Tagged Questions
5
votes
0answers
75 views
Complex differentiable but not analytic on circle of convergence
I'm trying to get a better handle on behavior of complex power series on the boundary of their maximal disk of convergence.
I'm reading Bak-Newman's Complex Analysis, Chapter 18.1.
A regular point ...
2
votes
2answers
56 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
1answer
76 views
Show that some $C^\infty$ real function is analytic
Consider the function $f:(a,b) \rightarrow \Bbb R $, which is $C^\infty$ class. It is required to show that if the function $f$ satisfies: $\forall_{n\in\Bbb N_+} \forall_{x\in(a,b)} f^{(n)}(x) \ge 0 ...
3
votes
2answers
46 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 ...
5
votes
2answers
94 views
$\lim_{x\rightarrow\infty}\frac{f(x)}{e^x}$ for analytic functions
For some analytic function $f(x)=\lim_{n\rightarrow\infty}\sum^n_{r=0}c_rx^r$,
...
1
vote
0answers
35 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 ...
2
votes
1answer
165 views
Riemann Zeta Function Manipulation
The Riemann zeta function is defined on the $Re z> 1$ by $$\zeta(z)=\sum_{n=1}^\infty \frac{1}{n^z}$$
(i) show that for $Re z> 1$, we have $$(1-2^{1-z})\zeta(z)=\sum_{n=1}^\infty ...
4
votes
2answers
62 views
Number of times two rescaled, 'fully' monotonic functions can cross
Consider two functions $f: [0,1) \rightarrow \mathbb{R}$ and $g: [0,1) \rightarrow \mathbb{R}$. Suppose $f(x) > g(x)$ for all $x \in [0,1)$. Suppose further that $f$ and $g$ are infinitely ...
6
votes
3answers
235 views
Expressing the area of the image of a holomorphic function by the coefficients of its expansion
I have the following problem.
Let $f:D\to \mathbb C$ be a holomorphic function, where $D=\{z:|z|\leq 1\}.$ Let $$f(z)=\sum_{n=0}^\infty c_nz^n.$$ Let $l_2(A)$ denote the Lebesgue measure of a set ...
2
votes
1answer
168 views
limit of supremum of a sequence
Can any one help me with this?
Let $c$ be a real number. I would like to show that
$$ \limsup_{n \to ...
3
votes
1answer
278 views
Analytic functions and Fourier Series
I'm taking my first real analysis course and I'm trying to get a better feel about analytic functions. My understanding is that an analytic function is one which can be written as a power series. My ...
4
votes
3answers
279 views
composition of power series
Does anyone know how to derive a formula for the coefficients.
That is if,
$f(x)=\sum _{n=0}^{\infty } a_nx^n$ and $g(x)=\sum _{n=0}^{\infty } b_nx^n$
suppose the compostion is an analytic ...
