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Consider the following function: The Laplace Transform of the above function...

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Consider the following function:

$f left parenthesis t right parenthesis equals fraction numerator 1 over denominator square root of 2 end fraction u left parenthesis t minus 1 right parenthesis minus fraction numerator 1 over denominator square root of 3 end fraction u left parenthesis straight t minus 3 right parenthesis plus fraction numerator 1 over denominator square root of 5 end fraction u left parenthesis t minus 5 right parenthesis minus fraction numerator 1 over denominator square root of 7 end fraction u left parenthesis straight t minus 7 right parenthesis plus fraction numerator 1 over denominator square root of 11 end fraction u left parenthesis straight t minus 9 right parenthesis$

The Laplace Transform of the above function F(S) is given by

$fraction numerator 1 over denominator square root of 2 end fraction e to the power of s over s minus fraction numerator 1 over denominator square root of 3 end fraction e to the power of 3 s end exponent over s plus fraction numerator 1 over denominator square root of 5 end fraction e to the power of 5 s end exponent over s minus fraction numerator 1 over denominator square root of 7 end fraction e to the power of 7 s end exponent over s plus fraction numerator 1 over denominator square root of 11 end fraction e to the power of 9 s end exponent over s$

$fraction numerator 1 over denominator square root of 2 end fraction e to the power of negative s end exponent over s minus fraction numerator 1 over denominator square root of 3 end fraction e to the power of negative 3 s end exponent over s plus fraction numerator 1 over denominator square root of 5 end fraction e to the power of negative 5 s end exponent over s minus fraction numerator 1 over denominator square root of 7 end fraction e to the power of negative 7 s end exponent over s plus fraction numerator 1 over denominator square root of 11 end fraction e to the power of negative 9 s end exponent over s$

$fraction numerator 1 over denominator square root of 2 end fraction fraction numerator 1 over denominator s plus 1 end fraction minus fraction numerator 1 over denominator square root of 3 end fraction fraction numerator 1 over denominator s plus 3 end fraction minus fraction numerator 1 over denominator square root of 5 end fraction fraction numerator 1 over denominator s plus 5 end fraction minus fraction numerator 1 over denominator square root of 7 end fraction fraction numerator 1 over denominator s plus 7 end fraction minus fraction numerator 1 over denominator square root of 11 end fraction fraction numerator 1 over denominator s plus 9 end fraction$

$fraction numerator 1 over denominator square root of 2 end fraction e to the power of negative s end exponent over s squared minus fraction numerator 1 over denominator square root of 3 end fraction e to the power of negative 3 s end exponent over s squared plus fraction numerator 1 over denominator square root of 5 end fraction e to the power of negative 5 s end exponent over s squared minus fraction numerator 1 over denominator square root of 7 end fraction e to the power of negative 7 s end exponent over s squared plus fraction numerator 1 over denominator square root of 11 end fraction e to the power of negative 9 s end exponent over s squared$

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