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The first three Hermite polynomials are Ho(x) = 1, H₁(x) polynomials are defined using the inner product (f\g) = **f*(x)g(x)w(x)dx, = 2x, and H₂(x) = 4x² 2. These - with the weight function w(x) = exp(-x²). a) Using the Wronskian, show that Ho(x), H₁(x) and H₂(x) are linearly independent. b) Show that these three Hermite polynomials are mutually orthogonal if we choose the Gaus- sian weight function given above. c) Explain what goes wrong if we instead use a unit weight function, w(x) = 1, in the definition of the norm. You may find the following integral useful: L where n is an integer and (2n-1)!! = 1.3.5 (2n-1) is the double factorial of 2n - 1. .. x²n exp(-x²) dr = (2n-1)!!√√/2", Page 3

The first three Hermite polynomials are Ho(x) = 1, H₁(x) polynomials are defined using the inner product (f\g) = **f*(x)g(x)w(x)dx, = 2x, and H₂(x) = 4x² 2. These - with the weight function w(x) = exp(-x²). a) Using the Wronskian, show that Ho(x), H₁(x) and H₂(x) are linearly independent. b) Show that these three Hermite polynomials are mutually orthogonal if we choose the Gaus- sian weight function given above. c) Explain what goes wrong if we instead use a unit weight function, w(x) = 1, in the definition of the norm. You may find the following integral useful: L where n is an integer and (2n-1)!! = 1.3.5 (2n-1) is the double factorial of 2n - 1. .. x²n exp(-x²) dr = (2n-1)!!√√/2", Page 3

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The first three Hermite polynomials are Ho(x) = 1, H₁(x) = 2x, and H₂(x) = 4x22. These polynomials are defined using the inner product (f\g) = **f*(x)g(x)w(x)dx, with the weight function w(x) = = exp(-x²). a) Using the Wronskian, show that H₁(x), H₁(x) and H₂(x) are linearly independent. b) Show that these three Hermite polynomials are mutually orthogonal if we choose the Gaus- sian weight function given above. c) Explain what goes wrong if we instead use a unit weight function, w(x) = 1, in the definition of the norm. You may find the following integral useful: ∞ exp(-x²)dx= (2n-1)!!√√/2", where n is an integer and (2n − 1)!! = 1.3.5. (2n - 1) is the double factorial of 2n - 1. x2n Page 3 2/9
Transcribed Image Text:The first three Hermite polynomials are Ho(x) = 1, H₁(x) = 2x, and H₂(x) = 4x22. These polynomials are defined using the inner product (f\g) = **f*(x)g(x)w(x)dx, with the weight function w(x) = = exp(-x²). a) Using the Wronskian, show that H₁(x), H₁(x) and H₂(x) are linearly independent. b) Show that these three Hermite polynomials are mutually orthogonal if we choose the Gaus- sian weight function given above. c) Explain what goes wrong if we instead use a unit weight function, w(x) = 1, in the definition of the norm. You may find the following integral useful: ∞ exp(-x²)dx= (2n-1)!!√√/2", where n is an integer and (2n − 1)!! = 1.3.5. (2n - 1) is the double factorial of 2n - 1. x2n Page 3 2/9
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