theorem, can one calculate the Euler characteristic of ...
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13 views

Question about a example of Schwarz-Christoffel Transformation

My question is 1.What the initial point mean for the first integration, and can it be $-\infty$? 2.For the integration of $A \int_{ 1 }^{ \infty } {\frac{dt}{(t^2-1)^\frac{2}{3}}} $,why it ...
4
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1answer
114 views

What is a good, hi-tech textbook on complex analysis?

I am looking for an introductory textbook for Complex Analysis that is hi-tech. All the books I have looked at suffer from the same problem; they're only assuming that the reader is familiar with is ...
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18 views

Entire function with constant argument on the unit circle

What is the easiest way to show that the only entire functions with constant argument on the unit circle (i.e., $\arg(f(z))=const$ for all $z$ on the unit circle) are the constant functions?
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1answer
22 views

Application of Minimum Modulus Principle

How do I use the minimum modulus principle to characterize the family of holomorphic functions $f:\mathbb{D}\to \mathbb{D}$ which have a continuous extension to the closure of $\mathbb{D}$ and map the ...
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0answers
31 views

Misplaced complex analysis intuition on Riemann Surfaces

Next week I will be giving a lecture, based on Chapter 2.6 from Jost's book Compact Riemann Surfaces. He states the following theorem: Theorem 1 (Jost Theorem 2.6.2) Let $S$ and $\Sigma$ be Riemann ...
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1answer
50 views

Hermite functions as eigenvectors of Fourier transform

In order to find an orthogonal basis of eigenvectors of the Fourier transform operator $F:L_2(\mathbb{R})\to L_2(\mathbb{R})$, $f\mapsto\lim_{N\to\infty}\int_{[-N,N]}f(x)e^{-i\lambda x}d\mu_x$ for ...
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3answers
12 views

Not sure how this inequality is formed - $\bigg|\int_0^{2\pi}\frac{e^{p(R+iy)}}{1+e^{R+iy}}idy\bigg| \le \frac{e^{pR}}{e^R - 1}2\pi$

I have the following inequality in my notes - $$\bigg|\int_0^{2\pi}\frac{e^{p(R+iy)}}{1+e^{R+iy}}idy\bigg| \le \frac{e^{pR}}{e^R - 1}2\pi$$ We can start as follows ...
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vote
1answer
60 views

Evaluate $\int_0^{2\pi} \frac{d\theta}{\left(1+\beta \cos (\theta )\right)^2}$

I am trying to evaluate the integral $$\int_0^{2\pi} \frac{d\theta}{\left(1+\beta \cos \left(\theta\right)\right)^2}$$ via change of variables and applying Cauchy's Residue Theorem. Here is how I'm ...
4
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4answers
125 views

Similar to Cauchy inegral formula

Let $f=u+iv$ be an analytic function in disk $\mathbb{D}$ and $0<r<1$. Can you help me to prove that $$\pi{r}f'(0)=\int_{0}^{2\pi}\frac{u(re^{i\theta})}{e^{i\theta}}d\theta\;\;\;?$$ I tried ...
3
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1answer
79 views

Simple proof that there is no isomorphism between any two of $ Aut(\hat{C}) $,$ Aut(H^+) $(upper half plane) and $ Aut(C) $

Referring the groups of automorphisms (holomorphic bijections) of the respective domains. This is a homework problem. Is a basic course, so sophisticated answers may not be of help (it has a simple ...
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2answers
18 views

Uniform convergence of the series $\sum_{r=1}^{\infty} \frac{1}{(r-z)^2}$ in a neighborhood

I am asked to show that the function $f(z)=\sum_{r=1}^{\infty} \frac{1}{(r-z)^2}$ defined on $\mathbb{C} \setminus \mathbb{N}$ is holomorphic assuming that the series $\sum_{r=1}^{\infty} ...
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1answer
62 views

Exercise - analytic function

Assume that $f$ is an analytic function on $|z|<2$ $f(0)=f'(0)=0$ , $f''(0) \not= 0$, $f(1/3)=i/12$ , $|f(z)|\le3$ for $|z|<2$ then find the value of $f(2i/3)$. Thank you
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23 views

Center of real projective line or Riemann sphere

I have recently encountered the ideas of the real projective line and the Riemann sphere, and it seems to me that in any circle (representing the real projective line) or sphere, the center is a ...
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1answer
48 views

Transcendental solution to system of equations

Suppose $A$ is a set of polynomials:$$P_1(x,y_1,\dots,y_n)=0,$$ $$P_2(x,y_1,\dots,y_n)=0,$$ $$\vdots$$ $$P_k(x,y_1,\dots,y_n)=0$$ is a system of equations with coefficients over $\mathbb{Z}$, and ...
0
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2answers
20 views

Why does ray for trigonometric functions not need to be cut?

My question is for complex variables, I understand that ray of log Z needs to be cut starting from the origin (since log 0 does not exist) and give a domain for the theta values, so we can have ...
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0answers
21 views

Is the error function in the complex plane bounded? [on hold]

I have to show that the $ erf (\sqrt{(\lambda / 2) }r(t) x)$ is bounded where $r(t)$ is only bounded when $\lambda < 0$
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2answers
57 views

Evaluate $\displaystyle\int_{-\infty}^{\infty}\frac{dx}{(1+x^2)^2}$ using residue method [on hold]

This is a real integral but I want to evaluate it using residue integration method $$\int_{-\infty}^{\infty}\frac{dx}{(1+x^2)^2}$$
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1answer
33 views

Find a T such that T maps the real axis onto itself and the imaginary axis onto the circle $|w-\frac{1}{2}|=\frac{1}{2}$

Find a linear fractional transformation T such that T maps the real axis onto itself and the imaginary axis onto the circle $|w-\frac{1}{2}|=\frac{1}{2}$ I have no idea how to do this kind of ...
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0answers
14 views

classifying singularities of functions

(a) $$\frac{\pi}{tan \pi z}$$ (b) $$\frac{z^2-z}{1 - sin z}$$ -- for part (a), I found the singularities to be $z = n$ $\space$ $\forall n \in \mathbb{Z}$ and for part (b), $z = \frac{\pi}{2} + ...
0
votes
2answers
36 views

Showing a certain complex function is surjective

Can you help me show that $f(z)=z+e^{z}$ is surjective onto $\Bbb C$? The idea is to show that for any $z$, we can construct a closed curve $C$ around $z$ such that $z$ is contained in $f(C)$ with ...
1
vote
1answer
20 views

Prove of an addition theorem for the general binomial coefficients

Prove that: $\sum_{k=0}^n \binom{s}{k} \binom{t}{n - k} = \binom{s + t}{n}$ for all $s, t \in\Bbb C $, $n \in N\cup {0}$. That's pretty much all I'm given, and therefore, I haven't come quite far ...
0
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0answers
17 views

How to determine the limit of a complex function

It is easy to show that a complex function doesn't have a limit as it approaches a certain point, but is there any way to know for sure whether any given complex function has a limit as it approaches ...