Questions tagged [complex-analysis]

For questions mainly about theory of complex analytic/holomorphic functions of one complex variable. Use [tag:complex-numbers] instead for questions about complex numbers. Use [tag:several-complex-variables] instead for questions about holomorphic functions of more than one complex variables.

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3 votes
1 answer
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If $f(z) = \sum_{n=0}^{\infty} a_n(z-z_0)^n$ has radius convergence $ R > 0$. if $f(z) = 0$ for all $z$ $|z-z_o| < R$ show that $a_0 = a_1 = ... =0$.

If $f(z) = \sum_{n=0}^{\infty} a_n(z-z_0)^n$ has a radius of convergence $R > 0$ and if $f(z) = 0$ for all $z$ $|z-z_o| < R$ show that $a_0 = a_1 = ... =0$. Proof Attempt: If it has a radius of ...
1 vote
0 answers
14 views

$|e^{\mp[f(x) - f(s)]}| < 1 $

Consider $f$ a suitable function so that for $x$ small enough, say $x \in [0,\epsilon]$, $$ |\text{Re}f(x) |= \left\{ \begin{array}{rcl} \epsilon_{1} \omega^{1-\delta},& \mbox{when} & 1 - \...
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0 votes
0 answers
36 views

Complex fuction approximation

If differentiable complex function f satisfies f(0)=0 and we already know f(0.001)=a+bi than How can i approximate f(0.001i) value? What i tried was to think like linear approximation in real function ...
2 votes
0 answers
29 views

How to solve this phasor question without some sort of approximation?

Question: Write $2\cos(100t + \frac{1}{3}) - \sin(100t-1)$ in the form $A\cos(\omega t + \phi)$. Find A, $\omega$, $\phi$ . (Hint: Phasor approach may simplify your task.) (Remark: Leave your answers ...
0 votes
0 answers
22 views

Stein complex analysis chapter 3 problem 2

Let $u$ be a harmonic function in the unit disc that is continuous on its closure. Deduce Poisson's integral formula $$u(z_0) = {1\over 2\pi}\int_0^{2\pi}{1-|z_0|^2\over|e^{i\theta}-z_0|^2}u(e^{i\...
2 votes
0 answers
87 views

Does it mean $\displaystyle\lim_{z=0}\oint\frac{d}{dz}\frac{z}{f(z)^n}dz=0$?

Source of my question My question comes from the book "Introduction to Analysis" (by 高木貞治)  → Chapter 7 "Sequel to Differentiation (Implicit Functions)"  → Section 85 "...
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0 votes
1 answer
29 views

Why is it the case that $|z-\alpha|\leq n\left|\frac{p(z)}{p'(z)}\right|$ for a polynomial $p$ of degree $n$ with root $\alpha$?

I was reading this article which claimed the following: $$|z-\alpha|\leq n\left|\frac{p(z)}{p'(z)}\right|$$ Where $p$ is a polynomial with degree $n$, and $\alpha$ is the zero of $p$ closest to $z$. ...
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0 votes
1 answer
34 views

$\sum{z^n\over (n!)^\alpha}$ is an entire function of order $1/\alpha$ [duplicate]

The problem is from Stein Complex analysis Chapter 5 Problem 3. Show that $\sum {z^n\over (n!)^\alpha}$ is an entire function of order $1/\alpha$. The problem was already posted here, but it seems ...
1 vote
0 answers
17 views

References regards complex analysis on Banach spaces

I'm looking for a references on complex analysis in Banach spaces. What are good books on this subject? Any help is welcome. Thanks in advance.
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1 vote
2 answers
79 views

Evaluate the following complex integral $\displaystyle \int_0^{2\pi} e^{ie^{it}}e^{-it} dt$

I need help evaluating the following complex integral $$\int_0^{2\pi} e^{ie^{it}}e^{-it} dt$$ I know that it evaluates to $2\pi i.$ My try was to use differentiation inside integral sign, with $$I(x) =...
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1 vote
2 answers
64 views

Sheaf of sections supported at a point is coherent?

Let $\mathcal{F}$ be a coherent analytic sheaf over some open subset of $U ⊆ \mathbb{C}^n$. I read in a book that if $p \in U$ then the subsheaf $\mathcal{G}$ defined by \begin{equation} \mathcal{G}(...
0 votes
0 answers
37 views

Solving Laplace's Equation with Complex Analytic Functions?

Let $R$ be the annulus $1/4\leq x^2+y^2\leq 4$ and $u$ satisfy the boundary conditions $u\left(r=2,\theta\right) = \cos3\theta$ and $u\left(r=1/2,\theta\right) = 1$. Find a suitable analytic function ...
0 votes
3 answers
39 views

complex sequence convergence problem

Let $i=\sqrt{-1}$ and $\{i^{\frac{1}{n}}\}_{n=1}^\infty$ then $i^{\frac{1}{n}}\to 1$ Is the convergence correct? I've tried $\epsilon>0$ given and Let's take $N=... \in \Bbb{N}$, $n>N$ and $\|i^{...
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1 vote
1 answer
53 views

Among the following sets, which are open?

Given $U$, an open subset of $\mathbb C$ such that $0\in U$, the question was to select the correct options among the following. (A) $\{e^z:z\in U\}$ is an open subset of $\mathbb C$ (B) $\{\mid e^z\...
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0 votes
0 answers
29 views

How can we prove this function has isolated zeros on a restricted domain?

Given a multi-variable function: $f(x_1, \dots, x_n, y)=\sum_{i=1}^{n}x_iz_{i}^y$, where $z_{i}\in\mathbb{C}$ are fixed constants and the domain of this function is $\mathbb{C}^{n+1}$. Then we further ...
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1 vote
0 answers
56 views

Conditions for an improper integral of a real-analytic function to be real-analytic

Let $U$ be a complex domain and suppose $f:U \times \mathbb{R} \rightarrow \mathbb{R}$ is continuous and, for all $t$, is real-analytic in $s$. That is, $f(s, t)$ is continuous and $g_t(s):=f(s, t)$ ...
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0 votes
0 answers
22 views

Question regarding holomorphic functions. [duplicate]

I have seen this proposition : $f$ holomorphic then partial derivatives of $u(x,y)$ and $v(x,y)$ are continous and satisfy Cauchy-Riemann (page 2 of https://math.berkeley.edu/~vvdatar/m185f16/notes/...
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18 views

A series in fractional powers

I am trying to expand the given function of $r$ in a series: $$\frac{1}{\left (t \left (t-k \right) + i r k \right)^\delta}$$ where $k$ is a finite positive number, $\delta$ lies between $0$ and $1$ ...
0 votes
0 answers
20 views

What is $\omega'$ in the definition of the Kramers-Kronig relation? [closed]

What is omega prime in the definition of the Kramers-Kronig relation?$$\begin{align}\chi_1(\omega)&=\frac{1}{\pi}\mathcal{P}\int_{-\infty}^\infty\frac{\chi_2(\omega')}{\omega'-\omega}d\omega'\\\...
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0 votes
0 answers
28 views

showing the image of unit disc on Complex plane under $f(z)=z^2$ is open

I want to use the fact that $|f(z)|=|z|^2$ and $|z|<1$(so basically the map f maps onto some subset of the unit disc that includes {-1,1,i,-i}) to construct an open disc for any arbitrary point in ...
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2 votes
1 answer
35 views

Show that $\sum_{n=1}^\infty\left({1\over z-a_n}+{1\over a_n}+{z\over a_n^2}+\cdots+{z^{p-1}\over a_n^p}\right)$ is meromorphic

Let $p\geq 1$ be an integer. If $\sum_{n=1}^\infty 1/|a_n|^{p+1}$ converges, show that $$\sum_{n=1}^\infty\left({1\over z-a_n}+{1\over a_n}+{z\over a_n^2}+\cdots+{z^{p-1}\over a_n^p}\right)$$ is ...
0 votes
0 answers
27 views

Integral formula for a complex function

I am working through some problems in complex analysis, and am stumped by the following: Let $f$ be an analytic function in the disk $D=\{z\in\mathbb{C}:|z|<R\}$ and $\gamma$ be the ...
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2 votes
3 answers
93 views

Show that $z\sin\frac{1}{z}$ is unbounded on the unit disc

Let $f(z)=z\sin\frac{1}{z}$ for $z\neq0$ and show that if $|z|<1$ the function is unbounded, here is what I've tried, let $z=x+iy$ $|z\sin\frac{1}{z}|=|z||\frac{e^{\frac{i}{z}}-e^{\frac{-i}{z}}}{2i}...
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0 votes
0 answers
28 views

Let $f(z)=\frac{1}{(z-1)(z+3)}$ be defined on $B_1(1) \setminus \{1\}$ does there exist a function on this domain, $F(z)$ such that $F'(z)=f(z)$?

Let $f(z)=\frac{1}{(z-1)(z+3)}$ be defined on $B_1(1) \setminus \{1\}$ does there exist a function on this domain, $F(z)$ such that $F'(z)=f(z)$? My thoughts are yes because we can split this using ...
2 votes
0 answers
20 views

ML bound conditions

I was wondering under which conditions do the ML bounds apply. For example I was asked to compute $\int_{-\infty}^{\infty}\frac{1}{x^2+1}dx$ using complex analysis. From calculus I know that the ...
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0 votes
0 answers
28 views

Power Series $(\sin(z))^2$ [duplicate]

Power Series $(\sin(z))^2$ for complex $z$: Knowing that $\sin(z) = \sum [(-1)^n z^{2n+1}] /[(2n+1)!] $. If I were to square it, then this is equivalent to multiplying the two series together $\sin(z) ...
0 votes
1 answer
33 views

Calculate $\int_{\gamma}\cot z dz$ where $\gamma=C(0,3)$

I've just started doing complex integrals with residues, and I'm struggling a bit with finding the singularities. The first integral is: $$\int_{\gamma}\cot z dz$$ where $\gamma=C(0,3)$. My solution ...
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0 votes
0 answers
31 views

Find an example of unbounded subset of complex set.

Find an example of unbounded subset of complex set. Attempt: The set $A=\{e^{z}: z \in \Bbb C\} \subseteq \Bbb C$ is unbounded over $\Bbb C$. Proof Let $M>0$ be an arbitrary real number. Then there ...
0 votes
1 answer
28 views

Complex number in polar representation with an unknown variable

For $a \in \Bbb R$ and $z=\frac{a+3i}{5-3i}$, compute Re(z) and Im(z) and express in polar representation. The $Re$ part and $Im$ part is easy enough $Re(z)= \frac{5a-9}{34}$ and $Im(z)=\frac{3a+15}{...
0 votes
1 answer
37 views

Problem with understanding how to use this theory in a concrete example

Im hoping this is a question that is okey to ask, without having the thread removed. Im having trouble understanding how to use a theorem in the book we are using in the complex analysis course. The ...
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2 votes
0 answers
46 views

the zeros of analytic function on $\mathbb{T}^d$

An analytic function $f(\theta) $ on $\mathbb{T}^d$, we can write it $f(\theta)=\sum a_ne^{i \langle n,\theta\rangle}$, we know the zeros of $f(\theta)$ is non dense, and the measure of zeros is $0$. ...
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0 votes
1 answer
38 views

$f(z) =\prod_{n \in\Bbb Z\setminus\{0\}}\left(1+{z\over a+n}\right)e^{-z/(n+a)}$ is an entire function

If $a$ is not an integer, show that $$f(z) =\prod_{n \in\Bbb Z\setminus\{0\}}\left(1+{z\over a+n}\right)e^{-z/(n+a)}$$ is an entire function. Using the series expansion of $e^{-z/(n+a)}$, we get \...
0 votes
0 answers
63 views

How to prove that $f’(z) = e^{\alpha z^{2} + \beta z + \gamma}$?

Given that $f(z)$ is an analytic function of $z$ such that $f^{\prime}(z) \ne 0$ and $$\nabla^{2} \log|f^{\prime}(z)| = 0$$ where $$\nabla^{2} = 4\frac{\partial^{2}}{\partial z \partial \bar{z}}$$ If $...
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1 vote
1 answer
49 views

How to prove that $\log(|f^{\prime}(z)|)$ is Harmonic?

Given that $f(z)$ is an analytic function of $z$ such that $f^{\prime}(z) \ne 0$. I want to show that $$\nabla^{2} \log|f^{\prime}(z)| = 0$$ where $$\nabla^{2} = \frac{\partial^{2}}{\partial x^{2}} + \...
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1 vote
0 answers
35 views

Are the poles of the $\frac{1}{e^{-s}-c}$ term simple?

I'd like to calculate the residues of a contour encircling $$\frac{e^{st}}{s(e^{-s}-c)}$$ in terms of $s.$ There is one simple pole at $s = 0,$ but, if we set the other factor in the denominator $(e^{-...
1 vote
0 answers
27 views

Complex integral different answers for different branch cuts

I tried solving the integral over the circle with radius 2. $$ \int_{|z|=2}{\bar{z}^{\frac{1}{2}}}dz $$ By using the branch $(-\pi,\pi)$ I obtain $8\sqrt{2}i$ Namely $$ \int_{|z|=2}{\bar{z}^{\frac{1}{...
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1 vote
0 answers
35 views

Rudin's RCA: $2.14$ theorem step $X$

This is the definition which we need in the proof: source : Rudin proof of theorem 2.14 RCA inequality explanation Proof. Clearly, it is enough to prove this for real $f$. Also, it is enough to ...
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0 votes
1 answer
73 views

Derive solution to $\int_{0}^{+\infty }\left( \frac{1}{1+s^{2}}\right) ^{\nu }ds$ where $\nu$ is noninteger real

Can anyone derive the solution to $$\int_{0}^{+\infty }\left( \frac{1}{1+s^{2}}\right) ^{\nu }ds$$ where $\nu$ is a non-integer positive real? The derivation has been provided under this and this ...
0 votes
0 answers
7 views

Obtaining a uniform expression for a family of inverses of perturbations of the identity

Let $h$ be a $C^2$ function with compact support. Define the family of functions $h_t(x) = x + th(x)$. They look like pertubations of the identity. For small $t$ we can show that $h_t$ is a $C^2$ ...
0 votes
1 answer
34 views

The Riesz representation theorem Rudin's RCA book: step $X$

This is the definition which we need for the theorem: There is the theorem: Let $X$ be a locally compact Hausdorff space, and let $\Lambda$ be a positive linear functional on $C_{c}(X)$. Then there ...
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0 answers
19 views

Intuition behind Laurent series expansion

I know Laurent series has to do something when function has singularities.I am trying to get idea why Laurent series in such way.. For that i read at somewhere... "A taylor series requires the ...
0 votes
0 answers
32 views

How do we evaluate the inverse laplace transform?

Here's two answered and upvoted questions that do not mention anything about a region of convergence in their premises. Usage of inverse Laplace transform Inverse Laplace Transform properities It is ...
0 votes
0 answers
18 views

Looking for interesting math functions that alternate between multiple different trigonometric parameters

I am looking for functions that do something like f(x) = { log(x) , if x is a multiple of 3; sin(x) , if x is not a multiple of 3 } But not being super random like the above (preferrably somewhat ...
0 votes
1 answer
23 views

Finding the Laurent series of the function $f(z) = \frac{1 - 2z}{(z^2 - z)^2}$ on the annulus $\Delta^*(0,1)$ (i.e. $0 < |z| < 1$)

I am tasked to find the Laurent series of the function $f(z) = \frac{1 - 2z}{(z^2 - z)^2}$ on the annulus $\Delta^*(0,1) = \{z \in \mathbb{C}\mid 0 < |z| < 1\}$. The hint I have been given is to ...
0 votes
2 answers
57 views

Finding poles or removable singularities of $f(z)=z\cot(z)$

Im trying to find pole or removable singularities, and also determining their order of following function: $f(z)=z\cot(z)$ My solution so far is: $f(z)=z\cot(z)=z\frac{\cos z}{\sin z}$ $\Rightarrow$ ...
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1 vote
1 answer
39 views

How to invert linear operators with functional coefficients?

So in this episode we described a way to invert linear operators with constant coefficients, that is operators of the form $O[f] = c_0f + c_1 f' + c_2 f'' ... = \sum_{n=0}^{\infty} c_n f^{(n)}$. Now ...
0 votes
2 answers
41 views

How to invert a linear operator with constant coefficients?

Given a linear operator $O[f] = c_0 f + c_1 f' + c_2 f'' + c_3 f''' ... = \sum_{n=0}^{\infty} c_n f^{(n)} $, where all the $c_i$ are constant is it possible to find a nice closed form for the inverse ...
1 vote
0 answers
62 views

What is a "continuous" Hilbert space?

The book I'm reading about quantum mechanics uses the term "continuous" Hilbert space, whereby apparently in such a space one has a "continuous" basis, e.g. $\{ |x\rangle \}_{x\in \...
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0 votes
0 answers
14 views

The tangency between a non-constant holomorphic map and a a singular foliation on the complex projective plane?

Let $\mathcal{F}$ be a Singular One-dimensional Holomorphic Foliation on the Complex Projective Plane $\mathbb{C} \mathbb{P}^2$. Let $f:\mathbb{C} \longrightarrow \mathbb{C} \mathbb{P}^2$ be a Non-...
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0 votes
0 answers
44 views

Lipschitz continuity for holomorphic function on unit disk

For the open unit disk $\mathbb{D}$ and $A \subset \mathbb{D}$ a subset I want to look at a conformal equivalence (a holomorphic bijection with $f'(z) \neq 0$ on all of $\mathbb{D}$) $$ f: \mathbb{D} \...
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