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2014-08-17: Feynman's missing method for third-orders?

2014-07-31: CIA spies even on congress

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2014-06-21: Kurtosis, 4th order diffusion, and wave speed

2014-06-20: Random dispersal speeds invasions

2014-05-06: Preservation of information asymetry in Academia

2014-04-16: Dual numbers are really just calculus infinitessimals

2014-04-14: More on fairer markets

2014-03-18: It's a mad mad mad mad prisoner's dilemma

2014-03-05: Integration techniques: Fourier--Laplace Commutation

2014-02-25: Fiber-bundles for root-polishing in two dimensions

2014-02-17: Is life a simulation or a dream?

2014-01-30: PSU should be infosocialist

2014-01-12: The dark house of math

2014-01-11: Inconsistencies hinder pylab adoption

2013-12-24: Cuvier and the birth of extinction

2013-12-17: Risk Resonance

2013-12-15: The cult of the Levy flight

2013-12-09: 2013 Flu Shots at PSU

2013-12-02: Amazon sucker-punches 60 minutes

2013-11-26: Zombies are REAL, Dr. Tyson!

2013-11-22: Crying wolf over synthetic biology?

2013-11-21: Tilting Drake's Equation

2013-11-18: Why $1^\infty != 1$

2013-11-15: Adobe leaks of PSU data + NSA success accounting

2013-11-14: 60 Minutes misreport on Benghazi

2013-11-11: Making fairer trading markets

2013-11-10: L'Hopital's Rule for Multidimensional Systems

2013-11-09: Using infinitessimals in vector calculus

2013-11-08: Functional Calculus

2013-11-03: Elementary mathematical theory of the health poverty trap

2013-11-02: Proof of the area of a circle using elementary methods

Integration techniques: Fourier--Laplace Commutation

Once upon a time, I was studying Laplace transforms as they related to generating functions and other applied-math linear transforms. One of the really fascinating examples that came up was the calculation of \begin{gather} \int_{0}^{\infty} \frac{ e^{-st-\frac{x^2}{2Dt}} }{ \sqrt{2\pi D t} } dt. \end{gather} This transformation comes up when one is working to solve certain examples of the diffusion equation \[ \frac{\partial n}{\partial t} = \frac{D}{2} \frac{\partial^2 n}{\partial x^2} \] What is interesting about this integral is that it provides an example of an unusual integration technique that isn't often taught in calculus classes. It can be solved by using Fourier transforms! Making the problem harder at first, actually eventually makes it easier!

The Fourier transform of a function $f(x)$ is \begin{gather} \mathscr{F}[f] := \int_{-\infty}^{\infty} e^{2 \pi i \omega x} f(x) \; dx \end{gather} The inverse Fourier transform \begin{gather} \mathscr{F}^{-1}[\hat{f}] := \int_{-\infty}^{\infty} e^{-2 \pi i \omega x} \hat{f}(\omega) \; d\omega \end{gather} The closely related Laplace transform \begin{gather} \mathscr{L}[f] := \int_{0}^{\infty} e^{-st} f(t) dt. \end{gather}

If the Laplace and Fourier transforms commute, then \[ \mathscr{L}[f] = \mathscr{F}^{-1}[ \mathscr{L}[ \mathscr{F}[f]]].\] Making this explict, it turns out that we can calculate the Laplace transform AFTER we do the Fourier transform, but not before. We then discover... \begin{gather} \int_{0}^{\infty} \frac{ e^{-st-\frac{x^2}{2Dt}} }{ \sqrt{2\pi D t} } dt = \int_{-\infty}^{\infty} e^{-2 \pi i \omega x} \int_{0}^{\infty} \int_{-\infty}^{\infty} \frac{ e^{-\frac{x^2}{2Dt}-st+2\pi i \omega x} }{\sqrt{2 \pi D t}} dx\; dt \; d\omega \\ = \frac{e^{\sqrt{\frac{2 s x^2}{D}}}}{\sqrt{2Ds}} \end{gather} This Laplace transform can be performed by first calculating the Fourier transform in $x$, then calculating the Laplace transform in $t$, and then inverting the Fourier transform. Residual calculus is helpful when doing the inversion.

This is just an illustrative example. I don't know how often this trick is actually useful. But it's similar in flavor to the old technique of differentiation under the integral sign.