June 21, 2024

Compute Integrals by Taking Derivatives!

When I first learned this technique in undergraduate Quantum Mechanics at the University of Minnesota — Minneapolis I was truly blown away.  Richard Feynman references this technique in his memoirs.  I am pretty sure it was in Surely You’re Joking Mr. Feynman!  When I have taught this technique to my physics students at SUNY Geneseo nd The College at Brockport, well they have shown impressive enthusiasm for the technique as well.  It is a very cool technique, and it applies not only to Gamma and related functions that appear in Quantum Mechanics.  It is also very useful for computing otherwise nasty integrals in four-dimensional space-time that appear in loop-level Feynman integrals in Quantum Field Theory.  Whether or not there is an immediate application, it is an extremely cool technique to explore!  Read more in the full article.

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Why Lissajous Figures?

Lissajous figures may be drawn by moving some type of marking tool back and forth in one direction with one frequency while simultaneously moving it in a perpendicular direction at a second frequency.  The resulting curve will be closed when the frequencies are in an integer ratio.  The results can be somewhat amusing, like a child’s Spirograph toy.  At the same time, these curves are rich with mathematical content interesting to mathematicians, physicists and engineers.  For a more in depth discussion visit my Lissajous Figures article.  The curve you see plotted here was generated by a Matlab program with a Graphical User Interface which is available for free download to users of Matlab.  The program is useful for investigating the properties of Lissajous curves and their derivatives.  The discussion is rich with the periodic functions sine and cosine with various parametrizations.  The curves are parametric curves, and the derivative of a Lissajous curve is another Lissajous curve.  Jules Antoine Lissajous was inspired by these curves to create a mechanical device for projecting them on a screen, and others improved upon the concept in the form of the harmonograph.

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