Convolution

The convolution product of f and g, whose transform is simply F(s)G(s), giving a solution formula for any admissible forcing function.

The ODE Project, Convolution

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9 exercises

Exercise 1

Calculate the convolution product of $f(t)$ and $g(t)$ in Convolution of Two Functions using the definition of the convolution of two functions.

$f(t) = 2$ and $g(t) = t$

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Exercise 7

Calculate the inverse Laplace transform of $F(s)$ in Convolution and the Inverse Transform using the convolution product.

$F(s) = \dfrac{1}{(s - 1)(s - 3)}$

2.5

Exercise 8

Calculate the inverse Laplace transform of $F(s)$ in Convolution and the Inverse Transform using the convolution product.

$F(s) = \dfrac{3}{s(s - 1)}$

2.5

Exercise 9

Calculate the inverse Laplace transform of $F(s)$ in Convolution and the Inverse Transform using the convolution product.

$F(s) = \dfrac{2}{s^2(s^2 + 2)}$

2.5

Exercise 10

Calculate the inverse Laplace transform of $F(s)$ in Convolution and the Inverse Transform using the convolution product.

$F(s) = \dfrac{1}{s^2 + 4s - 12}$

2.5

Exercise 17

Prove that the commutative property holds for the convolution product. That is, show that $f*g = g*f$ .

2.5

Exercise 18

Prove that the convolution product of two functions $f$ and $g$ distributes multiplication over addition. That is, show that $f*(g + h) = f*g + f*h$ for functions $f$ , $g$ , and $h$ .

2.5

Exercise 19

Prove that the convolution product is associative. In other words, show that $f*(g*h) = (f*g)*h$ for functions $f$ , $g$ , and $h$ .

2.5

Exercise 20

Prove that $0 *f = f*0 = 0$ .

2.5