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WindowTitle->14. Complex Inversion | Laplace Transforms in Theory and \
Practice
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sufficient\[LongDash]there is still a desire to find a formula for the \
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formula doesn\[CloseCurlyQuote]t guarantee an immediate determination of the \
final answer. For instance, in the Laplace transform formula, the challenging \
part was the integration process. Nevertheless, having a formula means that \
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 "Since both vertical lines are on the left of the dashed line that is in the \
region where ",
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analytic functions. It states that the integral along the closed contour ABED \
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Next, a few examples will be given that describe different scenarios of \
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theorem to evaluate the integral along the Bromwich contour (in blue):"
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Cell["\<\
One cannot simply close the contour to the left by adding a semicircle like \
in the previous examples, because it will cross the branch cut at some point \
and this will hinder us from applying Cauchy\[CloseCurlyQuote]s residue \
theorem (remember that in the theorem it is assumed that the function is \
analytic on the contour). Instead, one can choose the contour as follows:\
\>", "Text",
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Since the contour ABCDEFGHKM doesn\[CloseCurlyQuote]t contain any \
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integral is 0. This allows us to express the Bromwich integral along the \
vertical line as a sum of integrals along DE and FG:\
\>", "Text",
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The complex inversion formula, also known as the \
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Laplace transform to the Fourier transform and employing the inverse Fourier \
transform.\
\>", "SummaryList",
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The inversion formula is used to invert functions with poles, essential \
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\>", "SummaryList",
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Cell["\<\
Close the contour to the left, with all three poles inside it:\
\>", "SolutionAnswer",
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