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1.61 Laplace transforms of trig functions

Example.

Laplace transforms of $\sin$ and $\cos$ . For $f(x)=\sin ax$ and $g(x)=\cos ax$ the Laplace transforms converge for all $s>0$ and are

F(s)={{a}\over{s^{2}+a^{2}}}\qquad{\hbox{and}}\qquad G(s)={{s}\over{s^{2}+a^{2% }}}

respectively.

The first of these statements was proved in frame 1.47-48. The second can be proved in a similar way, or one can use the result for $\sin ax$ and Proposition 6.41. (See frame 6.42.)