Find the Laplace transform:$f(t) = \int_0^t {{e^{it}}\frac{{dt}}{{\sqrt {2\pi t} }}}$

Question

Find the Laplace transform of the function: $$\begin{array}{l} f(t) = \int_0^t {{e^{it}}\frac{{dt}}{{\sqrt {2\pi t} }}} \\ {\rm{Us}}e:\Gamma \left( {\frac{1}{2}} \right) = \sqrt \pi \\ \end{array}$$

Answer 1

$\mathcal{L}_{t\to s}\left\{\int_0^te^{it}\dfrac{dt}{\sqrt{2\pi t}}\right\}=\dfrac{\mathcal{L}_{t\to s}\left\{\dfrac{e^{it}}{\sqrt{2\pi t}}\right\}}{s}=\dfrac{\mathcal{L}_{t\to s-i}\left\{\dfrac{1}{\sqrt{2\pi t}}\right\}}{s}=\dfrac{1}{s\sqrt{2(s-i)}}$