All Questions
11
questions
0
votes
0
answers
52
views
Derivative of square root via power series
The power/Taylor series for the complex function $h(w)=\sqrt{w}$ is given by
$$h(w) = \sum_{k=0}^\infty \frac{a_k}{k!}(w-1)^k$$
where $a_0 = 1, a_{k+1} = (\frac{1}{2}-k)a_k$.
Now is my question: how ...
- 89
0
votes
1
answer
46
views
Cauchy residue formula question
The Cauchy formula can be used to find the $n$th derivative of an analytical function $f(z)$ as
$$
\frac{d^nf}{dz^n}\bigg|_{z=z_0}=\frac{n!}{2\pi i }\oint \frac{f(z)}{(z-z_0)^{n+1}}dz
$$
which happens ...
- 551
2
votes
4
answers
99
views
By differentiation or otherwise, prove that $(c(z))^2 +(s(z))^2 = 1$
Assume that the two power series $s(z) = \sum a_nz^n$ and $c(z) =\sum b_nz^n$ are convergent for all $z\in \mathbb{C},$ and that they satisfy the relations $s'(z) = c(z), c'(z) =-s(z).$ Deduce the ...
user670388
0
votes
1
answer
26
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Convergence of the derivative of a power series on the boundary
I'm wondering if the series $\sum a_nx^n$ converges at $x=R$, the radius of convergence, can we conclude that it's derivative also converges when x=R ?
- 39
0
votes
2
answers
241
views
Complex analysis power series / derivative
I need some help with this exercise: Show that there is no such power series $f(z)=\sum_{n=0}^{\infty}C_nz^n$ such that:
$f(z)=1$ for $z=\frac{1}{2}, \frac{1}{3}, \frac{1}{4},\ldots,$
$f'(0)>0$.
...
1
vote
2
answers
93
views
Uniform convergence of $\sum_{n=0} ^\infty \frac{(-1)^n}{2n+1} \left(\frac{2z}{1-z^2}\right)^{2n+1}$
I am given:
$$ f(z)=\sum_{n=0} ^\infty \frac{(-1)^n}{2n+1} \left(\frac{2z}{1-z^2}\right)^{2n+1}$$
And asked to show that this complex function series converges uniformly for $|z|\leq \frac{1}{3}$. ...
user459879
1
vote
3
answers
625
views
Prove if $k^\text{th}$ derivative of an entire function $f$ is polynomial, then $f$ itself is polynomial. Where's my mistake?
The exact wording of the question is as follows:
Let $f$ be an entire function. Suppose there exists a positive integer $k$ such that $k^\text{th}$ derivative $f^{(k)}$ is a polynomial. Prove that $...
- 133
1
vote
3
answers
71
views
Where can I find a quick proof that a function is analytic if and only if it admits a serie expansion?
I am currently studying functional analysis following the Aliprantis book. There, they define functions $f: O \longrightarrow X$ where $O$ is an open subset of $\mathbb{C}$ and $X$ is a Banach space. ...
- 108
1
vote
0
answers
56
views
I'm looking to find a power series for a solution f where $zf''(z)+f(z)=0$
I'm looking to find a power series for a solution f where $zf''(z)+f(z)=0$
and $f(0)=0$ and $f'(0)=1$
and with the assumption $\sum_{k=1}^∞{a_k z^k}=0$
So far I have done this:
$\sum_{k=1}^∞{k^2 ...
- 11
1
vote
1
answer
117
views
Derivative of exp with definition of differentiability
Prove with the definition of differentiability that $\exp(z)$ is differentiable in $\mathbb C$ and $(\exp(z))' = \exp(z)$ for all $z \in \mathbb C.$
I tried:
\begin{align*}
\frac{\exp(z+h) - \exp(z)}...
- 683
3
votes
1
answer
2k
views
How to prove that a complex power series is differentiable
I am always using the following result but I do not know why it is true. So: How to prove the following statement:
Suppose the complex power series $\sum_{n = 0}^\infty a_n(z-z_0)^n$ has radius of ...
- 683