Tagged Questions
5
votes
4answers
130 views
Are Taylor series and power series the same “thing”?
I was just wondering in the lingo of Mathematics, are these two "ideas" the same? I know we have Taylor series, and their specialisation the Maclaurin series, but are power series a more general ...
2
votes
2answers
45 views
Uniqueness of distribution with moments $M_n$ if $\limsup_{n\to\infty} \frac{1}{n}\sqrt[n]{M_n}$ finite
There's a theorem which states that the moments, i.e. $M_n = \mathbb{E}\left(X^n\right)$, of a distribution uniquely identify the distribution if $$
R := \left(\limsup_{n\to\infty} ...
1
vote
2answers
67 views
Expanding an analytic function to a powerseries
How would you expand the analytic function $$\frac{1}{1-z-z^2}$$ to a series of the form $$\sum_{k=0}^\infty a_k z^k \, \, ?$$
2
votes
4answers
73 views
Radius of Convergence of power series of Complex Analysis
I have come across the following few questions on past exams papers.. I know how to solve these type when it is of the form $a_nz^n$ but don't have a clue what to do with these. Any help would be ...
2
votes
2answers
46 views
Analytic functions of a real variable which do not extend to an open complex neighborhood
Do such functions exist? If not, is it appropriate to think of real analytic functions as "slices" of holomorphic functions?
2
votes
3answers
120 views
Difficulties performing Laurent Series expansions to determine Residues
The following problems are from Brown and Churchill's Complex Variables, 8ed.
From ยง71 concerning Residues and Poles, problem #1d:
Determine the residue at $z = 0$ of the function ...
2
votes
3answers
64 views
Radius of convergence of fast converging power series
Suppose $a_i\ge 0$ and $a_{n+1}+a_{n+2}+\cdots < 1/n!$. What can I say about the radius of convergence of
$$\operatorname{f}(x) = \sum_{n \ge 0} a_n x^n$$
The above condition gives that ...
7
votes
1answer
107 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}$ ...
1
vote
2answers
46 views
If $f$ is the sum of a convergent power series on a disk $D(0;R)$ prove that the integral of $f$ over any closed path $\gamma$ in $D(0;R)$ is zero
If $f$ is the sum of a convergent power series on a disk $D(0;R)$ prove that the integral of $f$ over any closed path $\gamma$ in $D(0;R)$ is zero.
How can I able to prove the above problem without ...
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 ...
0
votes
1answer
31 views
Intuition Behind Krantz Theorem
The theorem I'm referring to is as follows:
Let $z_0$ be a root of a nonzero holomorphic function $f$ , and let $n$ be the least positive integer such that, the $n$-th derivative of $f$ evaluated ...
1
vote
2answers
183 views
Power series by partial fractions
I am trying to find a power series centered at the origin for the function $f(z) = \frac {1}{1-z-2z^2}$ by first using partial fractions to express $f(z)$ as a sum of two simple rational functions. If ...
1
vote
2answers
89 views
How to construct this Laurent series?
How do I construct the following Laurent series (clipped off Wolfram Alpha)?
I know that the numerator can be written as $-1+\frac{\pi}2 z-...$
Alternatively (without the Laurent series), how can I ...
1
vote
1answer
79 views
using power series expansion to find a holomorphic function which solves a differential equation
Using power series expansions,
find a function $f$ which is holomorphic on the unit disk $D:=$ {$z\in\mathbb C:|z|<1$}
and solves the differential equation
$(1-z^2)f''(z)-4zf'(z)-2f(z)=0$
for ...
2
votes
1answer
68 views
Evaluate a complex integral using power series expansions
Using power series expansions, evaluate the integral
$$\int_{\gamma_r}\sin\left(\frac{1}{z}\right)dz.$$
where $\gamma_r:[0,2\pi]\rightarrow \mathbb C$ is given by $\gamma_r(t)=r(\cos t + i\sin ...
2
votes
2answers
115 views
Confused by Laurent series
A typical problem related to Laurent series is this:
For the function $\frac 1{(z-1)(z-2)}$, find the Laurent series expansion in the following regions: $\\(a) |z|<1, \\ (b) 1<|z|<2, ...
2
votes
1answer
51 views
Confused over analytic functions, point convergence of power series
It is well-known that a power series sums to a function that is analytic at every point inside its circle of convergence and that conversely, if a function is analytic on an open disc then its Taylor ...
3
votes
2answers
102 views
How do I obtain the Laurent series for $f(z)=\frac 1{\cos(z^4)-1}$ about $0$?
I know that $$\cos(z^4)-1=-\frac{z^8}{2!}+\frac{z^{16}}{4!}+...$$ but how do I take the reciprocal of this series (please do not use little-o notation)? Or are there better methods to obtain the ...
3
votes
3answers
71 views
What is the significance of this theorem (coefficients of power series as integrals)?
Isn't it easier to obtain the series' coefficicients by differentiation rather than by integration?
The above text uses this theorem as an intermediate step in obtaining the generalised Cauchy ...
0
votes
1answer
51 views
Can a power series always be integrated term-by-term inside the circle of convergence of its sum function?
Is it true that a power series can always be integrated term-by-term inside (i.e. in the interior of) the circle of convergence of its sum/limit function?
My complex analysis textbook merely states ...
0
votes
0answers
46 views
Complex analysis Laurent series evaluated on unit circle
Let $f(z)$ be a function analytic on an annulus that includes the unit circle $z=e^{i\theta}$. By taking that circle as the path of integration for the coefficients in the Laurent series, show that
$$
...
2
votes
0answers
70 views
Properties of Entire Functions
a).
Suppose an entire function f is bounded by M along $\vert z \vert = R$. Show that the coefficients $C_k$ in its power series expansion about $0$ satisfy $ \vert C_k \vert \leq \frac{M}{R^k} $.
I ...
5
votes
2answers
115 views
What is the order of this pole?
$$f(z)=\frac 1{\cos(z^4)-1}$$
$z=0$ is a pole of $f$, and I believe that the Laurent series centred at $0$ is $-\frac 2{z^8}-\frac 16+...$, which looks like the pole is of order $8$, but why does ...
0
votes
1answer
59 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 ...
5
votes
1answer
70 views
Does a bounded convergent power series on an open disc extend to the boundary?
Here is my question: Suppose that $|\sum_{n=0}^{\infty}a_nz^n| \leq M$ for all $z \in D_r$ (the open disc or radius $r$). Does this power series converge on $\partial D_r$?
2
votes
1answer
30 views
Specific question about the consequence of composing power series
Please bear with my possible abuse of notation/terminology. Consider the power-series composition f(g(x)). If g's range lies within f's interval of convergence, and if series g has a constant term 0, ...
9
votes
3answers
153 views
Can the the radius of convergence increase due to composition of two power series?
When composing power series, is the radius of convergence the minimum of that of the individual series, or is it like for multiplication and addition of power series where the resultant radius of ...
1
vote
1answer
96 views
There is no function $f$ on the open unit disk, defined by a convergent power series, such that $f(1/n)=(-1)^n/n^2$
Prove that there is no function $f$ on the open unit disk, defined by a convergent power series, such that $f(1/n)=(-1)^n/n^2$.
I'm not sure how to start... any hints would be appreciated!
0
votes
1answer
60 views
Uniform Convergence: Complex Analysis
To show that $f_k(z) = \frac{z^k}{k}$ converges uniformly for $|z| < 1 $ and that $f'_k(z)$ does not converge uniformly for $|z| < 1$, what must be done? What other things can be said about the ...
1
vote
1answer
41 views
Series expansion coefficients
Let $f$ be an entire function. We know that $f$ can be written as a series centered in $0$
$$
\sum_{k=0}^{\infty} a_k z^k
$$
where the coefficients $a_k=f^{(k)}(0)/k!$ are given by
$$
\frac{1}{2\pi ...
2
votes
4answers
69 views
Find the corresponding functions of the following power series
$\displaystyle\sum_{n=0}^\infty \frac{z^{4n}}{n!}$
$\displaystyle\sum_{n=0}^\infty n(n-1)z^n$
My thoughts:
I think it'll look something like the exponential function but I'm not sure what exactly ...
3
votes
2answers
41 views
Determining whether a family of power series is normal
How should I check whether a given family of power series forms a normal family? I am trying to apply Montel's theorem that says that a family of holomorphic functions is normal iff it is uniformly ...
1
vote
2answers
172 views
finding Laurent series for $\dfrac{1}{z(z-2)^3}$
I am trying to get the Laurent series for $\dfrac{1}{z(z-2)^3}$. I know there are poles at $z = 0$ and $z=2$, and so I am looking for expansions about the singularities. Using $\dfrac{1}{1-z} = ...
0
votes
1answer
137 views
Laurent Series and Taylor Series
I am trying to find the Laurent series of $\dfrac{1}{(1+x)^3}$; would this be the same as finding the Maclaurin series for the same function?
2
votes
1answer
39 views
Raise a Laurent series to a power
I want to know if there is a general rule by which a Laurent series can be raised to a power; in other words,I have the Laurent series for $f(z) =\dfrac{1}{z(z-1)(z-2)}$ about a certain pole, and I ...
1
vote
0answers
51 views
what are the borders of the convergence disks of series?
Let $\mathbb{T}=\{z\in \mathbb{C}\mid |z|=1\}$. For which $S\subseteq \mathbb{T}$, is there a sequence $(a_n)\subseteq \mathbb{C}$ such that the series:
$$\sum_{k=1}^\infty{a_kz^k}$$
is convergent on ...
1
vote
0answers
31 views
Existence of a certain oscillating power series?
Is there a power series with complex coefficients and is convergent around $0$ with radius $1$, but is unbounded on segment $[0,1)$, and there $\exists$ $\lbrace a_n\rbrace$, $a_n \in [0,1)$, ...
3
votes
0answers
129 views
Under which hypotheses is switching between sum and integral signs legit?
Which hypotheses are needed to change the order of sum and integral signs?
Concrete example: consider the expression
$$
...
0
votes
1answer
133 views
Complex geometric series
Prove that if $|z|=1$, the following series diverge :
$\displaystyle \sum_{n=0}^{\infty} z^k $
all the tests that I know failed here, so I used :
$\displaystyle z= \cos x+ i \sin x \rightarrow ...
1
vote
4answers
116 views
complex power series
Sorry for taking this from another question, but the second part was never answered, and I'm not sure how to get there. From: Prove the following equation of complex power series.
Show that for $|z| ...
1
vote
1answer
60 views
How to expand a fraction in powers of $z$ or $\dfrac{1}{z}$, and which to do, in determining Laurent series
I have a function $f(z)=\dfrac{12}{z(2-z)(1+z)}$, I'm trying to find the Laurent series for each of the three annuli. The singularities are at $z = 0$, $z = 2$, and $z = -1$, so I'm looking for three ...
2
votes
1answer
91 views
Problem with the Laurent series
Find The Laurent series for the following function on the annulus $1<|z|<2$ :
$\displaystyle f(z)=\frac{2z}{z^2+z-2}$
My work :
$\displaystyle f(z)=\frac{2}{3} \left( ...
0
votes
0answers
96 views
Complex series Convergence
Find the domain of convergence of :
$\displaystyle\sum_{n=1}^{\infty} \frac{e^{in}}{(z+1)^n} +\sum_{n=0}^{\infty} \frac{(z+1)^n}{e^{\frac{1}{2}+in}}\ \ \ \ (z\in\mathbb{C})$
I've found that it ...
2
votes
1answer
116 views
Prove the following equation of complex power series.
Show that for $|z| \lt 1$ with $z \in \Bbb C$, we have
$$ \sum_0^\infty \frac{{z^2}^k}{1-{z^2}^{k+1}} = \frac{z}{1-z} $$
$$ \sum_0^\infty \frac{2^k{z^2}^k}{1+{z^2}^{k}} = \frac{z}{1-z} $$
My guess ...
0
votes
3answers
50 views
Power series and recurrence
Please help me find the radius of convergence and the value of the following power series:
$\sum_{n=0}^{\infty} a_nz^n$, when $a_0=1,a_1=-1$, and $3a_n+4a_{n-1}-a_{n-2}=0$ for $n>1$.
1
vote
1answer
258 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 ...
1
vote
1answer
122 views
Fibonacci Generating Function of a Complex Variable
So I'm doing work on the Fibonacci Numbers, and I came across this problem for the generating function
for the recursive fibonacci numbers.
I have two questions:
1. Why is it useful to use a ...
2
votes
2answers
140 views
convergence of complex power series - infinite convergence radius
My books states that if the convergence radius of a complex power series is $+\infty$, then the power series is uniformly convergent over every 'disk' of the complex plane, although not necessarily ...
2
votes
1answer
133 views
Power Series With Bernoulli Numbers
The exercise reads "Express the power series for $\large \frac{z}{\sin (z)} = \frac{2 i z}{e^{iz} - e^{-iz}} $ in terms of Bernoulli numbers."
I am given in a previous exercise that the Bernoulli ...
1
vote
1answer
53 views
Radius of convergence of this series
Let $f$ an analytic function of complex variable, with radius of convergence $r$. What about the radius of $$x^pf(x),$$where $p$ is a complex parameter ?
Thanks in advance
