Eia eia 2a 2 2ai 2a lim f x x so lim a0 a0

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Unformatted text preview: . (2) g(ω ) = 1 2π +a −a 1 −iωx 1 sin aω e dx = . e−iωa − eiωa = 2a 2π (−2aiω ) 2πaω +∞ (∗) lim f (x) = δ(x), so lim a→0 a→0 −∞ (3) A1 e = 12 32 A2 e = 20 13 1 −1 2 f (x)e−x dx = 1. = = −e, so λ1 = −1. Similarly, = 1 −1 −1 1 2 −2 = 2e, so λ2 = 2. (∗) (A1 + A2 )e = A1 e + A2 e = −e + 2e = e, so (A1 + A2 )10 e = (A1 + A2 )9 e = · · · = e. The corresponding eigenvalue is 1. (4) The characteristic equation D 2 − 1 = 0 has roots ±1. Trying part...
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