All Questions
103
questions
-1
votes
2
answers
50
views
Determine whether the complex power series converges at a point
I need to determine if a series $$\sum\limits_{n=1}^{\infty} \frac{(z-1+i)^{2n-1}}{5^n(n+1)ln^3(n+1)}$$ converges at a point $z_1 = -1$
After substituting the point, I got: $$ \sum\limits_{n=1}^{\...
- 123
2
votes
1
answer
104
views
A confusion about the radius of convergence in Ahlfors' "Complex Analysis"?
On the third edition of Ahlfors' Complex Analysis, page 39 Theorem 2 it states: The derived series $\sum_{1}^{\infty}na_n z^{n-1}$ has the same radius of convergence, because $\sqrt[n]n \rightarrow 1$....
- 174
0
votes
2
answers
146
views
I find a "mistake" on p.40 in Ahlfors' "Complex Analysis"?
On the third edition of Ahlfors' Complex Analysis, page 40 Theorem 2 it states: we conclude that
\begin{equation*}
\left|\frac{R_n(z)-R_n(z_0)}{z-z_0}\right| \leq \sum_{k=n}^{\infty} k|a_k|\rho^{k-1}
\...
- 174
0
votes
1
answer
113
views
How to prove $f(z)=a_0+a_1z+\cdots+a_nz^n+\cdots$, which has infinite items, is an analytic function? [duplicate]
It is easy to prove the following four statements.
(i) If $f$, $g$ are analytic functions and $f^{\prime}$, $g^{\prime}$are continuous then $(f+g)(z)$ is analytic and $(f+g)^\prime(z)=f^\prime(z)+g^\...
- 174
2
votes
1
answer
116
views
Absolute convergence on boundary implies continuity of power series
Let $f(z) = \sum_{n=0}^{+\infty}c_nz^n$ be a complex power series with radius of convergence $R=1$. Suppose that the series of coefficients converges absolutely, i.e. $\sum_{n=0}^{+\infty}|c_n| < +\...
- 1,179
1
vote
1
answer
95
views
Find an explicit expression for $\sum_{n=0}^\infty \frac{1}{(z^2+1)^n}$
Let $z \in \mathbb{C}$, discuss the convergence of $$g(z)= \displaystyle\sum\limits_{n=0}^\infty \dfrac{1}{(z^2+1)^n}$$ and find an explicit expression for $g(z)$.
By the ratio test, I found that $z$ ...
- 410
2
votes
1
answer
49
views
Find an expression for the coefficient of a power series
I'm trying to find an expression for the coefficients of the Taylor series centered at zero of the function $$f(z)=\frac {e^{z^2}}{z-2}$$
As a hint I'm told to try and multiply through by $$z-2$$ and ...
- 175
0
votes
0
answers
95
views
Does the Laurent series of $e^{1/z}$ centred at $z=1$ have any non-zero terms of negative exponent? What is the radius of convergence?
I have been struggling with this problem for a while. So far, this is what I have managed to obtain. I have little faith in my solution. If my answer is wrong, could someone give me a hint?
We know ...
- 600
2
votes
3
answers
214
views
Laurent series of $e^{1/z}$ centred at $z=1$
If it were $e^z$ being expanded at $z=1$, it would be relatively easy. We would take advantage of the fact that
$$e^z =\sum_{-\infty}^\infty a_n(z-1)^n$$
where $a_n$ is given by the closed-loop ...
- 600
1
vote
2
answers
73
views
Find The Laurent Series of $\frac{z}{z-1} \sin{z}$ around $z=1$
We will start by finding the expansion of $\sin{z}$ around $1$.
$$\sin(z) = \frac{1}{2i} \left(e^{iz} - e^{-iz}\right) = \frac{1}{2i} \left(e^{i(z-1+1)} - e^{-i(z-1+1)}\right) = \frac{1}{2i} \left( e^{...
- 600
0
votes
1
answer
276
views
Find the power series expansion of $\frac{1}{1-z}$ centred at $3i$ over the domain $|z-3i|<3$
We can rewrite the region $|z-3i|<3$ as $\left| \frac{z-3i}{3}\right|<1$. Then using the fact that $\frac{1}{1-z} = \sum_{n=1}^\infty z^n$, we can obtain the following power series
$$\frac{1}{1-(...
- 600
2
votes
1
answer
60
views
Confusion regarding the use of the residue theorem in complex analysis.
I have been studying about the residue theorem and I am a little confused about the way the theorem is used.For example considering the function: $$f(z)=\frac {1}{1-z}+\frac{2}{2-z} $$ this function ...
- 306
11
votes
4
answers
337
views
Upper bound of $\sum_{n=1}^N |1-z^n|$ where $|z| \leq 1$
How to derive an upper bound of
$$\sum_{n=1}^N |1-z^n|$$ where $z\in\mathbb{C}$ and $|z| \leq 1$?
A trivial upper bound would be $2N$ since each $|1-z^n| \leq 2$. But I am hoping for tighter bounds. I ...
- 391
1
vote
3
answers
58
views
Struggling to compute a power series for a complex value function
I am struggling to compute the power series expansion of $$f(z) = \frac{1}{2z+5}$$ about $z=0$, where $f$ is a complex function. I tried comparing it to the geometric series as follows,$$ f(z) = \frac{...
- 145
0
votes
2
answers
50
views
Is it true that $\sum_{n=0}^{\infty} (-3-2z)^n =\frac{1}{2z+4}$ for $z\in\mathbb{C}$, $|-3-2z|<1$?
This problem requires us to find a power series expression for $\frac{1}{2z+4}$ where $z\in \mathbb{C}$.
ANSWER 1:
Let $S_k=\sum_{n=0}^{k} (-3-2z)^n$. Then we have the following:
$(-3-2z)S_k=(-3-2z)+...
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