7.4 Notes - Convolution

# 7.4 Notes - Convolution - evaluating the integral...

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Differential Equations Section 7.4 – Convolution Integrals Derivatives of Transforms If { } ) ( ) ( t f s F = and , , 3 , 2 , 1 = n then { } ( 29 ) ( 1 ) ( s F ds d t f t n n n n - = . Example: #6, Find { } t t cos 2 #10) 0 ) 0 ( , sin = = - y t te y y t Convolution of f and g If functions f and g are piecewise continuous on [ 29 , 0 , then a special product, denoted by g f , is defined by the integral ( 29 ( 29 τ d t g f g f t - = 0 and is called the convolution of f and g . Example: Find t e t 2

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The good news is that it is possible to find the Laplace transform of the convolution of two functions without actually
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Unformatted text preview: evaluating the integral. Convolution Theorem If functions f and g are piecewise continuous on [ 29 ∞ , and of exponential order, then { } { } { } ) ( ) ( ) ( ) ( s G s F t g t f g f = ℑ ℑ = ℑ . Example: #28 { } τ d t t ∫-ℑ ) cos( sin #26) { } d t ∫ ℑ sin #38) d t f t t f t ) ( sin 4 2 ) (--= ∫...
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7.4 Notes - Convolution - evaluating the integral...

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