Undergrad Laplace transformation of nested function

[chester20080]

Hello!
I want a formula (if there exists) to find the Laplace transformation of a nested function; a function within a function
For example what is the LT of θ(f(t)), where θ is the step function? Is there already a formula for such things or should I follow the definition integrating etc..?
I have searched similar tables online but I can't find anything so far..Thank you!

 

[lurflurf]

There is no hope for nested functions in general.
For step functions we have can do it as long as we can find all the steps.
After all a step function is just a sum of delayed constants.
Example f=1 sin x>0 0 sin x<0
$$\int_0^\infty \! f(t)e^{-s t}\,\mathrm{d}t=\sum_{k=0}^\infty (-1)^k \frac{1}{s} e^{-s k \pi}=\frac{1}{s(1+e^{-s \pi})}=\frac{e^{s \pi}}{s(1+e^{s \pi})}$$