All Questions
116
questions
2
votes
0
answers
104
views
Power Series with digits of $\pi$
Sorry if this has already been asked, but I haven't found a post.
Can anything be said about the function
$$f(z)=\sum_{n=0}^\infty a_n z^n$$
where the $a_n$'s are the digits of $\pi= 3.14159...$, so $$...
- 6,277
1
vote
3
answers
379
views
For holomorphic function $f: \Omega \to \mathbb{C}$ with $f^{(k)}(z_0) \in \mathbb{R}$ prove that $f(x) \in \mathbb{R}$ for every $(z_0 - r, z_0 +r)$
I have some difficulties with a question I have come across.
The question goes as follows:
Let $\Omega$ be an open set with $z_0 \in \Omega \cap \mathbb{R}$. Let $f: \Omega \to \mathbb{C}$ be ...
- 21
0
votes
0
answers
77
views
Series of Analytic Functions is Analytic
Let $0 \in \mathbf{N}$. Let $P_m(x): [0,1] \to \mathbf{C}$ be bounded analytic functions for every $m\in \mathbf{N}$. Formally, define
$$
f(x) = \sum_{m\in \mathbf{N}}c_m P_m(x)\overline{P_m}(x),
$$
...
- 147
1
vote
0
answers
73
views
Radius of Convergence of $\sum_{n=1}^{\infty}\frac{z^{3^n}-z^{2\cdot 3^n}}n$
Consider the power series
$$g(z)=\sum_{n=1}^{\infty} \frac{z^{3^n}-z^{2 \cdot 3^n}}{n}.$$
We can see that $\limsup_{n \to \infty} |a_n|^{1/n}=0$ so by Cauchy-Hadamard we know that the radius of ...
- 13
1
vote
2
answers
143
views
Expanding a power series at a different point
Let $f(z)=\sum_na_nz^n$ be a complex power series with convergence radius $R$, I want to show the function $f$ is analytic, i.e. can be expanded to a power series on a neighbourhood of any point in ...
- 1,434
2
votes
2
answers
83
views
Coefficients of a product of a Laurent polynomial and an infinite series
Let $N$ be an integer. Let us consider a Laurent polynomial in $q$ given by $\sum_{s=N}^{M} \gamma_s q^s$. Then consider the expression $(\sum_{s=N}^{M} \gamma_s q^s) \times \sum_{i=0}^{\infty}q^i$.
...
- 443
0
votes
0
answers
47
views
Why this equation holds?
Can someone help me why does this equation holds:
$\sum_{\rho}(1/2 + t -\rho)^{-s}= e^{i\pi s/2}\sum_{k=1}^{\infty}(\tau_k+it)^{-s}+e^{-i\pi s/2}\sum_{k=1}^{\infty}(\tau_k-it)^{-s}$ for $\rho=1/2 \pm ...
- 87
0
votes
1
answer
98
views
if $P(x_{0})=0$, but $\lim_{ x \to x_{0} }(x-x_{0})\left( \frac{Q(x)}{P(x)} \right) = \text{finite}$, then function is analytic at $x=x_0$
I'm currently learning about Differential Equations and the textbook I'm using (by DiPrima) says the following:
For polynomials $P(x)$ and $Q(x)$ that has no common factors, if $P(x_{0})=0$, but $\...
- 1,694
0
votes
1
answer
37
views
Power Series defined incorrectly.
The power series is defined as: $$\sum^\infty_{k=0}a_k(z-z_0)^k=f(z)$$Where $a_k$ are coefficients that make this true. But according to this definition, the sine function:$$\sum_{k\ge0}(-1)^k\frac{z^{...
- 6,549
1
vote
3
answers
354
views
Are there real analytic functions whose derivative is not the sum of the derivatives of the terms in its Taylor expansion?
In complex analysis we know that for every complex analytic function $f(z)$, that $f(z)$ has (by definition) a Taylor expansion around every point $z_0$: $\sum_{n=0}^{\infty} a_n (z-z_0)^{n}$ with ...
- 1,356
0
votes
1
answer
156
views
Does $\limsup_{n \to \infty }\left | \frac{a_{n+1}}{a_{n}} \right |=\limsup_{n \to \infty }\left | a_n \right |^{1/{n}}$? [duplicate]
Question: Does $\limsup_{n \to \infty }\left | \frac{a_{n+1}}{a_{n}} \right |=\limsup_{n \to \infty }\left | a_n \right |^{1/{n}}$?
I guess the identity is true, given $\limsup_{n \to \infty }\left | \...
- 109
3
votes
1
answer
83
views
Relation between roots of $f$ and roots of the partial sums of its power-series
It is well known we can write any holomorphic function as a local power series, for example
$$\exp(z) = \sum_{n=0}^\infty \frac{z^n}{n!}$$
Obviously each partial sums are polynomials of degree $n$ ...
- 901
4
votes
0
answers
402
views
Conditions and correct interpretation of Borel summation
Hello to the community.
In my line of research (theoretical particle physics) it is customary to apply the strategy of Borel summation to infinite power series in order to find closed forms and/or ...
0
votes
1
answer
804
views
Proof of the Cauchy–Hadamard theorem
I'm looking at the proof of Cauchy–Hadamard theorem.
Let $\sum_{n=1}^\infty a_nz^n$ be a power series where $a_n\in\mathbb{R}, z\in\mathbb{C}$. Let t = $\limsup|a_n|^{1/n}$, and let R = the radius of ...
- 1
1
vote
2
answers
69
views
Series of powers
Consider the sequences $a_k,b_k\in\mathbb R$ and let $N\in\mathbb N$. Do series of the form
$$f(n)=\sum_{k=1}^N a_k\,{b_k}^n$$
have a name? They seem quite a natural object to study but I'm not sure ...
- 631