Laplace Transform of $\sqrt{\frac{t}{\pi}}\cos(8t)$

[Econometricia]

1. find the Laplace transform of $$\sqrt{t/pi}$$cos(8t).



2. Tried to look at the tables and combine things but I'm not very sure where to start.

 

[Saladsamurai]

Do you know what $$L\{\sqrt{{t}/{\pi}}\}$$ is? Do you know what $$L\{\cos(8t)\}$$ is?

 

[jackmell]

Do you know what $$L\{\sqrt{{t}/{\pi}}\}$$ is? Do you know what $$L\{\cos(8t)\}$$ is?

I believe that requires evaluating a convoluted contour integral, which interestingly, equates to just it's residues but (rigorously) showing that seems pretty tough. I was just curious if the OP was prepared to do that type of analysis or if there is a more elementary way?

 

[fzero]

I believe that requires evaluating a convoluted contour integral, which interestingly, equates to just it's residues but (rigorously) showing that seems pretty tough. I was just curious if the OP was prepared to do that type of analysis or if there is a more elementary way?

The integral can be expressed in terms of a gamma function.

 

[Econometricia]

Yes, I do know those transforms. I am now trying to express the integral as a gamma function.

 

[Saladsamurai]

Yes, I do know those transforms. I am now trying to express the integral as a gamma function.

Why don't you show what you have done so that we can better assist you?

I believe that requires evaluating a convoluted contour integral, which interestingly, equates to just it's residues but (rigorously) showing that seems pretty tough. I was just curious if the OP was prepared to do that type of analysis or if there is a more elementary way?

The integral can be expressed in terms of a gamma function.

I am finding the Laplace transform of t^(1/2) in tables without use of the gamma function (there is a 'pi' factor in it which suggests it has already been evaluated for us). So it seems that if

$$L\{\sqrt{{t}/{\pi}}\} = f(t)$$

and

$$L\{\cos(8t)\} = g(t)$$Then we need to find (f * g)(t); that is, find the "convolution" of f(t) and g(t).

 

[Econometricia]

Why don't you show what you have done so that we can better assist you?
I am finding the Laplace transform of t^(1/2) in tables without use of the gamma function (there is a 'pi' factor in it which suggests it has already been evaluated for us). So it seems that if

$$L\{\sqrt{{t}/{\pi}}\} = f(t)$$

and

$$L\{\cos(8t)\} = g(t)$$Then we need to find (f * g)(t); that is, find the "convolution" of f(t) and g(t).

$$L\{\sqrt{{t}/{\pi}}\} = \frac{1}{s^(3/2)}$$

and

$$L\{\cos(8t)\} = \frac{s}{s^2 + 8^2}$$

So we are looking for ( 1 / (pi^(1/2)) $$\int (t-v)^(1/2) cos(8v) dv$$ Integrating from O to t