Question

Evaluate the integral $$f(t)=\int_{0}^{\infty} \frac{\sin \omega t}{\omega} d \omega$$ by finding the Laplace transform and changing the order of integration. Express the result for both $t>0$ and $t<0$ in terms of the theta function. (You will need some results from Problem 16.2.)

          Evaluate the integral
$$f(t)=\int_{0}^{\infty} \frac{\sin \omega t}{\omega} d \omega$$
by finding the Laplace transform and changing the order of integration. Express the result for both $t>0$ and $t<0$ in terms of the theta function. (You will need some results from Problem 16.2.)

Added by Ashley R.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Hunza Gilgit

Step 1

The Laplace transform of $f(t)$ is given by $$F(s) = \mathcal{L}\{f(t)\} = \int_0^\infty e^{-st} f(t) dt$$ Substitute the given function $f(t)$: $$F(s) = \int_0^\infty e^{-st} \left(\int_0^\infty \frac{\sin \omega t}{\omega} d\omega\right) dt$$ Show more…

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Evaluate the integral $$f(t)=\int_{0}^{\infty} \frac{\sin \omega t}{\omega} d \omega$$ by finding the Laplace transform and changing the order of integration. Express the result for both $t>0$ and $t<0$ in terms of the theta function. (You will need some results from Problem 16.2.)