Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n x^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex (or more generally an element of a Banach algebra).
9
votes
0answers
498 views
any pattern here ? (revised)
for any positive number $k$, I have a $(k+1)*(k+1)$ matrix. I wonder if these matrices follow any "obvious" pattern. My goal is to guess the elements for matrix with $k=5$ and above (most probably in ...
5
votes
0answers
246 views
Is there a reasonably simple, non-recursive formula for this sequence?
I have been curious about the following sequence of rational numbers for some time. I identified the numerator and denominators on the Online Encyclopedia of Integer sequences, but there was not much ...
4
votes
0answers
168 views
What is known about the transformation of a power series in which $z^n$ is replaced with $z^{n^2}$?
Say we have the function $$G(z) = \sum_{n \geq 0} g_n z^n.$$
Is there a name for the transform T defined so that $$(T(G))(z) = \sum_{n \geq 0} g_n z^{n^2}?$$
Is there anything known about this ...
3
votes
0answers
30 views
Formal integration of a series of the type $-f(x-a)=\sum_{n=0}\frac{C_n}{n!}\delta^{(n)}(x-a)$
This question is inspired from an answer given to this question in the physics stackexchange, specifically the integration step going from (12) to (13). We have a distribution given as
...
3
votes
0answers
81 views
Function composition and Bell polynomials
Suppose that we have the Taylor expansions :
$$f(x)=\sum_{n=1}^{\infty}\frac{a_{n}}{n!}x^{n}$$
$$g(x)=\sum_{n=1}^{\infty}\frac{b_{n}}{n!}x^{n}$$
Then we have the standard result :
...
3
votes
0answers
130 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
$$
...
3
votes
0answers
149 views
domain of convergence of a multivariable taylor series
consider the rational function :
$$f(x,z)=\frac{z}{x^{z}-1}$$
$x\in \mathbb{R}^{+}/[0,1]\;\;$, $z\in \mathbb{C}\;\;$ .We wish to find an expansion in z that is valid for all x and z. a Bernoulli-type ...
3
votes
0answers
121 views
Chebyshev Diff EQ
Find a power series solution about $x_0=0$ for the Chebyshev differential equation
$$(1-x^2)y''-xy'+n^2 y=0,$$
as a function of of the integer $n$. Show that the solutions form a terminating ...
3
votes
0answers
199 views
Series of nested double integrals
This is kind of a follow-up of my previous question. I'm investigating the following infinite series of nested two-dimensional integrals
$$\sigma(t,t^\prime) = 1 - \int_{t^\prime}^t\mathrm dt_1 ...
3
votes
0answers
124 views
Holonomic ideals and D-finite power series
I would like to understand the connection between the term D-finite power series
(in n variables) and the term of a holonomic module over the Weyl algebra A_n.
A power series $f \in K[[x_1,...,x_n]]$ ...
3
votes
0answers
66 views
Enhancing the monoid structure over a finite alphabet to prove Arden's rule
Suppose you have a finite-state, deterministic automaton, that you wish to convert to a regular expression. A common method, perhaps easier to apply by hand that Yamada's algorithm, is to reduce the ...
2
votes
0answers
18 views
Bivariate generating functions and diagonal like recurrences
I'm trying to solve recurrences of the type
$$a(n,m) = \sum_{k=0}^{m} a(n-k,k), \qquad a(n,0)= a(0,m)=1 \qquad (A_0)$$
with the help of generating functions, but I get stuck quite early on.
If I ...
2
votes
0answers
159 views
power series of arcsin(x) centered at x = 0
I am trying to prove that the Taylor expansion of $\arcsin(x) = \sum\limits_{n=0}^\infty \cfrac{(2n!)x^{2n+1}}{(2^nn!)^2(2n+1)}$.
Sorry about the notation, I'm not sure what syntax to use. S stands ...
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 ...
2
votes
0answers
266 views
Prove that sum is finite with the help of generating function
Please help me to prove that the following sum is finite
$$
\sum_{j=2l-2}^{\infty}j!\, a_j^{(l)},
$$
here the generating function of $\displaystyle{a_j^{(l)}}$ is ...
2
votes
0answers
170 views
Approximation of integral using series expansion of the integrand.
I have a smooth function $x \rightarrow f_\epsilon (x)$ on $x\in[-1\ldots 1]$ (dependent on the continuous parameter $\epsilon$) and I want to approximate the integral
$$
I=\int_{-1}^1 f_\epsilon ...
2
votes
0answers
92 views
Identification of the limit of an exponential-like power series
I have come across the series:
$$\sum_{j=1}^\infty \frac{x^j (j-1)}{j! \sqrt{2j - 1}}$$
which is easily seen to be absolutely convergent everywhere (e.g. ratio test). It seems that it should be very ...
2
votes
0answers
79 views
Solve $x^2u''+xu'-(x^2+\frac{1}{4})u=0$ using power series
I stumbled upon this question in an old exam (I'm preparing for an exam of a course about ODEs). I didn't have much difficulty solving the Legendre and Hermite equations using power series, but this ...
2
votes
0answers
130 views
Confusion! Power series and integration
Consider the below power series:
$\sum\limits_{n=1}^\infty \dfrac{x^{n}}{n^{2}}$
I know that it converges for $x\in [-1,1]$ and the sum $s(x)$ of the series is given by:
$s(x) = - ...
2
votes
0answers
243 views
Convergence of complex power series
I'll post the full problem before I'll show my (rather limited) progress:
i) Find all $z \in \mathbb{C}$ so that the following power series converge around $0$: a) $\sum_{k=0}^\infty z^k$, b) ...
2
votes
0answers
86 views
Convergence of a power series in the complex plane — does this argument work?
I'm interested to know for which $z \in \mathbb{C}$ the power series $\sum_{n=0}^{\infty}\frac{\cos(nz)}{n!}$ converges and what its value is when it does.
I've come up with the following argument, ...
2
votes
0answers
314 views
How is Lagrange's inversion theorem derived?
I am interested in the complex-analysis version of deriving Lagrange's inversion theorem:
If $y=f(x)$ with $f(a)=b$ and $f'(a)\neq 0$, then
$$x(y)=a+\sum_{n=1}^{\infty} \left(\lim_{x\to ...
2
votes
0answers
145 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
0answers
44 views
Taylor series of Fourier series of triangle wave
Odd triangle wave $\text{t}(x)$ with angles at $(2x+1)\in\mathbb{Z}$ can be represented by Fourier series:
...
1
vote
0answers
54 views
Expansion in powers
Let $n=2k, k \in Z_+$. Let
$$P_k\left(\frac{t}{\sqrt n}\right)=n!\sum_{\begin{smallmatrix} n_1+\ldots+n_k=n \\ ...
1
vote
0answers
76 views
Relating terms in differential equation with power series
Having problems with a task on a differential equation containing a power series.
Given
$$\frac{dx}{dt} = \lambda x + \sum_{n=2}^\infty b_n x^n$$
$$\frac{dy}{dt} = \lambda y$$
$$x(y) = y + ...
1
vote
0answers
48 views
Power Series and Matrices
I am trying to prove that if a function $f(x)$ can be written as a power series in the form
\begin{equation}
f(x)=\sum_{n=0}^{\infty}c_n(x-x_0)^n
\end{equation}
such that $|x-x_0|<r$, then
...
1
vote
0answers
50 views
inequality for series
Let $j \in Z_+$. Set
$$
a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!}
$$
and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$.
Let $X(i)=|a^{(2i)}_j|j!$. Verify that $X(i)\leq X(1)$ for ...
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 $\exists$ $\lbrace a_n\rbrace$, $a_n \in [0,1)$, ...
1
vote
0answers
41 views
Use of natural logarithm transformation on weighted index series
I have a value computed as sum of powers, e.g. $x^5+y^8+z^2$.
The exponent represents the weight for variables, $x, y$ and $z$ in the example above.
Applying natural logarithm on $x^5+y^8+z^2$, I get ...
1
vote
0answers
285 views
Using power series to evaluate the integral of a piecewise function
I was self-studying Calculus, and the book I'm using asked me to solve the following integral using the technique of integration of power series:
$$\int_{0}^{0.25} g(x)dx \text{ where }
...
1
vote
0answers
59 views
What are the Puiseux series for the inverse of $exp(z)(z-a)(z-b)(z-c)$ expanded at the singularities?
Let $a,b,c$ be real variables.
Let $z$ be a complex number and $g(z) = exp(z)(z-a)(z-b)(z-c)$.
Let $f(z)$ be the functional inverse of $g(z)$ such that $f(g(z)) = g(f(z)) = z$.
Now $f(z)$ must have ...
1
vote
0answers
34 views
How to show that $f(x,y)$ are real analytical
Given a real function $f(x,y)$ on $R^2$, we know that (from wiki) it is called real analytic if it is locally given by a convergent power series.
My question is that whether there has some principle ...
1
vote
0answers
36 views
Power series over integral change of variables
Let $y$ be given by
$$y=\int_x^\infty \frac{dx'}{(1+x'^2)^\alpha}$$
where $\alpha>1$. Is it possible to express the following as a series:
$$x(1+x^2)^{\alpha-1}=\sum a_n\left(\frac{1}{y}\right)^n$$
...
1
vote
0answers
88 views
generalizing superfunctions of entire functions
Let $z$ be complex and $x$ real.
Define $f(z,0) = f(z)$ where $f(z)$ is an entire function.
Define $f(z,x)$ as the $x$ th superfunction of $f(z)$.
We know that $f(z,x-1) = f(f^{-1}(z,x)+1)$ where ...
1
vote
0answers
54 views
series index manipulation of a polylogarithm product
I've been playing around with the product of polylogarithms, and have come across the following triple sum:
$$
\sum_{n=1}^\infty \sum_{m=1}^{\infty} \sum_{v=1}^{\infty} ...
1
vote
0answers
108 views
How to derive to inverse z transform of $\sqrt{\frac{1-a^2}{1-\frac{a}{z}}}$ from Laguerre differential equation?
How can I derive the inverse z-transform of:
$$\sqrt{\frac{1-a^2}{1-\frac{a}{z}}}$$
If Maple is not the way, how to derive manually?
With Maple code I encounter some problems
...
1
vote
0answers
42 views
What, in general, can I expect to be the restrictions and/or limitations to this alternative process to rewriting?
This is a specific idea I have to rewriting $x$ as $x y$, which I recently asked about in this question.
Suppose we have a power series
$$f(x) = \sum_{i=0}^\infty{c_i x^i} = c_0 x^0 + c_1 x^1 + c_2 ...
1
vote
0answers
74 views
subrings A of the ring of power series k[[t]] with the condition (A : k[[t]]) $\neq${0} and k $\subset$ A
I would like to understand the structure of the subrings A of the ring of formal power series k[[t]] (where k is a field) which satisfy the condition (A : k[[t]]) $\neq$ {0} and k $\subset$ A. Are ...
1
vote
0answers
71 views
series look up site
Is there a site for looking up a series to see some of the associated functions. (In the spirit of Encyclopedia of Integer Sequences OEIS.)
In particular I am looking for functions related to $ \sum ...
1
vote
0answers
52 views
How fast can $[x^n] f(x)$ be calculated, given some restrictions?
If we suppose that $[x^n] f(x)$ represents the coefficient of $x^n$ in a power series or generating function, I'd like to know how fast it can be calculated.
There are some restrictions that I'd like ...
1
vote
0answers
92 views
Inequalities from power series
Suppose we have a power series $\sum a_n x^n$ with some positive radius of convergence whose coefficients are known. Let $f(x) = \sum a_n x^n$ within the radius of convergence. When truncating the ...
1
vote
0answers
70 views
Power series expression of x in terms of y
Let $R$ be a commutative ring with identity and $y=a_1x+a_2x^2+a_3x^3+.....$ be a power series in $R[[x]]$ such that $a_1$ is an unit in $R$.
Does there exists a way to express x as a power series ...
1
vote
0answers
294 views
Finding a radius of convergence
Let $\sum_0^{\infty} a_n z^n$ have radius of convergence $R$ with $0< R< \infty$. Let $\alpha>0$. Find the radius of convergence of $\sum_0^{\infty} |a_n|^{\alpha} z^n$.
I tried to start ...
0
votes
0answers
27 views
Solving $m$ in $m = \lim_{n\to\infty}\prod_{k=x+1}^n\, 1+\dfrac{(k+x)^2}{2^{k-x}}$ from $n$ and $x$
How should one proceed in order to solve $m$, where $x$ is an integer
$$m = \lim_{n\to\infty}\prod_{k=x+1}^n\, 1+\dfrac{(k+x)^2}{2^{k-x}} $$
from $n$ and $x$ in an unconditional form, such as, for ...
0
votes
0answers
72 views
Is there a power series that converges to the function $f(x)= \lvert x\rvert$ for all $x$?
Is there a power series that converges to the function $f(x)= \lvert x\rvert$ for all $x$?
I am pretty lost on how to even start this.
0
votes
0answers
43 views
Solutions of Chebyshev equation about $x = 1$
I need to find two solutions of the Chebyshev equation: $(1-x^2)y'' - xy' + a^2y = 0$, where a is some constant. Attached is a picture of what I have so far. I apologize if its kind of hard to read. ...
0
votes
0answers
65 views
What is the bound of coefficients of the series $e^t$?
I'd like to get a bound of the coefficients of the series:
$$e^t = 1 + \frac{t^1}{1!} + \frac{t^2}{2!} + \dots + \frac{t^n}{n!} + \dots$$
In other words, after $n$ terms we get a term
...
0
votes
0answers
40 views
Taylors Inequality to evaluate $f(x) = x\sin(x)$ when $a = 0$ and $-1\le x\le1$
Trying to calculate the error of this function when you use a Taylor expansion to degree 4.
I keep getting $.039$ when the answer in the back of the book is $.042$.
I take the fifth derivative of ...
