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using convolution to find Laplace transform
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(Example)
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We start from the relations (see the table of Laplace transforms)
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(1) |
where the curved arrows point from the Laplace-transformed functions to the original functions. Setting $\alpha = a^2$ and dividing by $\sqrt{\pi}$ in (1), the convolution property of Laplace transform yields $$\frac{1}{(s\!-\!a^2)\sqrt{s}} \;\;\curvearrowright\;\; e^{a^2t}*\frac{1}{\sqrt{\pi t}} \;=\; \int_0^t\!e^{a^2(t-u)}\frac{1}{\sqrt{\pi u}}\,du.$$ The substitution $a^2u = x^2$ then gives $$\frac{1}{(s\!-\!a^2)\sqrt{s}} \;\curvearrowright\; \frac{e^{a^2t}}{\sqrt{pi}}\int_0^{a\sqrt{t}}\!e^{-x^2}\!\cdot\!\frac{a}{x}\!\cdot\!\frac{2x}{a^2}\,dx
\;=\; \frac{e^{a^2t}}{a}\!\cdot\!\frac{2}{\sqrt{\pi}}\int_0^{a\sqrt{t}}\!e^{-x^2}\,dx \;=\; \frac{e^{a^2t}}{a}\,{\rm erf}\,a\sqrt{t}.$$ Thus we may write the formula
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(2) |
Moreover, we obtain $$\frac{1}{(\sqrt{s}\!+\!a)\sqrt{s}} \;=\; \frac{\sqrt{s}\!-\!a}{(s\!-\!a^2)\sqrt{s}} \;=\, \frac{1}{s-a^2}-\frac{a}{(s-a^2)\sqrt{s}} \;\curvearrowright\; e^{a^2t}-e^{a^2t}\,{\rm erf}\,a\sqrt{t} \;=\; e^{a^2t}(1-{\rm erf}\,a\sqrt{t}),$$ whence we have the other formula
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(3) |
One can utilise the formula (3) for evaluating the improper integral $$\int_0^\infty\frac{e^{-x^2}}{a^2\!+\!x^2}\,dx.$$ We have $$e^{-tx^2} \;\curvearrowleft\; \frac{1}{s\!+\!x^2}$$ (see the table of Laplace transforms). Dividing this by $a^2\!+\!x^2$ and integrating from 0 to $\infty$ , we can continue as follows:
Consequently, $$\int_0^\infty\frac{e^{-tx^2}}{a^2\!+\!x^2}\,dx \;=\; \frac{\pi}{2a}e^{a^2t}\,{\rm erfc}\,a\sqrt{t},$$ and especially $$\int_0^\infty\frac{e^{-x^2}}{a^2\!+\!x^2}\,dx \;=\; \frac{\pi}{2a}e^{a^2}\,{\rm erfc}\,a.$$
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"using convolution to find Laplace transform" is owned by pahio.
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Cross-references: improper integral, convolution property of Laplace transform, functions, table of Laplace transforms
There are 3 references to this entry.
This is version 9 of using convolution to find Laplace transform, born on 2009-01-14, modified 2009-05-05.
Object id is 11504, canonical name is UsingConvolutionToFindLaplaceTransform.
Accessed 896 times total.
Classification:
| AMS MSC: | 44A10 (Integral transforms, operational calculus :: Laplace transform) | | | 26A42 (Real functions :: Functions of one variable :: Integrals of Riemann, Stieltjes and Lebesgue type) |
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Pending Errata and Addenda
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