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Unformatted text preview: 10 CHAPTER 2. MATHEMATICAL PREREQUISITES the following multiples of sin( x ) are all normalized: sin( x ) / √ π , (for α = 0), − sin( x ) / √ π , (for α = π ), and i sin( x ) / √ π , (for α = π/ 2). Answer: A multiple of sin( x ) means c sin( x ), where c is some complex constant, so the magnitude is  c sin( x )  = radicalBig ( c sin( x )  c sin( x ) ) = radicalBigg integraldisplay 2 π ( c sin( x )) ∗ ( c sin( x )) d x You can always write c as  c  e i α where α is some real angle, and then you get for the norm:  c sin( x )  = radicalBigg integraldisplay 2 π (  c  e − i α sin( x )) (  c  e i α sin( x )) d x = radicalBigg integraldisplay 2 π  c  2 sin 2 ( x ) d x =  c  √ π So for the multiple to be normalized, the magnitude of c must be  c  = 1 / √ π , but the angle α can be arbitrary. 2.3.7 Solution dotg Question: Show that the functions e 4i πx and e 6i πx are an orthonormal set on the interval ≤ x ≤ 1. Answer: You need to show that both functions are normalized, vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle e 4i πx vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = 1 and vextendsingle vextendsingle...
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 Fall '11
 GARVIN
 Derivative, Sin, Eigenvalue, eigenvector and eigenspace, Eigenfunction, α

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