Home page for accesible maths 6 Chapter 6 contents 6.42 Laplace transform of the derivative 6.44 Table of Laplace transforms

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6.43 Laplace transform of $\cos ax$ .

Prop. 6.42 can sometimes provide an easy way to calculate the Laplace transform of a function.

Example.

To deduce the Laplace transform of $\cos ax$ from the Laplace transform of $\sin ax$ .

Solution. Setting $f(x)=\sin ax$ , we have $f^{\prime}(x)={a\cos ax}$ in the previous theorem. Thus, for $s>a$ the Laplace transform of $a\cos ax$ equals

{-f(0)+s\frac{a}{s^{2}+a^{2}}}\,{=\frac{as}{s^{2}+a^{2}}.}

It follows that:

s>a

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6.43 Laplace transform of cos⁡a⁢x\cos ax.

Example.

6.43 Laplace transform of $\cos ax$ .