Laplace Transform of $\cos \sqrt t$

Question

Please provide me with a solution(with complete steps using power series only)

Answer 1

We have $$ \int_0^{ + \infty } {e^{ - xt} \cos \sqrt t dt} = \int_0^{ + \infty } {e^{ - xt} \left( {\sum\limits_{k = 0}^\infty {( - 1)^k \frac{{t^k }}{{(2k)!}}} } \right)dt} = \sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }}{{(2k)!}}\int_0^{ + \infty } {e^{ - xt} t^k dt} } \\ = \sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }}{{(2k)!}}\frac{{k!}}{{x^{k + 1} }}} = \frac{1}{x}\sum\limits_{k = 0}^\infty {( - 1)^k \frac{{k!}}{{(2k)!}}\frac{1}{{x^k }}} = \frac{{\sqrt \pi }}{x}\sum\limits_{k = 0}^\infty {\frac{1}{{\Gamma \left( {k + \frac{1}{2}} \right)}}\left( { - \frac{1}{{4x}}} \right)^k } \\ = \frac{1}{x} - \frac{{\sqrt \pi }}{{2x\sqrt x }}\exp \left( { - \frac{1}{{4x}}} \right)\operatorname{erfi}\left( {\frac{1}{{2\sqrt x }}} \right). $$ Here $\operatorname{erfi}$ denotes the imaginary error function. It is an entire function.