1. ## Hermite Polynomials

how do I calculate the first three Hermite Polynomials ?

thank you

2. Originally Posted by iLikeMaths
how do I calculate the first three Hermite Polynomials ?

thank you

Hermite polynomials - Wikipedia, the free encyclopedia

http://kiwi.atmos.colostate.edu/grou...rmitePolys.pdf

Hermite Polynomial

Hermite polynomials - ALGLIB

3. thanks for the link, so the polynomials are
• 1
• 2x
• 4x^2 - 2 etc
I know that they are orthogonal but how do I prove/verify that they are?

Thanks

4. Originally Posted by iLikeMaths
thanks for the link, so the polynomials are
• 1
• 2x
• 4x^2 - 2 etc
I know that they are orthogonal but how do I prove/verify that they are?

Thanks
Read the links carefully, do a little more research and see if you can find a proof. If you turn up nothing I'll post a proof in due course (but I'm betting you'll turn up something).

5. ive got to the point where i am integrating
1 and 2x and the weighting function e^x^-2 (sori i dont know how to use LaTex) between the intervals infinity and minus infinity but i dont understand how the integral equals to zero if the intervals do not have a value. i have integrated the polynomials and i got -e^x^-2, hope you understand thanks

6. Originally Posted by iLikeMaths
ive got to the point where i am integrating
1 and 2x and the weighting function e^x^-2 (sori i dont know how to use LaTex) between the intervals infinity and minus infinity but i dont understand how the integral equals to zero if the intervals do not have a value. i have integrated the polynomials and i got -e^x^-2, hope you understand thanks
To solve $\displaystyle \int_{-\infty}^{+\infty} (1) (2x) e^{-x^2} \, dx$ make the substitution $\displaystyle u = -x^2$ and deal with the improper integral in the usual way (have you been taught about improper integrals and how to treat them?).

7. Originally Posted by mr fantastic
To solve $\displaystyle \int_{-\infty}^{+\infty} (1) (2x) e^{-x^2} \, dx$ make the substitution $\displaystyle u = -x^2$ and deal with the improper integral in the usual way (have you been taught about improper integrals and how to treat them?).
Or just note that the integrand $\displaystyle 2xe^{-x^2}$ is an odd function, and hence :
$\displaystyle \int_0^\infty 2xe^{-x^2} ~ dx=-\int_{-\infty}^0 2xe^{-x^2} ~ dx$

Since $\displaystyle \int_{-\infty}^{\infty}=\int_0^\infty+\int_{-\infty}^0$, the integral is 0.

8. i still dont understand , can i post the actual question because i think i am not explaining the question properly

9. Originally Posted by iLikeMaths
i still dont understand , can i post the actual question because i think i am not explaining the question properly
What do you not understand in post #6?

Please feel free to post the actual question.

It looks to me like you will need to revise and consolidate some of the background mathematics that your question expects you to understand and use.

10. [LEFT]The Actual Question

Using the definition of orthogonality, verify that the polynomials found in
1.(i.e. $\displaystyle 1,2x, 4x^2 -2$[FONT=CMR12])are mutually
orthogonal. You may use the following integrals without proof
$\displaystyle \int_{-\infty}^{\infty}e^{-x^2}~dx = \sqrt\pi$

i think my actual problem is that i missed the class on this topic and i am totally confused, the book i am using is showing me a different way to do it, if you could please explain in a very simple way, thanks

11. Originally Posted by iLikeMaths
The Actual Question

Using the definition of orthogonality, verify that the polynomials found in 1.(i.e. $\displaystyle 1,2x, 4x^2 -2$)are mutually
orthogonal. You may use the following integrals without proof
Orthogonal with respect to

1. The weighting function $\displaystyle e^{-x^2}$

2. over the interval $\displaystyle -\infty < x < + \infty$

???

One of the required calculations has already been shown to you.