# Questions tagged [analytic-continuation]

For questions related to analytic continuation

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### Are all complex algebraic varieties in 2+ variables unbounded as a result of Hartogs' extension theorem?

Hartogs' extension theorem states that for any $n\geq 2$, $U\subseteq\mathbb C^n$, $K\subset U$ compact (in $\mathbb C^n$) and a holomorphic function $f:U\setminus K\to\mathbb C$, it can always be ...
1 vote
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### Is there some link between analytic continuation and potential theory?

It seems like the analytic continuation of a function has a lot in common with the process of trying to define a potential for a vector field (or a differential 1-form). In particular, an analytic ...
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### Finding the natural boundary of $\sum_{n=0}^{\infty}\frac{z^n}{n^k}$ for $k \ge 2$

Let $k\ge 2$, given $$f(z)=\sum_{n=1}^{\infty}\frac{z^n}{n^k}$$ It is easy to see that it converges for $|z|\le 1$, but how can it be analytically continued beyond the unit circle? Hadamard proved ...
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1 vote
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### Proposition 16.5.4 in Ireland-Rosen

We aim to show that if $\chi$ is a complex Dirichlet character mod $m$, then $L(1, \chi) \neq 0$. Assuming otherwise, we easily prove that if $F(s) = \prod_{\chi}L(s, \chi)$, where the product is over ...
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### solution verification: Is $K(s)$ holomorphic on $\Bbb C$?

Consider the Mellin integral $$K(s)=\int_{1/2}^1 \zeta\bigg(-\frac{1}{\log x}\bigg)~x^{s-1}~dx$$ Where $\zeta(\cdot)$ is the Riemann zeta function defined for real $1/e<x<1.$ $K(s)$ is ...
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### Proof of analytic continuation on manifold

I was reading the theorem below in this article. My question concerns the passages in bold. Specifically, why would Z be open? Is it because we can use germs to define a topology where Z would be the ...
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### Meromorphic continuation of Euler product

Short version: What can be said about the meromorphic continuation of the Euler product $$\prod _{p}\left (1+\frac {p^{-s}}{p-2}\right )?$$ Longer version: I realise I have some misconceptions about ...
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### analytically continue functions on independent planes in $\Bbb R^3$

I know techniques to analytically continue some functions off the positive real line. I am less sure given I have 2 independent euclidean planes sitting in $\Bbb R^3$ with some function $f$ defined on ...
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1 vote
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### Funny lattice equivalence between additive and subtractive extended square roots

I am interested in the behavior of the "extended square root", such as: $$\sqrt{2+\sqrt{2+\sqrt{2+\sqrt2}}} ...$$ $$\sqrt{n\pm\sqrt{n\pm\sqrt{n\pm\sqrt{n\pm}}}} ...$$ I've noticed after ...
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### How to find the divergent renormalization of $\sum_{n=1}^{\infty} \frac{2^n}{n}$?

Consider the divergent series $$\sum_{n=1}^{\infty} \frac{2^n}{n}$$ This can be seen as arising from the function $f(z) = -\ln(1-z)=\sum_{n=1}^{\infty} \frac{z^n}{n}$ and 'evaluating' that power ...
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### How to evaluate $\int_{-\infty}^\infty \sin^3(x)/x^3 dx$ without finding an analytic continuation, still using complex analysis.

There is a similar post regarding the integral $\int_{-\infty}^\infty \sin^3(x)/x^3 dx$ on Stack. The reason I have some trouble with this post is because it gives an analytic continuation of the ...
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### Showing that $f\in C^{n}$ if and only if $\frac{\partial^n f}{\partial z^k\partial\bar{z}^{n-k}}$ is continuous

Suppose that $D$ is an area, $n$ is a natural number. Show that: $f\in C^{n}$ if and only if $\frac{\partial^n f}{\partial z^k\partial\bar{z}^{n-k}}$ is continuous on $D$. It's easy to prove ...
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### Analytic continuation of function given by Moser-de Bruijn sequence

I was wondering about the function $$F(x) = \prod_{n=0}^{\infty}{(1+x^{4^{n}})} = 1+x+x^4+x^5+x^{16}+x^{17}+...$$ where the exponents in the resulting power series are given by the Moser-de Bruijn ...
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### How to interlopate the pickover factorial?

Question I want to be able to interlopate the pickover factorial, defined as: $$n = ^{n!}n!$$ where $^xx$ represents tetration. I want to be able to interlopate this function. Context The reason I ...
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### Analytic extension of $\text{Li}_0^{(1,0)}(z):=-\displaystyle\sum_{n=1}^{\infty}\ln(n)z^{n}$ for $|z|>1$

I would like to extend the domain of the following function: $$\text{Li}_0^{(1,0)}(z):=-\sum_{n=1}^{\infty}\ln(n)z^{n}\qquad\text{where }|z|<1$$ The part in red is the series, the part in green is ...
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### Extending a holomorphic function on $\mathbb{C}\backslash K$ to an entire function

Consider a compact set $K\subset \mathbb{R}$ with positive measure (i.e. $\mu(K)>0$), and for $z\in\mathbb{C}\backslash K$, define the holomorphic function $f$ on $\mathbb{C}\backslash K$ by \begin{...
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### For all complex $|z| \neq 1$ : $\sum_{n=0}^{\infty} w_n \frac{z^n}{1+z^n+z^{2n}+z^{3n}+z^{4n}} = \sum_{n=0}^{\infty} u_n \frac{z^n}{1+z^n+z^{2n}}$?

Ok I am a bit confused. So here comes a question, Consider a maclaurin series for $f(z)$ $$f(z) = \sum_{n=0}^{\infty} f_n z^n$$ where $f(z)$ has a radius of exactly $1$. $f(z)$ may or may not have a ...
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### Description of Riemann surface of polynomial inverse

My question is about page 4 of the pdf of the following paper , one does $\textbf{not}$ need to read pages 1-3 of the paper to understand my question ( the $\textbf{only}$ part that needs to be read ...
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I found this function: $$\zeta(\alpha+i\beta)*\zeta(\alpha-i\beta)=\prod \frac{1}{1-\frac{2cos(\beta ln{p})}{p^\alpha} + \frac{1}{p^{2\alpha}}}=\Re(\zeta(\alpha+i\beta))^2 +\Im(\zeta(\alpha+i\beta))^... • 135 4 votes 1 answer 112 views ### Understanding Vladimir Maz'ya's Problem 72 what is |dy|? In the article "Seventy Five (Thousand) Unsolved Problems in Analysis and Partial Differential Equations" by Vladimir Maz’ya Problem 72 on page 36 is given as follows: Let C be the unit ... • 17.5k 1 vote 0 answers 64 views ### Simple example of analytic continuation along a path I'm trying to understand the following example of analytic continuation given in a physics paper. In the last paragraph of page 16 the authors define the function$$f(z,\bar z)=(z\bar z)^{-\Delta}$$... 0 votes 0 answers 161 views ### Riemann continuation theorem Let f\rightarrow \mathbb{C} \setminus \{ 0 \} \rightarrow \mathbb{C} be holomorphic. Show that the following are equivalent: There exists g : \mathbb{C} \rightarrow \mathbb{C} holomorphic, such ... 0 votes 1 answer 66 views ### Analytic continuations for (-1)^x over the real number line The function f(x)=(-1)^x can only be mapped to the real numbers when x\in\mathbb{N}. So I wonder what analytic continuation can expand the domain of f to all real numbers without changing the ... • 6,549 0 votes 0 answers 30 views ### Useful Partial Sums The following formula:$$\sum_{k=m}^nf(k)=c(n-m)+\sum_{k=m}^\infty(f(k)-f(k+n))$$(Where f\rightarrow c) can be proven by telescoping the infinite sum in the RHS. The use of this formula is to expand ... • 6,549 0 votes 0 answers 26 views ### can there be fractional values in the Veblen hierarchy? I know that \varphi_0(0)=\omega and \varphi_1(0)=\epsilon_0, but what would the value of \varphi_{\frac{1}{2}}(0) be. it seems intuitive that it would be smaller than \epsilon_0, being$$\frac{...
I have the following integral: $$F(s) = \int\limits_{\alpha}^{\delta} t^{s-3} f(t) dt.$$ The function $f(t)$ is real-valued, continuous, monotonically increasing on $[0,\delta]$, bounded, and is non-...