Laplace Transform of the product of two functions

Question

What is the Laplace Transform of the product of two functions? Specifically, my function is $$\sin(5t) \cdot \cos(5t)$$, but I'd like to know a general principle if it's available. What's the easy way to compute this with a general formula without integrating?

Answer 1

If you have $$\sin(a)\times\cos(a)$$ then it is really as $$\frac{1}2\sin(2a)$$ ans so you can use the certain formula. But if you have $\sin(a)*\cos(a)$, there will be another different story. In fact $f*g$ means the convolution of $f$ and $g$ and so needs another approach. See here.

Answer 2

$\sin(5t) \cos(5t) = \sin(10t)/2$ You can take the transform of the above.

There is no general straight forward rule to finding the Laplace transform of a product of two functions. The best strategy is to keep the general Laplace Transforms close at hand and try to convert a given function to a linear combination of those forms.

Methods like partial fractions, writing sine, cosine as exponents .. etc, help.

Answer 3

For two functions $f$ and $g$ that share the same abscissae of convergence (i.e., the Laplace transform of $f$ is defined for $\Re(p)\in (a_1, a_2)$ for some $a_1,a_2\in\mathbb R$ and that of $g$ for $\Re(p)\in (b_1, b_2)$ for some $b_1, b_2\in\mathbb R$), the Laplace transform of their product is defined for $\Re(p)\in I= (\sup\{a_1,b_1\},\inf\{a_2,b_2\})$ and is the result of the convolution of their Laplace transforms as $(F\ast G)(p) = \int_{D_a}F(z)G(p-z)\mathrm d z$ where $a\in I$ and $D_a=]a-\imath\infty,\,a+\imath\infty[$ is a Bromwich contour.

An integral is needed to compute the convolution, though. If this is to be avoided, one must use (less general in terms of Laplace transforms) trigonometric identities and linearity of the transform, then identify standard Laplace transforms within tables.