Discussion
Laplace transform of \(J_{0}(a\, t)\) where \(J_{0}\) denotes the Bessel function of zero order is :
Answer: B
The Laplace transform is \(\dfrac{1}{\sqrt{s^{2} + a^{2}}}\).
Using the series \(J_{0}(at) = \sum (-1)^{k}(at/2)^{2k}/(k!)^{2}\) and \(L\{t^{2k}\} = (2k)!/s^{2k+1}\), the transform sums to
\[ \begin{aligned} &\frac{1}{s}\left(1 + \frac{a^{2}}{s^{2}}\right)^{-1/2} \\ &= \frac{1}{\sqrt{s^{2} + a^{2}}} \end{aligned} \]
by the binomial series. \(1/(s^{2} + a^{2})\) is (up to a factor \(a\)) the transform of \(\sin at\), and the forms with \(s^{2} - a^{2}\) belong to the modified Bessel function \(I_{0}(at)\) and to \(\sinh at\).
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