MHB Hermite Function: Writing x^2r in Polynomial Form

  • Thread starter Thread starter Another1
  • Start date Start date
  • Tags Tags
    Function
Click For Summary
To express x^{2r} in Hermite polynomial form, one can utilize the orthogonality of Hermite polynomials. The coefficients a_n can be calculated using the integral a_n = ∫_{-∞}^{∞} x^{2r} (H_n(x))^2 e^{-x^2} dx. This method allows for the determination of the series term by term. For a more general expression, employing the generating function is recommended to derive the (H_n(x))^2 term. The discussion emphasizes the importance of these mathematical tools in achieving the desired polynomial representation.
Another1
Messages
39
Reaction score
0
How can $$x^{2r}$$ be written in hermite polynomial form?
 
Mathematics news on Phys.org
Another said:
How can $$x^{2r}$$ be written in hermite polynomial form?
The Hermite polynomials are an orthogonal set so if you are looking for [math]x^{2r} = a_0 H_0(x) + a_1 H_1 (x) + \text{ ...}[/math], then
[math]a_n = \int_{-\infty}^{\infty} x^{2r} \left ( H_n (x) \right ) ^2 e^{-x^2} ~ dx[/math]

That will get you the series term by term. If you want a more general expression you'll likely have to use the generating function to get the [math]\left ( H_n (x) \right )^2[/math] expression.

-Dan
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
View full post »

Similar threads

Undergrad Does Transforming Hermite Polynomials Affect Their Orthogonality?
  • · Replies 2 ·
Replies
2
Views
2K
MHB Proving $x-1$ is a Factor of $P(x)$ with Polynomials
  • · Replies 1 ·
Replies
1
Views
1K
Hermite Polynomials: What Are the Initial Values?
  • · Replies 4 ·
Replies
4
Views
1K
MHB How are the Hermite Polynomials Defined and Calculated?
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad What type is a polynomial function?
  • · Replies 7 ·
Replies
7
Views
2K
MHB Understanding Gauss Quadrature for Approximating Integrals
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Proof for interpolating polynomial and error approximation
  • · Replies 6 ·
Replies
6
Views
2K
MHB Find Cubic Function & Polynomial of Degree 3
  • · Replies 5 ·
Replies
5
Views
1K
Undergrad A doubt about the multiplicity of polynomials in two variables
  • · Replies 8 ·
Replies
8
Views
3K
MHB Is this Trigonometric Expression a Constant Function of x?
  • · Replies 2 ·
Replies
2
Views
1K