All Questions
104
questions
2
votes
0
answers
34
views
Laurent Series question for Exponentials
I must find the Laurent series for $f(z) = \frac{e^z}{z^2}$ in powers of $z$ for the annulus $ |z| > 0$.
I wrote $f(z) = \frac{1}{z^2} \sum_{n=0}^{\infty} \frac{z^n}{n!} = \sum_{n=0}^{\infty} \frac{...
- 247
2
votes
1
answer
104
views
A confusion about the radius of convergence in Ahlfors' "Complex Analysis"?
On the third edition of Ahlfors' Complex Analysis, page 39 Theorem 2 it states: The derived series $\sum_{1}^{\infty}na_n z^{n-1}$ has the same radius of convergence, because $\sqrt[n]n \rightarrow 1$....
- 174
0
votes
2
answers
146
views
I find a "mistake" on p.40 in Ahlfors' "Complex Analysis"?
On the third edition of Ahlfors' Complex Analysis, page 40 Theorem 2 it states: we conclude that
\begin{equation*}
\left|\frac{R_n(z)-R_n(z_0)}{z-z_0}\right| \leq \sum_{k=n}^{\infty} k|a_k|\rho^{k-1}
\...
- 174
0
votes
1
answer
113
views
How to prove $f(z)=a_0+a_1z+\cdots+a_nz^n+\cdots$, which has infinite items, is an analytic function? [duplicate]
It is easy to prove the following four statements.
(i) If $f$, $g$ are analytic functions and $f^{\prime}$, $g^{\prime}$are continuous then $(f+g)(z)$ is analytic and $(f+g)^\prime(z)=f^\prime(z)+g^\...
- 174
1
vote
2
answers
248
views
Prove that If singular part of Laurent series has infinite many terms, then $\lim_{z\to z_0}(z-z_0)^mf(z)$ doesn't exist for all nautural number $m$.
Given $f$ an analytic function in open $D \subset \mathbb C$, $z_0$ is an isolated singularity defined as $B(z_0;r)\backslash\{z_0\} \subset D$, then know that $f$ can be written as an expansion of ...
user1194243
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
2
votes
1
answer
80
views
Radius of convergence for a power series where coefficients are given by recursive relation
Problem: Let $f(z) = \sum_{n \geq 0} a_n z^n$ be the formal power series with coefficients given by the following recursive relation: given $\alpha, \beta \in \mathbb{R}$,
$$a_0 = 0$$
$$a_1 = 1$$
and
$...
- 312
0
votes
3
answers
83
views
Jameson complex analysis Exercise 1.2.5: $\sum_{n=0}^{\infty} a_n z^n = s(z)$ is real for all real $z$, prove $a_n$ are real
How to solve the following problem? I can't use differentiation to solve it because this exercise is presented before the section that introduces differentiation in the book.
Suppose that $\sum_{n=0}^{...
- 451
2
votes
2
answers
66
views
Prove that the complex numbers for which the series $\sum_{k=1}^{\infty} \frac{3^n}{z^n+z^{-n}}$ converges is an open set of $\mathbb{C}$
My ideas was to find the radius of convergence of the series using the ratio test that says
$$R = \frac{1}{\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|} = \lim_{n\to\infty}|\frac{a_n}{a_{n+1}}|$$
However, I ...
- 23
1
vote
1
answer
74
views
Computing integral over unit circle
Compute \begin{equation}I=\frac{1}{2\pi i}\int_{C(0,1)}z^n\exp\left(\frac{2}{z}\right)\textrm{d}z\end{equation} where $C(0,1)$ is the unit circle centred at $0$ oriented anticlockwise, for integer ...
user1022331
1
vote
0
answers
36
views
Is f holomorph on the $B \left( a, \dfrac{R}{e} \right) $?
E and F be a Banach spaces. Let U be an open subset of E. A mapping $f :U \longrightarrow F$ is said to be holomorphic if for each $a \in U $ there exist a ball $ B(a,r) \subset U $ and a sequence of ...
user1014033
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:$$\sum_{k\ge0}(-1)^k\frac{z^{...
- 6,549
1
vote
0
answers
50
views
Half of a Mean-Value Theorem
For an entire function $$f(z)=\sum_{n=0}^\infty c_n z^n \, ,$$
we simply have that
$$\frac{1}{2\pi} \int_0^{2\pi} f(Re^{it}) \, {\rm d}t = c_0$$
$\forall R>0$. The same is true if $R\rightarrow 0$, ...
- 6,277