Problem Set 4 Solution

2 given t he r ecursion f ormula f or t he h ermite p

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Unformatted text preview: n the equation above is shown in the figure below. 6-term expansion of f(x) € 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.5 1.0 1.5 2.0 x The dashed line shows the actual step function on the interval between x=0 and 2. The six-term expansion crudely reproduces the step function. Clearly, many more terms are needed in the expansion in order to reproduce the step function more accurately. 2 . Given t he r ecursion f ormula f or t he H ermite p olynomials, an+2 = 2n + 1 − 2ε a, ( n + 1)( n + 2) n and H 4 (Q) = 16Q 4 − 48Q 2 + 12 , verify that the coefficients of the v = 4 polynomial are correct 4 starting w ith a 4 = 2 = 16. € € Since ε = v + 1, 2 we can write € an+2 = € ( 2n + 1 − 2 v + ( n + 1)( n + 2) 1 2 ) an . For v=4, the recursion formula is € an+2 = 2n − 8 an . n + 1)( n + 2) ( Substituting n=2 into the recursion formula, € −4 a2 3⋅ 4 1 = − a2 , 3 = − 3a 4 . a4 = or a 2 4 Substituting a 4 = 2 = 16, we have € a 2 = − 3a 4 = − 3 ⋅ 16 a 2 = − 48 . € Substituting n=0 into the recursion formula, € −8 a0 2 = − 4a0 , 1 =− a2...
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