Analytical continuation by contour rotation

[muzialis]

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.

 

[jvicens]

Hello,

Thank you for sharing your question about Langer's paper on the Theory of the Condensation Point. Analytical continuation is a powerful technique used in complex analysis to extend the domain of a function beyond its original definition. In this case, Langer is using analytical continuation to extend the function $f(H) = \int_{0} ^{\infty} \mathrm{d} t \quad t^2 e^ {-\frac{A}{H}(t^3+t^2)}$ from positive, real values of H to negative, real values of H.

The reason for rotating the contour in the complex plane is to avoid the singularity at 0. As you mentioned, the function has a singularity at 0 for negative values of H. By rotating the contour, Langer is able to move the singularity to a different location in the complex plane, allowing him to continue the function without any singularities. This is a standard technique used in analytical continuation, and it ensures the function remains well-defined and continuous as H is varied.

I hope this explanation helps to clarify the reasoning behind rotating the contour in Langer's paper. If you have any further questions, please let me know. Thank you.