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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.

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# Analytical continuation by contour rotation

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