I need to apply one Laplace transform formula while I have no idea how to prove it: $$\int_0^\infty e^{-st}e^{a k}e^{a^2 t}erfc(a \sqrt{t}+\frac{k}{2 \sqrt{t}})dt=\frac{e^{-k \sqrt{s}}}{\sqrt{s}(\sqrt{s}+a)}, k>0 \& a \in \mathbb{C}, $$ where $erfc(t)=\frac{2}{\sqrt{\pi}}\int_t^\infty e^{-x^2}dx$.

Could anyone help me with it? Thanks in advance.

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