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laplace_transform [2016/03/08 16:19] nikolaj |
laplace_transform [2016/03/08 16:22] nikolaj |
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| === Discussion === | === Discussion === | ||
| == Action on monomials == | == Action on monomials == | ||
| - | With $u=E/T\implies dE=Tdu$ and the definition of the [[Gamma function]], we find | + | In this article we use $\beta=\frac{1}{T}$. |
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| + | >todo: clean this $\beta$ vs. $T$ Thing up. Maybe drop $T$ altogether. | ||
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| + | With $u=\beta\,E=E/T\implies dE=Tdu$ and the definition of the [[Gamma function]], we find | ||
| $\int_0^\infty \left(\frac{1}{n!}E^n\right){\mathrm e}^{-E/T}dE = T^{n+1} \frac{1}{n!} \int_0^\infty u^{(n+1)-1}{\mathrm e}^{-u}du = T^{n+1},$ | $\int_0^\infty \left(\frac{1}{n!}E^n\right){\mathrm e}^{-E/T}dE = T^{n+1} \frac{1}{n!} \int_0^\infty u^{(n+1)-1}{\mathrm e}^{-u}du = T^{n+1},$ | ||
