Tagged Questions
2
votes
4answers
81 views
Radius of Convergence of power series of Complex Analysis
I have come across the following few questions on past exams papers.. I know how to solve these type when it is of the form $a_nz^n$ but don't have a clue what to do with these. Any help would be ...
1
vote
3answers
101 views
showing $1 + z + z^2 + \dots $ uniformly converges to $\frac{1}{1-z}$ for $|z| < 1$
What test can I use to show that $1 + z + z^2 + \dots $ uniformly converges to $\frac{1}{1-z}$ for $|z| < 1$.
I know $\displaystyle 1 + z + z^2 + \dots +z^n = \frac{1-z^{n+1}}{1-z}$ and as $n \to ...
3
votes
2answers
69 views
Convergence of Roots for an analytic function
Show that the roots of
$$
f(z) = z^n+z^3+z+2 =0
$$
converge to the circle $|z|=1$ as $n \to \infty$.
0
votes
1answer
30 views
Study the convergence of this sequence of functions
I have the following sequence of function:
$$f_n(\lambda)=\bigg[\alpha-i\bigg(\lambda+\frac{1}{n}\bigg)\bigg]^{-1}-\bigg[\alpha-i\lambda\bigg]^{-1},\,\,\,\alpha\neq 0$$
and I have to study its ...
0
votes
1answer
26 views
how to show uniform convergence for sequence $u_n(z) = n z e^{-nz^2}$ such that $\Re[z^2] > 0$
How to show uniform convergence for $u_n(z) = n z e^{-nz^2}$ such that $\Re[z^2] > 0$
Here is my attempt letting $z = x + iy$:
$$
\begin{align*}
|u_n - 0| &= |nz e^{-nz^2}| \\
&\le n ...
0
votes
1answer
43 views
help for convergence of an integral
Does this integral converges:
$$\int_{-\infty}^{\infty}\frac{\mathrm e^{ -\mathrm{i}\omega t}}{\sqrt{(-1)^\frac{1}{3}(\mathrm{i}\omega)^\frac{4}{3}}}\,\mathrm{d}\omega$$
and why ?
0
votes
1answer
29 views
Zeros of the analytic limit of complex rational function
For $n\in\mathbb{N}$ let $r_n,\ s_n$ be two polynomials of $O(n)$ degrees with real positive coefficients and set $f_n=r_n/s_n$.
Suppose there exists $c>0$ such that
$\bullet$ if $z\in\mathbb{C}$ ...
0
votes
1answer
38 views
Zeros of the analytic limit of complex polynomials
For $n\in\mathbb{N}$ let $p_n$ be a polynomial of degree $n$.
Suppose there exists $c>0$ such that
$\bullet$ if $z\in\mathbb{C}$ is a zero of a $p_n$, then $|z^2+c|\leq c$ (note that in particular ...
7
votes
1answer
111 views
Does a convergent power series on a closed disk always converge uniformly?
If I have a power series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i} \in\mathbb{C}[[z]]$ with radius of convergence $r>0$ and I know that the series $\displaystyle\sum_{i=0}^{+\infty} {a_iz^i}$ ...
0
votes
0answers
28 views
Proof that a sequence of numbers $A_i$ is an infinite product of complex residues correct? [closed]
Is the above proof correct? Assume that theorem 2 is true without proof
Thanks
Edit: At the end it should be $2^n$, not $2n$
1
vote
1answer
56 views
Behavior at infinity.
Classify the behavior at $\infty$ for $$f(z)=\frac{\sin z}{z^2},\,g(z)=\frac{1}{\sin z},\,h(z)=\exp\left(\tan\frac{1}{z}\right).$$
So I just considered $f(1/z),g(1/z),h(1/z)$ at $z=0$. For $f$ I ...
1
vote
1answer
82 views
Radius of convergence of Maclaurin series for $\frac1{\sin z}-1/z+\frac{2z}{z^2-\pi^2}$
What is the radius of convergence of the Taylor series about $z=0$ for $h(z)=\frac1{\sin z}-1/z+\frac{2z}{z^2-\pi^2}$?
Here's a plot
...
1
vote
1answer
55 views
Absolute convergence of $\sum\limits_{n=0}^{\infty} \frac {z} {(z+n)^2}$
I want to check the absolute convergence of $\displaystyle\sum_{n=0}^{\infty} \frac {z} {(z+n)^2}$ in the half plane $\Re(z)>0$, and to see if the convergence is uniform or locally uniform. How do ...
0
votes
1answer
58 views
Radius of convergence of $\sum_{-\infty}^{\infty}3^{-|n|}z^{2n}, z \in \mathcal{C}$
I want to find out the radius of the following power series of a complex variable:
$\sum_{-\infty}^{\infty} 3^{-|n|} z^{2n}, z \in \mathbb{C}$
The ration test $\lim_{n \to ...
2
votes
2answers
124 views
Confused by Laurent series
A typical problem related to Laurent series is this:
For the function $\frac 1{(z-1)(z-2)}$, find the Laurent series expansion in the following regions: $\\(a) |z|<1, \\ (b) 1<|z|<2, ...
2
votes
1answer
53 views
Confused over analytic functions, point convergence of power series
It is well-known that a power series sums to a function that is analytic at every point inside its circle of convergence and that conversely, if a function is analytic on an open disc then its Taylor ...
3
votes
2answers
103 views
How do I obtain the Laurent series for $f(z)=\frac 1{\cos(z^4)-1}$ about $0$?
I know that $$\cos(z^4)-1=-\frac{z^8}{2!}+\frac{z^{16}}{4!}+...$$ but how do I take the reciprocal of this series (please do not use little-o notation)? Or are there better methods to obtain the ...
0
votes
1answer
63 views
Convergence radius of power series
I am trying to solve an exercise, but i am not sure that the result i get at the end is correct...May i kindly ask you for a little help or a remark?
Find the radius of convergence of the following ...
1
vote
1answer
91 views
Existence of Riemann-Liouville Integral
The Riemann Liouville integral is defined as:
$\frac{1}{\Gamma\left(\nu\right)}\int\limits _{h}^{t}\left(t-\xi\right)^{\nu-1}f\left(\xi\right)d\xi$
It is supposed it does exist for all $\nu>0$ and ...
0
votes
2answers
131 views
Is the alternating sum of exp(-exp(n z)) analytic?
Define $f(z) = \frac{1}{\exp(\exp(z))} - \frac{1}{\exp(\exp(2z))} + \frac{1}{\exp(\exp(3z))} - \frac{1}{\exp(\exp(4z))} + ...$
$f(z) = \sum_{n = 1}^{\infty} (-1)^{n-1} \exp(-\exp(n z))$.
Is $f(z)$ ...
7
votes
1answer
252 views
Is there a $k$ such that $a_n=\frac{n^k!}{(n^k!!)^2}$ converges?
Lately I have been playing around with the sequence $$a_n(k) := \frac{n^k!}{(n^k!!)^2}.$$
For $k=1$, it does not look much like it converges.
I don't know $k=2$ it converges, but it doesn't really ...
10
votes
2answers
267 views
Is $\frac{1}{\exp(z)} - \frac{1}{\exp(\exp(z))} + \frac{1}{\exp(\exp(\exp(z)))} -\ldots$ entire?
Let $z$ be a complex number. Is the alternating infinite series $ f(z) = \frac{1}{\exp(z)} - \frac{1}{\exp(\exp(z))} + \frac{1}{\exp(\exp(\exp(z)))} -\ldots$ an entire function ? Does it even converge ...
-2
votes
0answers
166 views
Fibonacci Numbers - Complex Analysis [duplicate]
Possible Duplicate:
Complex Analysis - Integral over a circle of radius R
Hey guys~ Does anyone know where to find the solutions to this problem set on page 106 involving the fibonacci ...
1
vote
1answer
262 views
Radius of convergence for the exponential function
I'm studying physics and am currently following a course on complex analysis and in the section on analytic functions, the radius of convergence $R$ for power series was introduced. The Taylor ...
3
votes
2answers
109 views
Divergence of an infinite product
How can I prove that the infinite product
$$\displaystyle\prod_{n=1}^{+\infty}(1+z^{2n})$$
diverges for
$|z|>1$?
1
vote
1answer
190 views
Convergence of infinite product
This could be something which is already somewhere in the website, but I am unable to locate any.
Prove $$\prod_{n=1}^{\infty} (1-z^n)$$ converges absolutely and uniformly on each compact subset of ...
15
votes
3answers
396 views
A question on convergence of series
Suppose $(z_i)$ is a sequence of complex numbers such that $|z_i|\to 0$ strictly decreasing. If $(a_i)$ is a sequence of complex numbers that has the property that for any $n\in\mathbb{N}$
$$
...
1
vote
0answers
55 views
Uniform convergence of complex exponent derivative
I'm trying to prove the following:
Let $\Re z > 0$. Then $$\lim_{\varepsilon \to 0} \frac{t^{z + \varepsilon} - t^z}{\varepsilon} = t^z \log t$$ uniformly in $t \in [0,1]$.
I've tried to ...
1
vote
2answers
86 views
Find the limit of $\frac{\bar{z}}{z}$ as $z$ goes to $0$.
I put it in exponential form to get $\dfrac{re^{-i \theta}}{re^{i \theta}}$ but I think I'll get $\frac{0}{0}$ which isn't defined and isn't a good enough proof to say it doesn't have a limit.
0
votes
0answers
79 views
Presentation Topic: Convergence rate of Power Series
I am required to do a 10-15 minutes presentation in a Complex Analysis class on the following idea:
"Discuss convergence rates of power series, and numerical aspects of how to write good algorithms ...
0
votes
2answers
80 views
Where's the singular point of the series $\sum_{v\ge 1}v^{-2}z^v$?
The radius of convergence of the series $\sum_{v\ge 1}v^{-2}z^v$ is 1, so there must be a singular point on the boundary. But for every $|\zeta|=1$,
$|\sum_{v\ge 1}v^{-2}\zeta^v|\le\sum_{v\ge ...
0
votes
1answer
99 views
Infinite product of entire functions
Is it true that "an infinite product of entire functions $\{ f_{n}(z) \}_{n=1}^{\infty}$ is again an entire function"?
$$\prod_{n=1}^{\infty}f_{n}(z)$$
(entire function is a function which is ...
1
vote
1answer
234 views
Error bound for Taylor approximation to $\tan(x)$
I'm looking to bound the error for the Taylor series of $\tan(x)$ so that I will know how many terms I need to go out to get a desired precision. I've already searched and came across this, but the ...
1
vote
1answer
279 views
Complex analysis: Radius of convergence of power series
$$
\sum_{n=1}^\infty \frac{\cos n \theta}{(\sqrt{13})^{n+1}}x^n
$$
Find the radius of convergence for the above series. I have learnt to use the root test and ratio test but neither of them seem to ...
6
votes
0answers
112 views
$L^{2}(\mathbb R)$- norm of entire function
Let $f(z)$ be an entire function defined by $$f(z)=\prod_{n=1}^{\infty}\bigg(1-\frac{z^{2}}{a_{n}^{2}}\bigg),\qquad z\in \mathbb C$$
where $\{a_{n}\}_{n=1}^{\infty}$ is a sequence of positive real ...
6
votes
1answer
264 views
Uniform convergence of infinite series
Suppose $f$ is a holomorphic function (not necessarily bounded) on $\mathbb{D}$ such that $f(0) = 0$. Prove the the infinite series $\sum_{n=1}^\infty f(z^n)$ converges uniformly on compact subsets ...
2
votes
1answer
143 views
Bounded power series without convergence on the boundary
Does there exist a holomorphic function $f\in H^\infty$ (i.e. holomorphic and bounded on the open unit disc) such that its power series expansion at $0$ doesn't converge anywhere in $\{|z|=1\}$?
0
votes
1answer
91 views
Convergence in the upper half-plane
I have a sequence $\{F_{n}(z)\}_{n=1}^{\infty}$ of analytic functions in the open upper half plane $\mathbb H$ and continuous on $\mathbb R$, such that $|F_{n}(z)|\leq 1$ for all $n\geq 1$, and all ...
3
votes
0answers
223 views
Uniform convergence of analytic functions
Let $f_{n}(z), g(z)$ be entire functions, for all $n\geq 1$. Suppose that $g(x)$ doesn't vanish on $\mathbb H\cup\mathbb R$ (so we have $\frac{f_{n}(z)}{g(z)}$ analytic on $\mathbb H\cup\mathbb R$). ...
1
vote
0answers
253 views
Convergence of sequence of analytic functions
Given $\{f_{n}(z)\}$, a sequence of analytic functions in the upper half plane $\mathbb C^{+}$, where each $f_{n}(z)$ has continuous extension to the real line, and $|f_{n}(z)|\leq 1$ for all $z\in ...
7
votes
1answer
123 views
Zeros in the complex plane and convergence
I'm doing some number theory which requires some work in $\mathbb{C}$, but unfortunately my complex analysis is a little rusty.
A text I am reading states the following:
...and given that ...
2
votes
1answer
73 views
Convergence to zero of a sequence
Let $c > 0$. I'm trying to show that the sequence $\displaystyle\sum\limits_{k=0}^n \left|\frac{n^k-\frac{n!}{(n-k)!}}{n^k}\right|\frac{c^k}{k!}$ converges to zero, as $n \to \infty$.
I know that ...
0
votes
2answers
101 views
Absolute convergence for all values except the values $z=\left( 1+\frac {a} {m}\right) e^{\frac {2k\pi i} {m}}$
I am trying to show that the series $$\sum _{n=1}^{\infty }\dfrac {nz^{n-1}\left( \left( 1+\dfrac {1} {n}\right) ^{n}-1\right) } {\left( z^{n}-1\right) \left( z^{n}-\left( 1+\dfrac {1} {n}\right) ...
7
votes
2answers
587 views
Determining precisely where $\sum_{n=1}^\infty\frac{z^n}{n}$ converges?
Inspired by the exponential series, I'm curious about where exactly the series $\displaystyle\sum_{n=1}^\infty\frac{z^n}{n}$ for $z\in\mathbb{C}$ converges.
I calculated
$$
...
3
votes
1answer
202 views
Absolute Convergence of an Infinite Product based on Weierstrass's Factor Theorem
I am trying to show that $\left\{ \left( 1-\dfrac {z} {\pi }\right) e^{\left( \dfrac {z} {\pi }\right) }\right\} \left\{ \left( 1+\dfrac {z} {\pi }\right) e^{\left( -\dfrac {z} {\pi }\right) }\right\} ...
4
votes
1answer
384 views
Every Cauchy sequence in $\mathbb{C}$ is bounded
Prove that every Cauchy sequence in $\mathbb{C}$ is bounded.
In $\mathbb{R}$, this is a sketch of the proof that I recall:
Let {${a_k}$} be Cauchy in $\mathbb{R}$, since $1\in\mathbb{R}$, ...
0
votes
1answer
172 views
Proof of Uniform Convergence
Prove that the sequence of functions $\{f_{n}(z) = (1+nz)^{-1}\mid n=1,2,...\}$ converges uniformly to $f(z)=0$ for $|z| \geq r > 0$. To answer the question, for a given choice of $\epsilon > ...
2
votes
0answers
146 views
Complex Analysis: Power Series and Convergence
Let $f(z)=\sum_{n=0}^{\infty}c_nz^n$ have radius of convergence $R$.
Problem
Prove that $\sum_{n=0}^{\infty}\overline{c_n}z^n$ has radius of convergence $R$ and that ...
1
vote
2answers
240 views
Complex Analysis: Convergence of Power Series
Problem
Let $p(z)=(z-a_1)(z-a_2)...(z-a_N)$, where $a_1, a_2, ..., a_N$ are distinct complex numbers. Let $M=\min_{1\le{k}\le{N}}|a_k|$. Prove that it is possible to express $\frac{1}{p(z)}$ as a ...
4
votes
2answers
346 views
Complex Analysis: Radius of convergence of Power Series
Let p be a polynomial of degree $k>0$. Prove that $\sum p(n)z^n$ has radius of convergence $1$ and that there exists a polynomial $q(z)$ of degree $k$ such that $$\sum_{n=0}^{\infty} p(n) ...
