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\def\R{{\bf R}}
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\input graphicx.tex %
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{\includegraphics[scale=0.57]{LaPlaceTable.pdf}}
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To calculate the {\bf inverse LaPlace transform}, you will use
the following  algebra techniques:
\vskip -6pt
\hskip 1.1in partial fractions, 
\us
\hskip 1.1in  completing the square,
\us
\hskip 1.1in  adding 0, and 
\us
\hskip 1.1in multiplying by 1.

\vskip -3pt
\underbar{\bf 1.  Look at denominator}
\vskip -3pt

i.)  Does your denominator equal one of the following?

\centerline{$s^n$,~~ $s - a$,~~ $s^2 \pm a^2$, ~~$(s-a)^{n+1}$}
\vskip -9pt
If so, use the appropriate formula.

ii.)  Can you factor denominator over the reals?  If so, factor and use partial fractions.



iii.)  Do you need to complete the square?
Example:  
\uss
$10s^2 + 60s + 91 = 10(s^2 + 6s) + 91$
\uss
\hskip .99in $= 10(s^2 + 6+ 9 - 9) + 91$
\uss
\hskip .99in $= 10(s^2 + 6+ 9 ) -90 + 91$
\uss
\hskip .99in $ = 10(s+3)^2 +1$

\underbar{\bf 2.  Look at the numerator}
\vskip -3pt
i.)  Do you need $s-a$?  Try adding 0.  For example to make $s + {3 \over 2}$ appear in $5s + 21$:
\vskip 5pt
\centerline{$5s + 21 = 5(s + {3 \over 2}) - {15 \over 2} + 21 = 5(s + {3 \over 2}) + {27 \over 2}$}


ii)  Do you need $b$?  Try multiplying by 1.  For example, if you need ${\sqrt{7 \over 4}} $, but you have 
${27 \over 2}$:
~~~~
${27 \over 2} = {27 \over 2}{\sqrt{4 \over 7}} {\sqrt{7 \over 4}} $




\end
