- #1
muzialis
- 166
- 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##, 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.
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.