Talk:Gudermannian function

Inverse gd = arcgd

Since the Gudermannian function is a trig function, it would follow that its inverse is an arc function, therefore $\operatorname {arcgd} (x)={\rm {gd}}^{-1}(x)\,$ should be a valid notation.  ~Kaimbridge~ 13:37, 23 February 2006 (UTC)

This page has multiple problems that need to be fixed. Most seriously, several of the definitions are only true for positive x and fail for negative x. For example, arccos(sech x) = |gd x|, not gd x.

The article on the tractrix no longer mentions the Gudermannian, so there is no reason to link to it. The connection is spelled out at the Wolfram site. It could be added to the article here, since it is rather interesting.

The wikipedia article on inverse hyperbolic functions prefers the name arcosh, not arccosh, and likewise for the other inverse hyperbolics.

gd is not a trig function and its inverse should not be designated by arc for the same reason given in the article above for not calling the inverse hyperbolic functions arcfunctions. An arcfunction should be something whose output is an arc, i.e., a number that could, in reasonable applications, be an angle in radians. In this sense, gd is an arcfunction, not its inverse.

Finally, the word "trigonometric" should be deleted in the first paragraph. The circular functions are another name for the trigonometric functions, and the hyperbolic functions are just so, and not "hyperbolic trigonometric". Hombre1729 (talk) 10:40, 3 April 2009 (UTC)

I've just done a bunch of edits that I think may fix all the problems above. It's not altogether impossible that I missed some points. Michael Hardy (talk) 18:45, 3 April 2009 (UTC)

I'm gratified that there was such a quick response to my "to do" list. I'm a little new at this. Now let me go into more detail.

I went ahead and deleted the word "trigonometric" altogether, since "circular trigonometric" is redundant. The link "circular functions" redirects to "trigonometric functions", but here the words "circular" and "hyperbolic" stand in a nice contrast to each other.

arccos(sech x), arcsec(cosh x), and arccot(csch x) must all be removed from the list of definitions of gd x. But, lest someone come along and add them back in, it may be useful to say, immediately after the definitions, "Some related formulas don't quite work as definitions because, for real x, arccos(sech x) = |gd x| = arcsec(cosh x) and arccot(csch x) = \cases gd x if x > 0 \\ pi + gd x if x < 0 \endcases (See inverse trigonometric functions.)" Note also that arccsc(coth x) fails to give a value when x = 0, but its limit there, at least, agrees with gd 0. Should we point this out or let it go?

Moving on to the definitions of gd inverse, we should change the first line to "The inverse Gudermannian function is defined, on the interval -pi/2 < x < pi/2, by" because ln|sec x + tan x|, among others, is defined more widely than that. Once that change is made, all the absolute value signs may be changed to parentheses or brackets in all the definitions, since the functions contained in them will each be positive.

Is the definition with sec x(1 + sin x) really needed? It is a rather trivial variation on sec x + tan x.

arsinh(tan x) and artanh(sin x) are perfectly good definitions of gd inverse and could be added back in. As you pointed out below that, arcosh(sec x) is not a complete definition, although there I would replace "values of x for which sec x > 0" by "0 le x < pi/2" because the former inequality can be true even when gd inverse is not defined.

Now that the definitions have been addressed, let me reiterate that the article on the tractrix no longer mentions the gudermannian. So, unless we want to go edit that article, the reference to tractrix under "see also" should be deleted. The connection, as I said, can be found on the Wolfram site, but now that I think about it more, there is an error there that we don't want to repeat. So here is what we could say.

"If a tractrix is parametrized by x = t - tanh t and y = sech t then the slope of the tangent line to the curve is given by -csch t. Therefore the angle that the tangent line makes with the y axis is gd t." Is this interesting enough to include here?

Perhaps a more important application that I didn't mention before is that, "pi/2 - gd x is the angle of parallelism function in hyperbolic geometry."

The article on the hyperbolic secant distribution also fails to mention gd, but the connection is pretty obvious. If a probability density function is defined by (1/2)sech(pi x/2), then its cumulative distribution function must be (1/ pi)gd(pi x/2) + 1/2.Hombre1729 (talk) 10:25, 5 April 2009 (UTC)

I'm going to go through what you wrote carefully.

In the mean time: you have a very odd way of using the word "definition", as if any characterization were a definition. "Definition" is one of those words that need to be used with complete precision in mathematics, normally introduced in about 9th grade and seen incessantly all the way through high school and then for the rest of one's career without much refinement beyond what one was taught initially. Michael Hardy (talk) 15:27, 5 April 2009 (UTC)

You make a valid point, although I believe my use of the word "definition" is in reaction to the way the article was already written and not characteristic of my usual way of doing things. This article has multiple definitions of both gd and gd inverse (or, at least, in the first case the phrase is "is defined by" and in the second case it is "is given by") and I don't see any compelling reason to change that. Had things been otherwise, though, we might agree to have one definition for each and then assert that the others can be derived as identities. Is this what you meant? Hombre1729 (talk) 03:44, 6 April 2009 (UTC)

• 
  arccos(sech x)
• 
  arccot(csch x)
• 
  arccsc(coth x)
• 
  arcsec(cosh x)
• 
  arcsin(tanh x)
• 
  arctan(sinh x)

after seeing these graphs of these functions, it is able to be said:

1.arccsc(coth x) , arcsin(tanh x) ,and arctan(sinh x) ,are perfectly good definitions of gd.

2.arccos(sech x) , arcsec(cosh x) aren't ,but in french wikipedia ,these two can be replaced with sgn(x)arccos(sech x) and sgn(x)arcsec(cosh x).

3.arccot(csch x) isn't perfectly good definitions of gd.

Akjz (talk) 17:15, 20 February 2015 (UTC)

• 
  arcosh(sec x)
• 
  arcoth(csc x)
• 
  arcsch(cot x)
• 
  arsech(cos x)
• 
  arsinh(tan x)
• 
  artanh(sin x)

after seeing these graphs of these functions, it is able to be said:

on the interval -pi/2 < x < pi/2

1.arcoth(csc x) , arcsch(cot x) , arsinh(tan x) ,and artanh(sin x) ,are perfectly good definitions of gd inverse.

2.arcosh(sec x) , arsech(cos x) aren't ,but in french wikipedia ,these two can be replaced with sgn(x)arccosh(sec x) and sgn(x)arcsech(cos x).

Akjz (talk) 17:27, 20 February 2015 (UTC)

References

I'm not familiar with how to edit the {reflist} on this page, so I was wondering if someone else could make this edit for me -- the book "Hyperbolic Functions", which is cited in Note 1, was written by George F. Becker and C.E. van Orstrand. (not "Georg", as it says now.) —Preceding unsigned comment added by 121.107.184.243 (talk) 05:32, 15 January 2011 (UTC)

Spring clean

I got rather over enthusiastic after carrying out a simple edit.

New info and improved references is uncontoversial.

I have restored arc for the inverse hyperbolic functions. This is good enough for NIST (see ref number 1). I suspect the six letter nomenclature dates from the days of early Fortran when all names were so restricted.

The statement about the unsuitability of some definitions needs to be reworded and clarified a little without unbalancing the paragraph.

Please check my changes carefully. Most are cosmetic.

Peter Mercator (talk) 18:09, 2 February 2015 (UTC)

Gudermanian not explicitly mentioned in Casimir reference

Good, Michael R. R.; Anderson, Paul R.; Evans, Charles R. (2013). "Time dependence of particle creation from accelerating mirrors". Physical Review D 88 (2): 025023. arXiv:1303.6756. Bibcode:2013PhRvD..88b5023G. doi:10.1103/PhysRevD.88.025023, does not explicitly reference the Gudermanian.  If someone knows where the implicit reference is, could you please clarify the reference?  — Preceding unsigned comment added by Hcarter333 (talk • contribs) 01:06, 20 July 2016 (UTC) 

Clarifying/unifying notation for identities

I wonder if it would make sense to pick 3 variable names for a circular-function argument, a hyperbolic function argument, and the stereographic projection (geometrically, these are twice the areas of a circular sector, twice the area of a hyperbolic sector, and twice the area of a triangle, respectively), and then consistently use those instead of calling the arguments all x. Then it’s easy to prove/demonstrate all of the identities here by (a) simple rational calculations in terms of the stereographic projection, and (b) geometric intersections.

But I don’t know what the right variable names are to pick. Common names for circular-function argument (coincidentally also the arc length of a circle) include θ and φ, and φ seems especially relevant in this context since it is typically used represent latitude or colatitude. In the figures here (which I would be happy to change) In the figures I just added, I used the variable ψ for hyperbolic argument but I’d be happy to swap it for something else, e.g. u (Cayley’s variable name), χ, or η. For the stereographic projection I could imagine using e.g. σ or h for half-tangent. I used a latin letter for contrast because all of the coordinates here in terms of stereographic projection are rational quantities, whereas in terms of circular/hyperbolic argument they are transcendental. But I’m not wedded to those.

Does anyone have any preferences / any pointers to standard symbols? Does using the same symbol for each thing for most of the article make sense? (In a section about complex definitions it could perhaps switch to the letters like z, ζ, w.) –jacobolus (t) 01:34, 29 June 2022 (UTC)

If just for the sake of example we kept the symbols ${\textstyle \psi ,\varphi ,s}$, I am imagining writing the “identities” section along the lines of:

$${\begin{aligned}s&=\tan {\tfrac {1}{2}}\varphi =\tanh {\tfrac {1}{2}}\psi \\[10mu]{\frac {2s}{1+s^{2}}}&=\sin \varphi =\tanh \psi \quad &{\frac {1+s^{2}}{2s}}&=\csc \varphi =\coth \psi \\[10mu]{\frac {1-s^{2}}{1+s^{2}}}&=\cos \varphi =\operatorname {sech} \psi \quad &{\frac {1+s^{2}}{1-s^{2}}}&=\sec \varphi =\cosh \psi \\[10mu]{\frac {2s}{1-s^{2}}}&=\tan \varphi =\sinh \psi \quad &{\frac {1-s^{2}}{2s}}&=\cot \varphi =\operatorname {csch} \psi \\[8mu]\end{aligned}}$$

And then there’s no need for separate identities sections for forward/inverse functions. –jacobolus (t) 01:59, 29 June 2022 (UTC)

Or we could use υ and κ for the greek letters for hyperbola and circle, respectively. –jacobolus (t) 03:19, 29 June 2022 (UTC)

Image error

The second image shows $\tan {\frac {1}{2}}\phi =\tan {\frac {1}{2}}\psi$, missing an h. --2607:FEA8:86DC:80E0:21E1:ADE:6B93:28DF (talk) 19:01, 18 July 2022 (UTC)