Question

Find Laplace transform F(p) for function f(t)=erf(sqrt(t))

Answer 1

$$e ^ {t} \rightarrow \frac {1}{p - 1}$$ $$\frac {1}{\sqrt {\pi t}} \rightarrow \frac {1}{\sqrt {p}}$$

By multiplication theorem:

$$\begin{array}{l} \frac {1}{p - 1} \cdot \frac {1}{\sqrt {p}} \rightarrow \int_ {0} ^ {t} e ^ {t - u} \frac {d u}{\sqrt {\pi u}} = \frac {2 e ^ {t}}{\sqrt {\pi}} \int_ {0} ^ {t} e ^ {- u} d (\sqrt {u}) = \\ = \frac {2 e ^ {t}}{\sqrt {\pi}} \int_ {0} ^ {t} e ^ {- v ^ {2}} d (v) = e ^ {t} \operatorname {erf} (\sqrt {t}) \\ \end{array}$$

By shift theorem:

$$\operatorname {erf} \left(\sqrt {t}\right)\rightarrow \frac {1}{p \sqrt {p + 1}}$$