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

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