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M334Lec28 - Math 334 Lecture#28 6.6 Convolution Question If...

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Math 334 Lecture #28 § 6.6: Convolution Question. If F ( s ) = L{ f ( t ) } and G ( s ) = L{ g ( t ) } , does L{ f ( t ) g ( t ) } = F ( s ) G ( s )? [This asks if pointwise function multiplication in our world corresponds to pointwise function multiplication in the Laplace world.] Answer. Well, F ( s ) G ( s ) = lim A →∞ A 0 e - f ( ξ ) lim B →∞ B 0 e - g ( τ ) = lim A →∞ lim B →∞ A 0 e - f ( ξ ) B 0 e - g ( τ ) = lim A →∞ lim B →∞ B 0 e - g ( τ ) A 0 e - f ( t ) = lim A →∞ lim B →∞ B 0 A 0 e - g ( τ ) e - f ( ξ ) dξdτ = lim A →∞ lim B →∞ B 0 A 0 e - s ( ξ + τ ) f ( ξ ) g ( τ ) dξdτ. For fixed τ , make the change of variable ξ = t - τ in the inner integral: B 0 A 0 e - s ( ξ + τ ) f ( ξ ) g ( τ ) dξdτ = B 0 A + τ τ e - st f ( t - τ ) g ( τ ) dtdτ. The region of integration in this double integral is a type τ region: it is the parallelogram in the -plane bounded by the lines τ = 0 , τ = B, t = τ, t = A + τ. [Sketch this region.] As a type t region, it is the region in the -plane bounded by the curves t = 0 , t = A + B, τ = bottom( t ) = 0 if 0 t A, t - A if A t A + B, τ = top( t ) = t if 0
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