MATH 308 Lecture 24

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Lecture Notes

In Boston for ICTCM Conference

Dirac Delta Function

Essentially the derivative of the Heaviside function:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta(x) = \begin{cases}\infty & x=0 \\ 0 & \text{otherwise}\end{cases}}

Integration property:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-\infty}^\infty f(x) \, \delta(x) \, \mathrm{d}x = f(0)} for any interval containing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = 0}

Laplace Transform

${\mathcal {L}}\left\{\delta (t-c)\right\}=\mathrm {e} ^{-s\,c}$