# Analytical continuation by contour rotation

• muzialis
In summary, Langer's paper discusses the use of 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. By rotating the contour in the complex plane, the singularity at 0 is avoided and the function remains well-defined and continuous. This is a standard technique in analytical continuation.
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.

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.

## 1. What is analytical continuation by contour rotation?

Analytical continuation by contour rotation is a mathematical technique used to extend the domain of a given function to a larger region. It involves rotating the contour of integration in the complex plane to a different angle, which can reveal new information about the function.

## 2. Why is analytical continuation by contour rotation important?

This technique is important because it allows us to better understand the behavior of a function in areas where it may not be defined or where it may be difficult to evaluate. It also helps us to make connections between different areas of mathematics, such as complex analysis and real analysis.

## 3. How does analytical continuation by contour rotation work?

Analytical continuation by contour rotation works by using the Cauchy-Riemann equations to rotate the contour of integration in the complex plane. This allows us to explore different paths and angles of integration, which can lead to new insights about the function.

## 4. What are some applications of analytical continuation by contour rotation?

This technique has many applications in physics, engineering, and mathematics. For example, it can be used to solve differential equations, calculate path integrals in quantum mechanics, and study the properties of complex functions in number theory.

## 5. Are there any limitations to analytical continuation by contour rotation?

Yes, there are some limitations to this technique. It may not work for all functions, and the rotated contour must still pass through points where the function is defined. It also requires some knowledge of complex analysis and the Cauchy-Riemann equations, which can be challenging for beginners.

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