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**Unformatted text preview: **27 Convolution “Convolution” is an operation involving two functions that turns out to be rather useful in many applications. We have two reasons for introducing it here. First of all, convolution will give us a way to deal with inverse transforms of fairly arbitrary products of functions. Secondly, it will be a major element in some relatively simple formulas for solving a number of differential equations. Let us start with just seeing what “convolution” is. After that, we’ll discuss using it with the Laplace transform and in solving differential equations. 27.1 Convolution, the Basics Definition and Notation Let f ( t ) and g ( t ) be two functions. The convolution of f and g , denoted by f ∗ g , is the function on t ≥ 0 given by f ∗ g ( t ) = integraldisplay t x = f ( x ) g ( t − x ) dx . ! ◮ Example 27.1: Let f ( t ) = e 3 t and g ( t ) = e 7 t . Since we will use f ( x ) and g ( t − x ) in computing the convolution, let us note that f ( x ) = e 3 x and g ( t − x ) = e 7 ( t − x ) . So, f ∗ g ( t ) = integraldisplay t x = f ( x ) g ( t − x ) dx = integraldisplay t x = e 3 x e 7 ( t − x ) dx = integraldisplay t x = e 3 x e 7 t e − 7 x dx 533 534 Convolution and Laplace Transforms = e 7 t integraldisplay t x = e − 4 x dx = e 7 t · − 1 4 e − 4 x vextendsingle vextendsingle vextendsingle t x = = − 1 4 e 7 t e − 4 t − − 1 4 e 7 t e − 4 · = − 1 4 e 3 t + 1 4 e 7 t . Simplifying this slightly, we have f ∗ g ( t ) = 1 4 bracketleftbig e 7 t − e 3 t bracketrightbig when f ( t ) = e 3 t and g ( t ) = e 7 t . It is common practice to also denote the convolution f ∗ g ( t ) by f ( t ) ∗ g ( t ) where, here, f ( t ) and g ( t ) denote the formulas for f and g . Thus, instead of writing f ∗ g ( t ) = 1 4 bracketleftbig e 7 t − e 3 t bracketrightbig when f ( t ) = e 3 t and g ( t ) = e 7 t , we may just write e 3 t ∗ e 7 t = 1 4 bracketleftbig e 7 t − e 3 t bracketrightbig . This simplifies notation a little, but be careful — t is being used for two different things in this equation: On the left side, t is used to describe f and g ; on the right side, t is the variable in the formula for the convolution. By convention, if we assign t a value, say, t = 2 , then we are setting t = 2 in the final formula for the convolution. That is, e 3 t ∗ e 7 t with t = 2 means compute the convolution and replace the t in the resulting formula with 2 , which, by the above computations, is 1 4 bracketleftbig e 7 · 2 − e 3 · 2 bracketrightbig = 1 4 bracketleftbig e 14 − e 6 bracketrightbig . It does NOT mean to compute e 3 · 2 ∗ e 7 · 2 , which would give you a completely different result, namely, e 6 ∗ e 14 = integraldisplay t x = e 6 e 14 dt = e 20 t ....

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