5—Fourier Series 1176 Same as the preceding, show that these functions are orthogonal:
e^2 iπx/L and e−^2 iπx/L, L−^14 (7 +i)x and L+^32 ix
7 With the same scalar product the last two exercises, for what values ofαare the functionsf 1 (x) =
αx−(1−α)(L−^32 x)andf 2 (x) = 2αx+ (1 +α)(L−^32 x)orthogonal? What is the interpretation
of the two roots?
8 Repeat the preceding exercise but use the scalar product
〈
f,g
〉
=
∫ 2 L
L dxf(x)*g(x).
9 Use the scalar product
〈
f,g
〉
=
∫ 1
− 1 dxf(x)*g(x), and show that the Legendre polynomialsP^0 ,P^1 ,
P 2 ,P 3 of Eq. (4.61) are mutually orthogonal.
10 Change the scalar product in the preceding exercise to
〈
f,g
〉
=
∫ 1
0 dxf(x)*g(x)and determine if
the same polynomials are still orthogonal.
11 Verify the examples stated on page 104.