Verified answer

Using power series,

L {cos(√t) / √t}

= L {(1/√t) Σ(n = 0 to ∞) (-1)^n (√t)^(2n)/(2n)!}

= L {(1/√t) Σ(n = 0 to ∞) (-1)^n t^n/(2n)!}

= Σ(n = 0 to ∞) (-1)^n L {t^(n - 1/2)} / (2n)!

= Σ(n = 0 to ∞) (-1)^n [Γ((n - 1/2) + 1)/s^((n - 1/2) + 1)] / (2n)!

= Σ(n = 0 to ∞) (-1)^n Γ(n + 1/2) / [s^(n + 1/2) (2n)!]

= Σ(n = 0 to ∞) (-1)^n [(2n)! √(π)/(4^n * n!)] / [s^(n + 1/2) (2n)!]

= √(π/s) Σ(n = 0 to ∞) (-1)^n /(4^n s^n * n!)

= √(π/s) Σ(n = 0 to ∞) (-1/(4s))^n / n!

= √(π/s) e^(-1/(4s)), via power series for exp.

Link:

http://en.wikipedia.org/wiki/Gamma_function#Genera...

I hope this helps!