#### Posts

2019-05-16: Glow worms return

2019-04-11: Original memetic sin

2019-01-31: The theory of weight

2018-11-06: Origins of telephone network theory

2018-10-24: Modern thought

2018-09-10: Feeding a controversy

2018-06-11: Glow worm distribution

2018-04-23: Outlawing risk

2017-08-22: A rebuttal on the beauty in applying math

2016-11-02: In search of Theodore von Karman

2016-09-25: Amath Timeline

2016-02-24: Math errors and risk reporting

2016-02-20: Apple VS FBI

2016-02-19: More Zika may be better than less

2016-01-14: Life at the multifurcation

2015-09-28: AI ain't that smart

2015-06-24: MathEpi citation tree

2015-03-31: Too much STEM is bad

2015-03-24: Dawn of the CRISPR age

2015-02-08: Risks and values of microparasite research

2014-11-10: Vaccine mandates and bioethics

2014-10-18: Ebola, travel, president

2014-10-12: Ebola numbers

2014-09-23: More stochastic than?

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

2014-07-31: CIA spies even on congress

2014-07-16: Rehm on vaccines

2014-06-20: Random dispersal speeds invasions

2014-04-14: More on fairer markets

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

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 $1^\infty != 1$

2013-11-14: 60 Minutes misreport on Benghazi

2013-11-09: Using infinitessimals in vector calculus

2013-11-08: Functional Calculus

## 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.