# Analytical continuation free energy

1. Jun 3, 2015

### 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 $H$, over the singularity at $0$..

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 contour $C_1$, 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 $f(H_2)$ 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. Jun 3, 2015

### mathman

I suggest you try one of the math forums. Analytic continuation is a math procedure.

3. Jun 4, 2015

### muzialis

Thanks for the advice, will do so.