As mentioned in class on Tucsday, the Legendre polynomials are a set of polynomial functions defined as follows: Po(r) – 1 P (r) – I Pa(x) = (32 – 1) 1 d 24 dr We claimed without proof that the inner product (P(r)|Pm(x)) = L', P(x)Pm(1)dr = 2/(21+ 1)dim, which means that the Legendre Polynomials are orthogonal but not orthonomal. (a) Show through explicit computation that the inner product (P(x)|P2(x)) – 0 and that the norm of P(r) is V2/(2 + 1 + 1). (b) Show through explicit computation that the inner product (P(r)|P;(1)) – 0 and that the norm of P(r) is V2/(2 *5+1). Hint: Use Mathematica or Wolfram Alpha to find P3(r) and P(r).
As mentioned in class on Tucsday, the Legendre polynomials are a set of polynomial functions defined as follows: Po(r) – 1 P (r) – I Pa(x) = (32 – 1) 1 d 24 dr We claimed without proof that the inner product (P(r)|Pm(x)) = L', P(x)Pm(1)dr = 2/(21+ 1)dim, which means that the Legendre Polynomials are orthogonal but not orthonomal. (a) Show through explicit computation that the inner product (P(x)|P2(x)) – 0 and that the norm of P(r) is V2/(2 + 1 + 1). (b) Show through explicit computation that the inner product (P(r)|P;(1)) – 0 and that the norm of P(r) is V2/(2 *5+1). Hint: Use Mathematica or Wolfram Alpha to find P3(r) and P(r).
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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Transcribed Image Text:As mentioned in class on Tucsday, the Legendre polynomials are a set of polynomial
functions defined as follows:
Po(r) = 1
P(r) = 1
1
P2(r) – (3r – 1)
P{(#) = ()
2! da:
d
(교2
We claimed without proof that the inner product (P(r)|Pm(x)) = L', P(x)Pm(1)dr =
2/(21 + 1)dim, which means that the Legendre Polynomials are orthogonal but not
orthonomal.
(a) Show through explicit computation that the inner product (P (r)|P2(x)) = 0 and that the
norm of P(1) is V2/(2 + 1+1).
(b) Show through explicit computation that the inner product (P(r)|P;(1)) = 0 and that the
norm of P3(r) is V2/(2 + 5+1). Hint: Use Mathematica or Wolfram Alpha to find P3(x)
and P;(r).
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