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Shift rule and partial fractions

The shift rule (s-shift) tells us what happens to the transform when $f(t)$ is multiplied by $e^{at}$ . Partial fraction decomposition is used to write a complicated fraction in $s$ as simpler terms found in the table.

\mathcal L\{e^{at}f(t)\} = F(s-a)

the s-shift rule

\frac{1}{s(s+1)} = \frac1s - \frac{1}{s+1}

partial fractions, example

Symbols

a

shift in s, from the exponential factor

e^{at}

Example

$\mathcal L\{te^{-4t}\} = \dfrac{1}{(s+4)^2}$ , and the inverse of $\dfrac{1}{s(s+1)}$ is $1-e^{-t}$ .

A time shift

f(t-a)

instead gives a factor

e^{-as}

- don't mix up the two rules.

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Part of Laplace, Fourier and PDEs: The Laplace transform.