Gaussian integral w/ imaginary coeff. in the exponential

In summary, there are two main approaches to solving the integral ∫e-i(Ax2 + Bx)dx. The first method involves treating it as a gaussian integral and completing the square, which is valid when A has a positive imaginary part. This results in an overall decaying magnitude of the integrand. However, if A has no imaginary part, then the integral may not strictly converge. In this case, one can replace A with A + iε and take ε to 0 at the end of the calculation. While this approach may lead to a finite answer, it may not be unique and depends on the curve being integrated along and whether x is real or complex.
  • #1
So I've seen this type of integral solved. Specifically, if we have

∫e-i(Ax2 + Bx)dx then apparently you can perform this integral in the same way you would a gaussian integral, completing the square etc. I noticed on wikipedia it says doing this is valid when "A" has a positive imaginary part. I can see why that might be important, we would then have an overall decaying magnitude of the integrand, as opposed to purely oscillating.

What if A has no imaginary part, how would one go about doing such an integral?
 
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  • #2
If A is real, then the integral does not strictly converge. However, sometimes you can get a finite answer by replacing ##A \to A + i \varepsilon## and then take ##\varepsilon \to 0## at the end of the calculation.

What you're doing here is interchanging the order of limits, which is not really allowed. The answer you get, although finite, might not be unique (i.e., there are many sequences of converging integrals that approach your diverging one, and we've only looked at one such sequence).
 
  • #3
You have not said anything about the curve you are integrating along or whether x is real or complex. For example: If x is complex and you integrate over a circle anywhere in the complex plane, the answer is 0 (since the integrand is analytic).
 

What is a Gaussian integral with imaginary coefficients in the exponential?

A Gaussian integral with imaginary coefficients in the exponential is a mathematical integral that involves the Gaussian function, also known as the bell curve, and an imaginary number in the exponent. It is used to solve various problems in physics, engineering, and statistics.

How is a Gaussian integral with imaginary coefficients in the exponential solved?

A Gaussian integral with imaginary coefficients in the exponential is solved using complex analysis techniques, such as contour integration and the residue theorem. These techniques involve manipulating the integral into a form that can be evaluated using known mathematical formulas.

What are the applications of a Gaussian integral with imaginary coefficients in the exponential?

A Gaussian integral with imaginary coefficients in the exponential has various applications in different fields. In physics, it is used to solve problems involving quantum mechanics, electromagnetism, and statistical mechanics. In engineering, it is used to model systems with damping or noise. In statistics, it is used in probability distributions such as the normal distribution.

What is the relationship between a Gaussian integral with imaginary coefficients in the exponential and the complex plane?

A Gaussian integral with imaginary coefficients in the exponential can be visualized in the complex plane as a contour integral. The imaginary number in the exponent corresponds to a rotation in the complex plane, which can be used to evaluate the integral using the residue theorem.

Are there any special cases of a Gaussian integral with imaginary coefficients in the exponential?

Yes, there are special cases of a Gaussian integral with imaginary coefficients in the exponential that can be solved using specific methods. For example, when the imaginary coefficient is zero, the integral reduces to the standard Gaussian integral. When the imaginary coefficient is purely imaginary, the integral can be solved using the substitution method.

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