# Questions tagged [analytic-continuation]

For questions related to analytic continuation

22 questions
680 views

### Alternating series test for complex series

I want to show that we can continue Riemann's zeta function to Re$(s)>0$, $s\neq 1$ by the following formula \begin{align} (1-2^{1-s})\zeta(s)&=\left(1-2\frac{1}{2^s}\right)\left(\frac1{1^s}+\...
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### Alternative analytic continuation to zeta, not giving $-\frac{1}{12}$ for sum of integers

Apologies if this has been asked already. Inspired partly by this answer where an $n e^{-\epsilon n}$ rather than $n^s$ regularization was made in the 'evaluation' of $\sum\limits_{n=1}^{\infty}n$ and ...
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### Does the Abel sum 1 - 1 + 1 - 1 + … = 1/2 imply $\eta(0)=1/2$?

If $\sum_{n=1}^\infty a_n$ is Abel summable to $A$, then necessarily $\sum_{n=1}^\infty a_n n^{-s}$ has a finite abscissa of convergence and can be analytically continued to a function $F(s)$ on a ...
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### Is $\int_0^\infty \frac{dt}{e^t-xt}$ analytic continuation of $\sum_{k=1}^\infty \frac{(k-1)!}{k^k} x^{k-1}$?

The following power series apparently converges only for $-e \leq x <e$: $$f(x)=\sum_{k=1}^\infty \frac{(k-1)!}{k^k} x^{k-1}$$ We can use it to define a real function $f(x)$, analytic in that ...
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### real part of analytic function

Assume $f(x) \in D$ is analytic, where $x \in R$ and $D$ denotes the unit circle, which means $x$ is real and $f(x)$ is complex. Then what kind of function is the real part of $f(x)$? harmonic or ...
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### Maximal analytically continued domain

Can a power series or a Laurent series always be analytically continued into a domain strictly larger than its convergent annulus of finite radii? Is there a way to find its maximal domain that it can ...
### Analytic continuation for $\zeta(s)$ using finite sums?
$\zeta(s)$ converges for $\sigma >1$ but not for $\sigma =1/2.$ But for some reason for $s = 1/2 + i t$ and fixed finite $N,~$ $\zeta_N(s) =\sum_{n=1}^N\frac{1}{n^s}$ is very close to $\zeta(s)$ ...