Solve Gaussian Integral Homework: Even/Odd Cases

In summary, the problem asks to solve for the integral \int_{0}^{\infty} x^n e^{-\lambda x^2} dx and gives hints to evaluate for different values of n and compare to the p-th derivative of Io. The solution involves applying the derivative recursively p times and using the even and odd cases to obtain a final answer.
  • #1
csnsc14320
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1

Homework Statement


Solve:

In = [tex] \int_{0}^{\infty} x^n e^{-\lambda x^2} dx [/tex]


Homework Equations





The Attempt at a Solution


So my teacher gave a few hints regarding this. She first said to evaluate when n = 0, then consider the cases when n = even and n = odd, comparing the even cases to the p-th derivative of Io.

For the Io case, I evaluated it and obtained [tex] I_o = \frac{1}{2} \sqrt{\frac{\pi}{\lambda}} [/tex]

Now, for the "p-th" derivative of Io, i got

[tex] \frac{d^p}{d \lambda^2} I_o = \frac{\prod_{p=1}^p (1 - 2p)}{2^{p+1}} \sqrt{\pi} \lambda^{-\frac{(2p + 1)}{2}} [/tex]

I don't see how this related to n = 2p (even case) where

I2p = [tex] \int_0^\infty x^{2p} e^{- \lambda x^2} dx [/tex]

And even when I do figure this out, does this all combine into one answer, or is it kind of like a piecewise answer?

Any help with what to do with the even/odd cases would be greatly appreciated

Thanks
 
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  • #2
Is this what you are asking?

[tex] \frac{\partial}{\partial \lambda} I_0 = \frac{\partial}{\partial \lambda} \int_{0}^{\infty} e^{-\lambda x^2} dx = \int_{0}^{\infty} -x^2 e^{-\lambda x^2} dx = -I_2[/tex]

You can apply this recursively p times to get [tex](-1)^p I_{2p}[/tex]
 
  • #3
nickjer said:
Is this what you are asking?

[tex] \frac{\partial}{\partial \lambda} I_0 = \frac{\partial}{\partial \lambda} \int_{0}^{\infty} e^{-\lambda x^2} dx = \int_{0}^{\infty} -x^2 e^{-\lambda x^2} dx = -I_2[/tex]

You can apply this recursively p times to get [tex](-1)^p I_{2p}[/tex]

Oh yeah I see the pattern if you take the derivative of Io in integral form instead of what it actually is.

I also did it for the odds and got nice cases for both :D

thanks.
 

1. What is a Gaussian integral?

A Gaussian integral is an integral of the form ∫-∞ e-x2 dx. It is named after the mathematician Carl Friedrich Gauss and is used in many areas of mathematics and physics.

2. How do you solve a Gaussian integral?

To solve a Gaussian integral, we use a technique called completing the square. This involves rewriting the integrand as a perfect square, which then allows us to easily evaluate the integral using standard integration techniques.

3. What is the difference between the even and odd cases of a Gaussian integral?

In the even case, the integrand is symmetric about the y-axis, meaning that the function is unchanged when we substitute x with -x. In the odd case, the integrand is antisymmetric about the y-axis, meaning that the function changes sign when we substitute x with -x. This leads to different approaches in solving the integral.

4. When do we use the even or odd case in solving a Gaussian integral?

We use the even case when the integrand is an even function, and the odd case when the integrand is an odd function. This is determined by the symmetry or antisymmetry of the function about the y-axis.

5. What are some applications of Gaussian integrals?

Gaussian integrals have applications in probability theory, statistics, and quantum mechanics. They are also used in solving differential equations and in signal processing. Additionally, Gaussian integrals are used in approximating many other types of integrals, making them a powerful tool in mathematics and science.

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