Analytic continuation Definition and 25 Threads

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent.
The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). They may alternatively have to do with the presence of singularities. The case of several complex variables is rather different, since singularities then need not be isolated points, and its investigation was a major reason for the development of sheaf cohomology.

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  1. S

    A Does the Z boson pole show up in the photon propagator?

    If I look at the photon propagator <A_mu (x) A^nu(0) > in momentum space, as I understand it I am to compute this by summing up all the self-energy diagrams of the photon, which look like: photon -> stuff -> photon In particular, since the photon shares the same quantum numbers as the Z, you...
  2. K

    I Pauli-Villars regularization for Vacuum Polarization

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  3. A

    I Analytic continuation of Dilogarithm

    I am trying to understand the branching geometry of the Dilogarithm function as described in Branching geometry of Dilogarithm. In Theorem 8.6, the following (dilogarithm) definition is given by letting ##n=2##: $$ \text{Li}_2(z)=\text{Li}_2^{(k_0,k_1)}(z)=\text{Li}_2^{(0)}(z)+\sum_{m=0}^1...
  4. benorin

    Insights The Analytic Continuation of the Lerch and the Zeta Functions

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  5. C

    I Analytic continuation of a dilogarithm

    The correct analytical continuation of the dilog function is of the form $$\text{lim}_{\epsilon \rightarrow 0^+} \text{Li}_2(x \pm i\epsilon) = -\left(\text{Li}_2(x) \mp i\pi \ln x \right)$$ I read this in a review at some point which I can no longer find at the moment so just wondered if this...
  6. nomadreid

    I Video (analytic continuation) seems to mix 4-D & 2-D maps

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  7. D

    Contour integration & the residue theorem

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  8. J

    Analytic continuation of Airy function

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  9. K

    Analytic continuation and physics

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  10. B

    Newest implementation of GW approximation needs analytic continuation?

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  11. alyafey22

    MHB Analytic continuation and Regularization simplified.

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  12. U

    Analytic Continuation along a Loop

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  13. A

    Analytic continuation to find scattering bound states

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  14. J

    Extending radius of convergence by analytic continuation

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  15. S

    Exploring the Riemann Hypothesis and Analytic Continuation

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  16. M

    Analytic continuation of an integral involving the mittag-leffler function

    greetings . we have the integral : I(s)=\int_{0}^{\infty}\frac{s(E_{s}(x^{s})-1)-x}{x(e^{x}-1)}dx which is equivalent to =I(s)=\frac{1}{4}\int_{0}^{\infty}\frac{\theta(ix)\left(sE_{s/2} ((\pi x)^{s/2})-s-2x^{1/2}\right)}{x}dx E_{\alpha}(z) being the mittag-leffler function and...
  17. G

    Analytic continuation of the zeta function

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  18. C

    How to find analytic continuation?

    The following is the problem from Fetter and Walecka (problem 3.7) If f(z) is defined to be the integration of rho(x) * (z-x)^(-1) from -infinity to +infinity. rho is in the following form rho(x)=gamma * ( gamma^2+x^2 )^(-1). Evaluate f(z) explicitly for Im(z)>0 and find its analytic...
  19. Z

    IR divergences by analytic continuation of parameters

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  20. L

    How do you DO analytic continuation?

    they talk about the existence of analytic continuation, but how do you find (the power series/product), calculate, compute the analytic continuation? how do you actually do analytic continuation on a function?
  21. T

    Analytic Continuation: Definition & Uses in QFT

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  22. benorin

    Exploring Analytic Continuation of Dirichlet Series

    I am writing my senior thesis (I am an undergrad math major at UCSB) on Dirichlet Series, which are, in the classical sense, series of the form \xi (s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s} where a_n,s\in\mathbb{C} and a_n is multiplicative, hence \forall n,m\in\mathbb{N}, \...
  23. S

    Can Analytic Continuation Determine the Metric Over a 4-D Manifold?

    Hello all, Can the method of analytic continuation [1] be applied to the calculation of the metric over a 4-d manifold? In other words, suppose that we are given the value of the metric g_ab as well as its power series at a single point p in a 4-dimensional manifold. Assume further that...
  24. benorin

    Convergence question on analytic continuation of Zeta fcn

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  25. C

    Why Apply Analytic Continuation in Complex Analysis?

    I don't understand the concept of analytic continuation, at any level (I have some small amount of experience with introductory undergraduate-level complex analysis from a long time ago, mostly forgotten): 1) Firstly, why would you want to apply analytic continuation on some complex function...
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