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Analytical continuation by contour rotation

  1. Jun 4, 2015 #1
    Hi All,

    reading a paper by Langer (Theory of the Condensation Point, Annals of Physics 41, 108-157, 1967), I came across an analytical continuation technique which I do not understand (would like to upload the paper PDF but I am not so sure this is allowed).
    Essentially, he deals with the integral
    ## \int_{0} ^{\infty} \mathrm{d} t \quad t^2 e^ {-\frac{A}{H}(t^3+t^2)} ##
    His aim is to analytically continue this function for negative H H , over the singularity at 0 0 ..

    He starts by considering the real part of ## t^2 + t^3 ##, showing two saddle point at ##0 ## and ## -2/3 ##.
    Then he says, let us stat with positive, real only H and move it around the origin in the complex plan, anticlockwise (the same will be repeated clockwise).
    The "array of three valleys and mountains, given by ## \frac{-t^3}{H^2}## for large ## t## , also moves anticlockwise. And as they say, so far, so good, but now the bit that puzzles me.
    "The analytical continuation of ## f(H)## is obtained, according to a standard and rigorous construction, by rotating the ##C_1## , going from ## 0 ## to infinity on the real axis, so that it always remains at the bottom of its original valley, and a picture shows the countour ## C_2 ## going from ## 0## to ## -2/3## along the negative real axis, and then up tp the top left along one of the "valleys". Once, he adds, ##H ## is moved to ## H_2 = e^{i \pi} H ##, the integrand has returned to its original form, but f(H2 ) f(H_2) is obtained integrating along the rotated countour ## C_2## . I understand that the integrand will return to its original form, but why rotating the contour?

    If anybody had a hint, that would be so appreciated, thanks.
     
  2. jcsd
  3. Jun 9, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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