All Questions
7
questions
0
votes
0
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86
views
Continuity on boundary of convergence power series
I’m stuck trying to prove the following: Given $f(z) = \log(2+z^2)$, I consider its power series representation around $z=0$, which is $i(z) = \log(2)+\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n2^n}z^{2n}$...
- 1
0
votes
1
answer
94
views
If $g: \mathbb{R}^n \to \mathbb{R}$ then there is a $h$ entire analytic function that extends $ g $
Let $g: \mathbb{R}^n \to \mathbb{R}$ be a continuous function, such that $g(x)>0$, for all $ x \in \mathbb{R}^n$. I want to prove that: there is an entire analytic function $h$ (this means that $...
- 1,657
2
votes
1
answer
38
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Show that $f(s)=\sum \frac{1}{n^s}$ is continuous for $Re(s)>1$
Show that $f(s)=\sum \frac{1}{n^s}$ is continuous for $Re(s)>1$. In my attempt i try to use Weierstrass Test, $\frac{1}{n^s}=\frac{1}{n^{Re(s)+iIm(s)}}=\frac{1}{e^{\log n^{\Re(s)+iIm(s)}}}=\frac{1}{...
- 1,251
1
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0
answers
145
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Step in the proof of the continuity of the Gauss Gamma function
Define the Gamma function as
\begin{align}
\Gamma(z):= \frac{1}{G(z)} \quad \forall \, z\in \mathbb{C}\setminus \{0, -1, -2, ... \}
\end{align}
where
\begin{align}
G: \mathbb{C} \rightarrow \mathbb{C}...
- 866
0
votes
0
answers
40
views
Uniform convergence of logarithm on a compact set.
Given that $\phi$ and $\phi_n,n \geq 1$ are continuous functions from $\mathbb{R}$ into $\mathbb{C}$ such that $ \phi(0) = \phi_n(0) = 1$, $\phi(x) \neq 0$ and $\phi_n(x) \neq 0$ for any $x$. Suppose ...
- 163
0
votes
2
answers
173
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example of a sequence of uniformly continuous functions on a compact domain converging, not uniformly, to a uniformly continuous function
I am looking for an example of a sequence of uniformly continuous functions, defined on a compact domain that converge pointwise, but NOT uniformly, to a uniformly continuous function.
At first, I ...
- 171
0
votes
0
answers
68
views
Non zero continuous path $[0,1]\to \mathbb C$ has continuous logarithm
Let $\gamma:[0,1]\to \mathbb C$ be continuous, and not passing through $0$. How can we prove that, using complex analysis, there is a continuous $G:[0,1]\to \mathbb C$ so that $\gamma=e^G$ ?
This can ...
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