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6.45 Shift formula

Theorem.

If the Laplace transform of $f(x)$ converges and equals $F(s)$ for all $s>s_{0}$ , then the Laplace transform of $e^{ax}f(x)$ converges for all $s>s_{0}+a$ , and

{\mathcal{L}}(e^{ax}f(x))(s)=F(s-a).

Proof. This follows immediately from taking limits in the equality:

\int_{0}^{R}e^{-sx}.e^{ax}f(x)\,dx=\int_{0}^{R}e^{-(s-a)x}f(x)\,dx.

Note that the third, sixth and seventh rows in the Table from the previous slide are special cases of the Theorem.