How Do I Resolve Cyclical Issues in Integration by Parts for Laplace Transforms?

[Dustinsfl]

Last two inner product questions.

The first one I am little confused on and the second one I don't know what to do.

See worked attached.

 

[vela]

What are you confused about?

 

[Dustinsfl]

I don't know how to do it so I am confused on what to do.

 

[vela]

If two vectors are orthogonal, what does that mean about their inner product? And what does it mean to say a vector is a unit vector?

 

[Mark44]

Last two inner product questions.

The first one I am little confused on and the second one I don't know what to do.

See worked attached.

For the first one, show that <cos(mx), sin(nx)> = 0. Also show that <cos(mx), cos(mx)> = 1 and that <sin(nx), sin(nx)> = 1.

For the second one, you're showing that the "1 norm" is a norm on Rⁿ. Verify that the formula satisfies all the defining properties of a norm: positive definiteness, etc. BTW, this norm simply adds the absolute values of the coordinates of a vector. For example, in R³, ||<2, -1, 3>||₁ = |2| + |-1| + |3| = 6.

 

[Dustinsfl]

How can I integrate cos(mx)*sin(nx) when they don't have the same angle?

 

[vela]

Use a trig identity to rewrite the product of a sine and a cosine.

 

[Dustinsfl]

For the second problem, I don't know how to start the proof.

 

[Mark44]

Integration by parts twice would probably work, or you could look in a table of integrals. Since this isn't about learning to integrate, but is instead and application of integration, it seems reasonable to me to look it up in a table of integrals.

 

[Dustinsfl]

On the integration by parts comment, I have an unrelated issue. I was doing a Laplace Transform of e^(-st+t)*sin(t) but after doing integration by parts, I had a form of the original with a 1/(s-1) time the integral so I can't subtract across to the other side. What can I do there since if I keep going it will an endless cyclical cycle?

 

[Dustinsfl]

I have attached a update that show cos(mx) and sin(nx) are orthogonal; however, showing cos(mx) cos(mx) equals 1 isn't quite working.

 

[vela]

It didn't say it explicitly in the problem statement, but you need to assume m and n are integers. What is sine evaluated at any multiple of 2π?

 

[Dustinsfl]

If they are integers, then of course sin goes to zero.

 

[Mark44]

On the integration by parts comment, I have an unrelated issue. I was doing a Laplace Transform of e^(-st+t)*sin(t) but after doing integration by parts, I had a form of the original with a 1/(s-1) time the integral so I can't subtract across to the other side. What can I do there since if I keep going it will an endless cyclical cycle?

Can you elaborate on this a bit more? You haven't said quite enough so that I'm not sure I understand what you're talking about. If you have a given integral on one side, and 1/(s - 1) times the same integral on the other side, add -1/(s - 1) times the integral to both sides, and then combine the two integrals using the rules of plain old fractions. At that point you can solve for the integral algebraically.

If that's not what you're talking about, set me straight, please.