M334Lec28 - 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

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Unformatted text preview: 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 →∞ Z A e- sξ f ( ξ ) dξ lim B →∞ Z B e- sτ g ( τ ) dτ = lim A →∞ lim B →∞ Z A e- sξ f ( ξ ) dξ Z B e- sτ g ( τ ) dτ = lim A →∞ lim B →∞ Z B e- sτ g ( τ ) Z A e- sξ f ( t ) dξ dτ = lim A →∞ lim B →∞ Z B Z A e- sτ g ( τ ) e- sξ f ( ξ ) dξdτ = lim A →∞ lim B →∞ Z B Z A e- s ( ξ + τ ) f ( ξ ) g ( τ ) dξdτ. For fixed τ , make the change of variable ξ = t- τ in the inner integral: Z B Z A e- s ( ξ + τ ) f ( ξ ) g ( τ ) dξdτ = Z B Z 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 tτ-plane bounded by the lines...
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This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.

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M334Lec28 - 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

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