All Questions
13
questions
2
votes
1
answer
132
views
How do you show that the exponential is not uniformly convergent on the entire complex plane? [duplicate]
I want to show that the series $\sum_{z=0}^\infty\frac{z^n}{n!}$ is not uniformly convergent on the entire complex plane. I understand that we just need to show that the supremum norm of the ...
- 153
1
vote
1
answer
133
views
Bound power series where each term is divided by $n!$
Suppose the radius of convergence, $R$, of the power series $\sum_0^\infty c_nz^n$ has a positive radius of convergence. How can I show that
$$\left| \sum_{1}^\infty \frac{c_nz^n}{n!} \right| \le C \...
- 11
1
vote
1
answer
83
views
Find the limit $\lim_{z\to2k\pi i}\frac{z}{e^z-1}$ where $k\in\Bbb{Z}$
If $k=0$, then ${e^z-1\over z}=\sum_{n\ge0}{z^n\over (n+1)!}$. So. $\lim_{z\to0}\frac{z}{e^z-1}=1$.
Now if $k\ne0$, then $\lim_{z\to2k\pi i}{e^z-1\over z}=\sum_{n\ge0}{(2k\pi i)^n\over (n+1)!}$, now ...
- 3,144
0
votes
2
answers
269
views
Justification for the power series representation of exponential function
In the book of Complex Analysis by Conway, on page $32$, it is given that
Consider the series $\displaystyle \sum _{n=0}^{\infty }\:\frac{z^n}{n!}$ ;we have that this series has radius of convergence ...
- 7,337
1
vote
2
answers
33
views
Radius of convergence of $\sum_{n=0}^\infty \frac{z^n}{e^{an}+1}, a \in \mathbb{C}$
$$\sum_{n=0}^\infty \frac{z^n}{e^{an}+1}, a \in \mathbb{C}$$
I have some doubts on the result because of that complex parametre. I think that the radius is $R=0$, since the negative real axis of the ...
- 353
2
votes
1
answer
974
views
Computing the Taylor expansion of the square root of cos(z),
Let $\large f(z)=\sqrt{cosz}$ with the branch of the square root chosen so that $f(0)=1$.
Consider the power series expansion of $f(z)$ in powers of $z$.
Part 1) Compute the first three non-zero ...
- 1
1
vote
0
answers
88
views
Analytic Function with Exponential Coefficients
I've been struggling with a weird complex analysis statement for a few days without making really any headway; I'd really appreciate a hint to get me past where I'm at. The theorem goes like this:
...
- 51
1
vote
1
answer
138
views
For every $z\in \Bbb C$, the exponetial series converges uniformly on every bounded subset of the complex plane
$$\operatorname{exp}(z)=\sum_{n=0}^\infty \frac{z^n}{n!}$$
This series converges uniformly on every bounded subset of the complex plane. What does this mean in simple terms?
- 1,443
3
votes
1
answer
1k
views
Exponential of a complex number converges absolutely
$$\operatorname{exp}(z)=\sum_{n=0}^\infty \frac{z^n}{n!}$$
This converges absolutely for every $z\in \Bbb C$. What does this mean to a layman?
- 1,443
4
votes
3
answers
2k
views
Find a closed form from the given power series
I have the power series $\sum_{n=0}^{\infty} {z^{2n}\over{n!}}$, how do I find the closed form for this power series.
I am aware that $e^z=\sum_{n=0}^{\infty} {z^{n}\over{n!}}$, so I tried to ...
- 3,454
1
vote
2
answers
144
views
Finding power series
I need to find the power series for $e^z + e^{az} + e^{a²z}$ where $a$ is the complex number $e^{2πi/3}$.
I know that $1 + a + a² = 0$.
I have tried to differentiate the expression and give values ...
- 157
2
votes
1
answer
1k
views
Proving that $\exp(z_1+z_2) = \exp(z_1)\exp(z_2)$ with power series [duplicate]
Probably a simple question, but I wonder about the following:
To prove that $\exp(z_1+z_2) = \exp(z_1)\exp(z_2)$, I use :
$$\exp(z_1+z_2) = \sum_{n=0}^{\infty}\sum_{k=0}^n\frac{1}{k!(n-k)!}z_1^kz_2^{...
- 729
6
votes
1
answer
13k
views
Radius of convergence for the exponential function
I'm studying physics and am currently following a course on complex analysis and in the section on analytic functions, the radius of convergence $R$ for power series was introduced. The Taylor ...
- 418