Unformatted text preview: y ( x )] 2 are both solutions of y 00 + p ( x ) y +2 y = 0. Find y ( x ) and the values of p ( x ) for which this is possible. 6. We will show that Z ∞-∞ δ ( x-a ) f ( x ) dx = f ( a ) . Let us assume that f has a Taylor series expansion about the point x = a . Consider two continuous sequences that approach the δ function: δ ± = 1 √ π± e-( x-a ) 2 /± δ ± = ± 1 / (2 ± ) | x-a | < ± | x-a | ≥ ± . In both cases, if you compute Z ∞-∞ δ ± ( x-a ) f ( x ) dx for ± 6 = 0 the result is not f ( a ) but instead f ( a ) with some corrections. Estimate the ﬁrst nonvanishing correction for each form of δ ± ....
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This note was uploaded on 03/16/2010 for the course MAE 294b taught by Professor Young,w during the Fall '08 term at UCSD.
- Fall '08