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Show that the functions 1 dn w(x) dxn (w(x)(1 — x²)^), Pn(x) = n = 0, 1, 2, ... are orthogonal to every polynomial of degree less than n with respect to the inner product (f,g) = f'¹₁ f(x)g(x)w(x)dx, (w(x) > 0).

Show that the functions 1 dn w(x) dxn (w(x)(1 — x²)^), Pn(x) = n = 0, 1, 2, ... are orthogonal to every polynomial of degree less than n with respect to the inner product (f,g) = f'¹₁ f(x)g(x)w(x)dx, (w(x) > 0).

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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Show that the functions 1 dn w(x) dxn (w(x)(1 — x²)n), Pn(x) = n = 0,1,2,... are orthogonal to every polynomial of degree less than n with respect to the inner product (f,g) = f₁ f(x)g(x)w(x)dx, (w(x) > 0).
Transcribed Image Text:Show that the functions 1 dn w(x) dxn (w(x)(1 — x²)n), Pn(x) = n = 0,1,2,... are orthogonal to every polynomial of degree less than n with respect to the inner product (f,g) = f₁ f(x)g(x)w(x)dx, (w(x) > 0).
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