How can the difficult Gaussian integral be solved using standard tricks?

In summary, the conversation is about a complicated statistics problem that involves solving difficult integrals. One integral in particular is proving to be challenging due to the presence of a Gaussian factor. The individual asking for help suggests using partial fraction decomposition and the Schwinger parametrization to simplify the integral. The solution involves redefining variables and using the fundamental theorem of calculus, resulting in the integral being expressed in a simpler form.
  • #1
Ben D.
4
0
Hi everyone,

in the course of trying to solve a rather complicated statistics problem, I stumbled upon a few difficult integrals. The most difficult looks like:

[tex] I(k,a,b,c) = \int_{-\infty}^{\infty} dx\, \frac{e^{i k x} e^{-\frac{x^2}{2}} x}{(a + 2 i x)(b+2 i x)(c+2 i x)} [/tex]

where [tex] a,b,c [/tex] are real positive numbers and [tex] k [/tex] is a real number. This integral cannot be done by simple contour integration because of the Gaussian factor. From the context, I expect error functions to appear in the solution.

Any clever ideas?

B.D.
 
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  • #2
Ben D. said:
Hi everyone,

in the course of trying to solve a rather complicated statistics problem, I stumbled upon a few difficult integrals. The most difficult looks like:

[tex] I(k,a,b,c) = \int_{-\infty}^{\infty} dx\, \frac{e^{i k x} e^{-\frac{x^2}{2}} x}{(a + 2 i x)(b+2 i x)(c+2 i x)} [/tex]

I think you can do this with a few standard tricks. First, assume that we can do partial fraction decomposition, so that we can consider three integrals of the form
$$I_{k,a} = \int_{-\infty}^{\infty} dx\, \frac{e^{i k x} e^{-\frac{x^2}{2}} x}{ x - ia/2 }.$$
Next, use the Schwinger parametrization
$$ \frac{1}{x - ia/2 } = -i \int_0^\infty du\, e^{i u(x - i a/2)},$$
so
$$I_{k,a} = -i e^{a/2} \int_0^\infty du\, \int_{-\infty}^{\infty} dx\, e^{i (u+k) x} e^{-\frac{x^2}{2}} x.$$
Next, the factor of ##x## can be handled by replacing it with a derivative ##-i d/dk##, so
$$I_{k,a} = -e^{a/2} \frac{d}{dk} \int_0^\infty du\, \int_{-\infty}^{\infty} dx\, e^{i (u+k) x} e^{-\frac{x^2}{2}} .$$
The Gaussian integral is done by completing the square:
$$I_{k,a} = - e^{a/2} \frac{d}{dk} \int_0^\infty du\, \sqrt{2\pi} e^{-(u+k)^2/2 } .$$
Finally, we redefine variable to ##u'=u+k## and use the fundamental theorem of calculus to find
$$I_{k,a} = \sqrt{2\pi} e^{(-k^2 + a)/2} .$$
This has the right behavior in the limit as ##k,a\rightarrow 0##.

Note that you can skip the partial fractions and just do a more complicated Schwinger parameterization in your original integral, but I'm not sure that would save too much effort.
 

Related to How can the difficult Gaussian integral be solved using standard tricks?

What is a Non-trivial Gaussian integral?

A Non-trivial Gaussian integral refers to a type of mathematical integral that involves a Gaussian function, also known as a bell curve. It is considered non-trivial because it cannot be solved using simple algebraic methods and requires more advanced techniques.

What are some common applications of Non-trivial Gaussian integrals?

Non-trivial Gaussian integrals have a wide range of applications in various fields such as physics, statistics, and engineering. Some common examples include calculating probabilities in statistical distributions, solving differential equations in quantum mechanics, and analyzing signal processing in electrical engineering.

How do you solve a Non-trivial Gaussian integral?

Solving a Non-trivial Gaussian integral involves using techniques such as substitution, integration by parts, and completing the square. It may also require the use of special functions such as the error function or the gamma function. The specific method used will depend on the complexity of the integral and the desired result.

What is the significance of the term "Non-trivial" in a Gaussian integral?

The term "Non-trivial" in a Gaussian integral indicates that the integral cannot be solved using basic algebraic methods such as the power rule or substitution. It requires more advanced techniques and may involve complex calculations. Non-trivial Gaussian integrals are often used to solve problems that have real-world applications and cannot be easily solved using simple methods.

Are there any special properties of Non-trivial Gaussian integrals?

Yes, Non-trivial Gaussian integrals have several special properties, including symmetry, convergence, and independence of the mean and variance. They also have a connection to the central limit theorem, which states that the sum of a large number of independent random variables will approximate a Gaussian distribution.

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