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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Does There exist a non constant analytic function satisfying these conditions?

Does There exist a non constant analytic function $f:\mathbb{C} \rightarrow \mathbb{C}$ such that $f(0)=1$ and for every $z\in \mathbb{C}$ such that $|z|\geq 1$ we have $|f(z)|\leq e^{-|z|}$?
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24 views

showing that set is measurable - Rudin Real and complex analysis 1.9

Let $ E=\{ x|f(x) =0 \} $ and $ f $ is a complex measurable function, $ f:X \to \mathcal{C} $. Then the text mentions that $ E $ is a measurable set. Assuming standard topology on the complex plane, ...
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37 views

Exchange series and integral in a complex context.

Let $a_{n},b_{n},c_{n}$ complex sequences and let $k>0$ a real parameter. Assuming that $$\sum_{n\geq1}\sum_{m\geq1}\left|\frac{a_{m}b_{n}}{c_{m+n+k}}\right|<\infty\tag{1} $$ if $k>1/2 $ ...
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1answer
24 views

Complex line integral over a square

Evaluate following complex line integral.Let $c=\{z|\max\{|\text{Re}(z)|,|\text{Im}(z)|\}=1\}$ be the square with orientation $+1$. Calculate $$\int_c \frac{z\ dz}{\cos(z)-1}$$ with ...
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2answers
64 views

Is the Hardy space $H^\infty$ closed under differentiation? [duplicate]

** As it's been pointed out, the underlying matter of this question (that $f$ holomorphic and bounded does not imply $f'$ bounded) has been already answered in other post. However, here it's seen as a ...
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37 views

Calculating Infinite Real Integrals Using Residues

I want to calculate the following real integral using residues and I am unsure how to proceed. $$\int_{-\infty}^{+\infty}\frac{1- x^2}{1+ x^4} dx$$ I know I must change this to a contour integral so ...
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28 views

Computation of a Residue (Complex Analysis)

I have attempted to compute this residue: $g(z) = \frac{e^z − 1}{z^3} $ at $z = 0$ and I found that: $\frac{e^z -1}{z^3} =\frac{1}{z^2} + \frac{1}{2! z} + \frac{1}{3!} + \frac{z}{4!} + \ldots $ ...
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27 views

Parametrising a straight line

Question: Parametrise $\Gamma_1=$ {$z\in \mathbb C:r\lt\lvert z \rvert\lt R, \arg(z)= \pi-\delta $} $\Gamma_2=$ {$z\in \mathbb C:r\lt\lvert z \rvert\lt R, \arg(z)= \delta - \pi $} ...
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49 views

What is the method to use the generalised Cauchy Integral Formula

Past Paper Question: a) State the generalized form of Cauchy’s integral theorem b)Evaluate $$\displaystyle f(z)=\int_{\gamma}\frac{z^2}{\biggr(z-\dfrac{\pi}{4}\biggl)^3} dz$$ where $\gamma$ ...
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36 views

Verification on finding the radius of convergence of a Laurent series, “the largest R”.

Question: Determine the largest number $R$ such that the Laurent series of $$f(z)= \dfrac{2}{z^2-1} + \dfrac{3}{2z-i}$$ about $z=1$ converges for $0<|z-1|<R$? Attempt: The radius ...
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29 views

Are homogeneous holomorphic functions necessarily polynomials?

Let $h: \mathbb{C}^n \rightarrow \mathbb{C} $ be holomorphic. If $$h (\lambda z)= \lambda ^d h (z) $$ Always for non zero $\lambda $, is h a homogeneous polynomial?
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25 views

Problem from complex analysis [duplicate]

How to solve such problem: Prove that polynomial $a_0 + a_1 z + ... + a_n z^n$ where $0 < a_0 < a_1 < ... a_n$ has $n$ roots in the circle $|z| < 1$
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2answers
25 views

If Cauchy-Riemann hold and the first order partials are continuous does that imply it is differentiable?

Suppose we have a function $f(z)$ where $f(z)$ satisfies Cauchy-Riemann only one point say $z=0$ then if the first order partials are continuous does that imply it is differentiable at $z=0$ or do we ...
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1answer
25 views

How to prove a complex limit with epsilon delta definition?

I have $$\lim_{z \to i} \frac {iz^3-1}{z+i}=0$$ To prove this I am trying to use the epsilon-delta definition. By saying that for any $\delta >0$ and any $\varepsilon >0$ then: ...
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38 views

Intersection of a smooth plane curve and a circle

Let $\gamma(t)=(x(t),y(t)):[0,2\pi] \rightarrow \mathbb{C}$ be a simple and closed $C^1$-curve. Prove that there is a small circle that intersects $\gamma$ only at two points?
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53 views

Complex analysis - $\bar{z}$ cannot be uniformly approximated by polynomials in $z$ on the closed unit disc in $\mathbb{C}$.

In reference to this question, I tried to prove that $\bar{z}$ cannot be uniformly approximated by complex polynomials in $z$ on the closed unit disc $D$. I came up with a proof, but I'm not entirely ...
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30 views

Laurent series of $f(z)=\frac{1}{z(z-1)(z-2)}$

I am trying to find the Laurent series of the function $$f(z)=\frac{1}{z(z-1)(z-2)}$$in the rings: 1) $0<|z-1|<1$, 2) $1<|z-1|$, 3) $1<|z-2|<2 $ First I expressed $f$ as ...
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16 views

Applications of Stein's Interpolation Theorem

Are there any neat applications for Stein's interpolation theorem, especially in the context of complex analysis? There's a lot of proofs online but I couldn't find many "corollaries" that follow from ...
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1answer
34 views

Uniform convergence of $\sum_{n=-\infty}^{\infty}\frac{1}{n^2 - z^2}$ on any disc contained in $\mathbb{C}\setminus\mathbb{Z}$

I'm currently revising some complex analysis, and need to show that the series $$\sum_{n=-\infty}^{\infty}\frac{1}{n^2 - z^2}$$ defines a holomorphic function on $\mathbb{C}\setminus\mathbb{Z}$. The ...
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1answer
27 views

Some of the Cauchy type integral property

Let $D$ - a simply connected bounded domain and $\phi(t) \in C(\partial D)$. Prove that $ \displaystyle \oint \limits_{\partial D} \frac{\phi(t)}{t - z}dt = 0 $ $ \forall z \notin \overline D $ if ...
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1answer
72 views

Analytic continuation of power series on the unit whose terms tends to 0

This problem is from complex analysis. Set $$f(z)=\sum_{n=0}^{\infty}a_nz^n$$ with convergence radius of 1, and $$\lim_{n \to \infty}a_n=0$$ Prove that if $z_0 \in \partial B(0,1)$ is not a singular ...
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1answer
21 views

Finding $z$ for Complex Convergence

I am having an issue understanding how to go about solving a problem regarding complex sequences. The problem is as follows: Find a $z$ for which the following sequence converges: $f_{n} (z) ...
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1answer
40 views

partial derivative with respect to $\overline{z}$

In my text on complex analysis, they give the definition of $\frac{\partial f}{\partial \overline{z}}$ for suitable $f : \mathbb{C} \rightarrow \mathbb{C}$. However, I do not understand how to make ...
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22 views

$\int \frac{\cos z}{z(z+2)}\mathop{\mathrm{d}z}$

$$\int \frac{\cos z}{z(z+2)}\mathop{\mathrm{d}z}$$ traversing the unit circle counterclockwise. So the singularities are $z=0$ and $z=-2$ but the second is outside the unit circle so it isn't ...
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3answers
47 views

$\int \frac{2+\sin(z)}{z} dz$

Please bear in mind that I am trying to teach myself complex integration having never taken a course in complex analysis, so assume I know very little. $\int \frac{2+\sin(z)}{z} dz$ traversing the ...
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1answer
24 views

Filling gaps in a proof

Consider the smooth map $\gamma:[a,b]\to\mathbb{C}$. Consider a partition of $[a,b]$ $$ P=\{a=t_0<t_1<\ldots,<t_m\} $$ Let's define $$ v=\sum_{k=1}^m|\gamma(t_k)-\gamma(t_{k-1})| $$ and $$ ...
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23 views

Holomorphic function which is not the derivative of a holomorphic function

I don't know how much knowledge of complex analysis is needed to find an open set $U\subset \mathbb{C}$ and a holomorphic function $f\colon U\to \mathbb{C}$, which is not the derivative of a ...
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25 views

Compute the order, type and genus of the entire function $\prod_{n=1}^\infty \left( 1-\frac{\sigma(n)}{n^3}z \right) $

Since $$\sum_{n=1}^\infty\frac{1}{(n^3/\sigma(n))}=\frac{\pi^2}{6}\zeta(3)$$ converges, where $\sigma(n)$ is the sum of divisor function (with maximal size a constant times $n\log\log n$), and ...
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1answer
42 views

how could we compute this infinite real integral using complex methods?

$\int^{\infty}_{-\infty} \frac{cos(x)}{x^4+1}dx$ I know a similar result, but I'm not sure if I can take it for granted, that $\int^{\infty}_{-\infty} \frac{cos(x)}{x^2+1}dx = \frac{\pi}{e}$ The ...
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51 views
+50

radius of convergence of Taylor series, function with branch cuts

Let $f(z) $ being the analytic continuation of some holomorphic function, having many branch points and isolated singularities at $\beta_1,\beta_2,\ldots,\beta_n,\ldots$ is the radius of convergence ...
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1answer
25 views

Suppose $f(z)$ is entire and $Re(f(z))$ is bounded. Show that $f$ is constant

There is a hint that states it might be helpful to consider $exp(f(z))$. I don't see why having a real part would imply that the function is constant.
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3answers
47 views

Finding the antiderivative of $\exp(\cos(z))$ on $\mathbb{C}$

Does this function have an antiderivative on the complex plane? How can this be proven? And if it does, how can we compute it?
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1answer
39 views

$\Gamma(1/2)$ and the Euler reflection formula

With Euler's reflection formula we can show that $\Gamma(1/2)^2=\pi$. Why can't $\Gamma(1/2)=-\sqrt \pi$ ?
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show that $f(z) = \overline{z}^2$ has no antiderivative in any nonempty region

It's been a few quarters since I've taken complex analysis and I'm reviewing for a comprehensive exam. I ran into this problem on a sample exam and it stumped me. I'm guessing I would have to do ...
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1answer
31 views

Characteristic function of a random variable $X$ is absolutely continuous

Let $(\Omega, \mathcal{F}, P)$ be a measurable space and $X:\Omega \to > \mathbb{R}$ a random variable. Assuming $E[|X|]<\infty$, prove that $\psi_X(t)=E[e^{itX}]$ is absolutely ...
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1answer
49 views

Determineing the largest number such that the Laurent series of converges for a trig function.

Question How to determine the largest number $R$ such that the Laurent series of $$f(z)= \dfrac{2sin(z)}{z^2-4} + \dfrac{cos(z)}{z-3i}$$ about $z=-2$ converges for $0<|z+2|<R$? Attempt : Its ...
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1answer
41 views

How to find the Laurent series expansion of an exp function.

Question: How to find the Laurent series expansion in powers of z of a) $f(z)= \dfrac{e^{z^2}}{z^3}$ $\text{where} \left| z \right| > 0$ Attempt: I know that the main idea is to ...
3
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1answer
127 views

Inequality with analytic functions on the unit ball

Let $g(z) = \sum_{n\geqslant 0} a_nz^n$ be an analytic function where $a_n$ only take values in $\{0,1\}$ (not sure if it is a necessary condition, it is just the case I'm considering). Let ...
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24 views

Find the complex integral over a path

I have to integrate the complex function $$z+1/z$$ which is parameterized by $\gamma(t), 0 \le t\le 1$ and satisfies $Im\gamma(t) > 0$, $\gamma(0) = -4+i$ and $\gamma(1) = 6+2i$. Can I assume the ...
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1answer
53 views

Integrating a Complex Exponential Function

Suppose $w=\exp(2i\pi/3)$. How would I go about integrating $$\int\frac{3dx}{e^x+e^{wx}+e^{w^2x}}$$ Is there a transformation i can use? This is an entire function; there is no $x$ that will ...
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1answer
37 views

Prove that $f^{2016}(z)=z(z-1)\ldots(z-2015)$ for an analytic $f$.

Let $\Omega= \mathbb{C}\setminus \{x+iy:~x\in \mathbb{N},~y\geq 0\}$. Prove there there is an analytic function $f$ on $\Omega$, such that $$f^{2016}(z)=z(z-1)\ldots(z-2015).$$ I don't know ...
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Question about using the Cauchy-Riemann equation and showing complex differentiability?

Define $f(x+iy)=u(x,y)+iv(x,y)=xy^3$ then we have $u(x,y)=xy^3$ and $v(x,y)=0$. It follows that $u_x=y^3,u_y=3xy^2$ and $v_x=v_y=0$ so the Cauchy-Riemann equations hold for $(x,0)$ where $x \in ...
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27 views

It is possible to prove that the inverse of every periodic function is a multivalued function?

As I state in the title, I wonder if is possibile to prove that the inverse of every periodic function is a multivalued function. First of all I can't found a counterexample for the statement, and ...
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23 views

Finding a function given its residues

I was given this question : The only singularities in C (the complex set) of the analytic function $f$, are simple poles at z=1 and z=2, with residues at these poles equal to -3 and 7 respectively. If ...
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27 views

Complex variable limit at infinity

Is $\lim\limits_{z\to\infty} \frac{4z^2}{(z-1)^2}$, $z\in\mathbb{C}$, evaluated the same way as a real variable function limit? Or does one need to show separate cases for $x\to\infty$ and ...
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3answers
36 views

Why does Dirichlet Series of Mangoldt Function has simple pole of order 1 at s = 1

Could someone explain why $\sum_{n=1}^\infty \frac{\Lambda(n)}{n^s} = -\frac{\zeta'(s)}{\zeta(s)}$ has a first order pole at $s=1$ with residue 1? That's what I found from Apostol's Introduction to ...
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1answer
29 views

$f_n$ converges uniformly on $\partial \Omega$ then $f_n$ converges uniformly on $\bar{\Omega}$

The problem states that $f_n$ is a sequence of functions which are continuous on the closure of $\Omega$ and holomorphic on $\Omega$ where $\Omega$ is a bounded region and were asked to show that if ...
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2answers
33 views

Laurent series and convergence [duplicate]

"Assume that a complex function $f(z)$ is regular in a neighborhood of $z = 0$ and satises $$f(z)e^{f(z)}= z$$ Write the polynomial expansion of $f(z)$ at $z = 0$ and find its radius of convergence." ...
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3answers
105 views

Is this an equivalent statement to the Fundamental Theorem of Algebra?

Is the following equivalent to the usual statement of the fundamental theorem of algebra: Let $$f(z)=c_nz^n+\cdots+c_1z+c_0$$ be a polynomial with complex coefficients. For all but finitely many ...
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0answers
25 views

Bounding of series

Please help - I tried writing the LHS term as a sum and was hoping to find a 'known' convergent sequence that would bound it, but haven't had any luck. Let R > 1. Show that there is some M > 0 such ...