I'm trying to find the convolution f*g where f(t)=g(t)=sin(t). I set up the integral and proceed to do integration by parts twice, but it keeps working out to 0=0 or sin(t)=sin(t). How am I supposed to approach it? integral (sin(u)sin(t-u)) du from 0 to t.

Question

I'm trying to find the convolution f*g where f(t)=g(t)=sin(t). I set up the integral and proceed to do integration by parts twice, but it keeps working out to 0=0 or sin(t)=sin(t). How am I supposed to approach it? integral (sin(u)sin(t-u)) du from 0 to t.

Answer 1

If f(t)=g(t)=sin(t)

The convolution would be
∫ sin(u)sin(t-u) du from 0 to t

Use the identity:
sin(x)sin(y)=(1/2)(cos(x-y)-cos(x+y))
where x=u, y=t-u

∫ sin(u)sin(t-u) du from 0 to t
=∫ (1/2)(cos(u-t+u)-cos(u+t-u))du
=∫ (1/2)(cos(2u-t)-cos(t))du
=(1/2)[(1/2)sin(2u-t)-ucos(t)] (from 0 to t)
=(1/2)[(1/2)sin(t-(1/2)sin(-t)-tcos(t)]
=(1/2)(sin(t)- t*cos(t))

Answer 2

I figured it out before I got your answer, and I used the trig identity sin(t-u)=sin(t)cos(u)-cos(t)sin(u). I ended up having to use four or five more trig substitutions before finally getting to the answer you have there. Your substitution is much easier to compute. Too bad I didn't check back here before going through all of that! Thank you!

Answer 3

You're most welcome!

Glad that it helped, and thank you for your feedback.

Answer 4

To find the convolution of the two functions f(t) = g(t) = sin(t), we can use the definition of convolution and evaluate the integral. However, the approach you tried using integration by parts twice is not suitable for this particular case.

Let's go through the correct approach step by step:

1. Start with the definition of convolution:
(f * g)(t) = ∫[0 to t] f(u)g(t-u) du

2. Substitute the given functions:
(sin(t) * sin(t)) = ∫[0 to t] sin(u)sin(t-u) du

3. Expand the product of sin(u)sin(t-u) using the trigonometric identity:
sin(u)sin(t-u) = (1/2) [cos(2u - t) - cos(t)]

4. Substituting back into the convolution integral:
(sin(t) * sin(t)) = (1/2) ∫[0 to t] [cos(2u - t) - cos(t)] du

5. Evaluate each term of the integral one by one:
(sin(t) * sin(t)) = (1/2) ∫[0 to t] cos(2u - t) du - (1/2) ∫[0 to t] cos(t) du

6. Simplify the second term using basic integration:
(sin(t) * sin(t)) = (1/2) ∫[0 to t] cos(2u - t) du - (1/2) [sin(t) - sin(0)]

7. Evaluate the second integral:
(sin(t) * sin(t)) = (1/2) ∫[0 to t] cos(2u - t) du - (1/2) [sin(t) - 0]

8. Simplify further:
(sin(t) * sin(t)) = (1/2) ∫[0 to t] cos(2u - t) du - (1/2) sin(t)

Now, we have reduced the convolution integral to a single integral that needs to be evaluated. Evaluating this integral will yield the convolution value.