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The Care and Feeding of anti-Gudermannians: Spotting Them in the Wild

I mentioned yesterday that the anti-Gudermannian had come up in several articles I’d read, but that the authors hadn’t recognized, or pointed out the anti-Gudermannians lurking in their formulas.  This is a brief set of instructions on how to recognize anti-Gudermannians in their natural state which can be pretty  messy looking.  The first thing you’ll need to know how to spot the multiplicative inverse of something known as the quotient function[5].  It looks like this, 


but may be frequently disguised as


The quotient function is very interesting in its own right and turns up all over in things like electrical transmission line formulas, quantum mechanical transmission and reflection coefficients, and in optically active material formulas.  For complete coverage of the quotient function, see Lindell in AJP[5], (sorry I couldn’t find an open access version, so you’ll have to head to your closest university library).  As cool as it is though, it’s only the start of the Gudermannian.

Next, you’ll want to look for a hyperbolic arctangent.  They often look more like quotient functions than hyperbolic arctangents:


Finally, if x in the above formulas represents the sin of an angle, then you’ve found yourself an anti-Gudermannian[3, 4 p. 14]:

The form shown above is how the anti-Gudermannian appeared in the research I’ve been studying.  In MacColl’s article on relativistic projectiles in gravitational fields, we have:



which shows how to find the angle for the maximum range using an anti-Gudermanian.  In Sarafian[7], it turns up looking very disguised as:


The expression inside the natural log can be massaged into a the proper quotient function.  Rather than do the math here, I’ll just reference another article by Dou and Staples[8] which has an expression for the arc length that is more immediately suggestive:


One last little note.  The angle found by both of these formulas, relativistic range, and classical arc length, (56.46 degrees), is also the maximum angle that a catenoid soap film can make with the wall of its containing cylinder, (imagine a horizontal cylinder between the rings in the picture shown below), before it will become unstable[9].  Catenoid’s are surfaces of rotation constructed with catenaries.



References:
1. Mercator Projection
https://en.wikipedia.org/wiki/Mercator_projection

2.  Hyperbolic functions
https://en.wikipedia.org/wiki/Hyperbolic_function#Inverse_functions_as_logarithms

3.  Gudermannian
https://en.wikipedia.org/wiki/Gudermannian

4.  All about hyperbolic functions
https://archive.org/details/hyperbolicfuncti020206mbp

5.  Quotient functions
http://dx.doi.org/10.1119/1.18845

6.  MacColl on Relativistic Projectiles in +Mathematical Association of America's AMM
http://www.jstor.org/stable/2302436

6.a.  More of interest on MacColl
http://publikationen.ub.uni-frankfurt.de/frontdoor/deliver/index/docId/2678/file/flow.pdf

7.  Sarafian on projectile motion
http://dx.doi.org/10.1119/1.880184

7.a.  More Sarafian on the Angles of Parabolic Trajectories
http://www.mathematica-journal.com/issue/v9i2/contents/MagicAngles/MagicAngles.pdf

8.  Dou and Staples
http://www.jstor.org/stable/2687203

9.  Ito and Sato on Catenoids
http://arxiv.org/pdf/0711.3256v5.pdf

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