Special Integrals Hermite(2n+1,x)*Cos (bx) and e^(-x^2/2)

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In summary, the conversation is about how to integrate the function Hermite(2n+1,x)*Cos (bx)*e^(-x^2/2) from 0 to infinity. The first response suggests checking out a Wikipedia page for information about the integral. The original poster clarifies that they are looking for the integral of Exp[-x^2/2]Cos[ b x] HermiteH[2n+1,x] from 0 to infinity. Another user offers a possible method for solving the integral, but the original poster mentions that the problem is complicated due to the interval of integration being from 0 to infinity. The final response acknowledges the difficulty of the problem and suggests using a specific software to solve it
  • #1
dongsh2
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Do some one know how to integrate the
Hermite(2n+1,x)*Cos (bx) and e^(-x^2/2), x from 0 to infinity?
 
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  • #2
Do some one know how to integrate the
Integrate [Hermite(2n+1,x)*Cos (bx)*e^(-x^2/2), {x,0, \infinity}]?
 
  • #4
Svein said:
Check out http://en.wikipedia.org/wiki/Gaussian_integral. It tells you most of what you need to know about that integral.

Thanks. However, what I want to calculate is : Integrate[Exp[-x^2/2]Cos[ b x] HermiteH[2n+1,x],{x,0,\infinity}]. I have checked some references but I cannot find the result.
 
  • #5
If we leave the Hermite polynoms for a moment, we can transform the rest: [itex] e^{-\frac{x^{2}}{2}}\cos(bx)[/itex] is the real part of [itex] e^{-\frac{x^{2}}{2}+ibx}[/itex]. The exponent can be further transformed: [itex] -\frac{x^{2}}{2}+ibx=-\frac{1}{2}(x^{2}-2ibx)=-\frac{1}{2}(x-ib)^{2}-\frac{1}{2}b^{2}[/itex]. Thus you end up with [itex]e^{-\frac{1}{2}(x-ib)^{2}}\cdot e^{-\frac{1}{2}b^{2}} [/itex], where the last part is constant. Now, put z = (x-ib), then dz = dx. Use that with the contents of the link I gave you and see where you end up.
 
  • #6
Thanks. However, it is not easy as what you thought. The problem is: x \in (0,\infinity). If we take z=x-i b and we have dz=dx, but the Hermite polynomial becomes
H[2n+1, z+ib] and integral interval becomes (-i b,\infinity). Using the mathematica, it still cannot find the solutions.
 
  • #7
dongsh2 said:
However, it is not easy as what you thought.
I did not say it was easy, I just gave you an idea of how to attack it.
 

FAQ: Special Integrals Hermite(2n+1,x)*Cos (bx) and e^(-x^2/2)

1. What is the formula for Special Integral Hermite(2n+1,x)*Cos (bx)?

The formula for Special Integral Hermite(2n+1,x)*Cos (bx) is given by:
∫(Hermiten(x) * cos(bx)) dx = (Hermiten+1(x) * sin(bx)) / b + C
where Hermiten(x) is the Hermite polynomial of degree n.

2. How is the Special Integral Hermite(2n+1,x)*Cos (bx) used in mathematical applications?

This special integral is used in various mathematical applications, such as in quantum mechanics, signal processing, and statistical mechanics. It is also used to solve differential equations and in the calculation of probabilities in statistics.

3. What is the relationship between Special Integral Hermite(2n+1,x)*Cos (bx) and e^(-x^2/2)?

The Special Integral Hermite(2n+1,x)*Cos (bx) is closely related to the Gaussian function e^(-x^2/2). In fact, the Hermite polynomials are orthogonal to the Gaussian function, which makes them useful in various mathematical calculations.

4. Can the Special Integral Hermite(2n+1,x)*Cos (bx) be evaluated analytically?

Yes, the Special Integral Hermite(2n+1,x)*Cos (bx) can be evaluated analytically using the formula mentioned in the first question. However, for higher values of n, the integration can become more complex and numerical methods may be used.

5. Are there any real-life applications of the Special Integral Hermite(2n+1,x)*Cos (bx)?

Yes, this special integral has many real-life applications. It is used in the calculation of wave functions in quantum mechanics, in the analysis of signals in signal processing, and in the calculation of probabilities in statistics. It is also used in modeling various physical phenomena in engineering and physics.

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