The Gudermannian function
Dave L. Renfro
11 September 2009 in another group, is intended
in the same spirit as the post I made yesterday,
which is at
http://groups.google.com/group/sci.math/msg/1472c60d9d643639
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[URL for archived version of the original post]
http://mathforum.org/kb/message.jspa?messageID=6841108
On Sept. 2 Jack Bross wrote (in part):
http://mathforum.org/kb/message.jspa?messageID=6832566
> If anyone has a way of deriving this [antiderivative of
> secant] that DOESN'T seem like pulling a rabbit out of
> a hat, I'd be interested to hear it.
On Sept. 9 Don Katze wrote:
http://mathforum.org/kb/message.jspa?messageID=6838361
> Can anyone give me the basis for the definition
> of sinh x to be (e^x - e^-x) / 2 ?
I'm surprised that no one has yet put these two
together and mentioned the Gudermannian function,
especially since a lot of the follow-up posts for
the secant antiderivative question dealt with the
relevance of this material in the presence of today's
computer algebra systems. I may write more about my
thoughts another time, but for now I'll just say that
some of the proponents of skipping all the calculations
and just using a computer algebra system seem to me
to be too concerned with "getting the answer to some
specific problem" at the expense of how the problem
and the techniques for solving it are related to
other problems and other techniques. I'm not suggesting
that this particular antiderivative is all that important
for calculus students to know [1], or even something that
calculus students should see derived. But what I do
think is important is to not project an attitude
towards students that math is all about getting the
answer to something (any more than literature is
all about being able to read or history is all about
dates).
[1] In my case, I often used it as a "gateway type
problem" on short differentiation quizzes, and I
sometimes just gave the formula on integration tests
(or some other "lesser known" antiderivative formula
picked more or less at random from my CRC table of
integrals) and asked them to use it to find the
antiderivative of something like 5x^2 * sec(2x^3).
Anyway, back to Don Katze's question. I assume he
was asking about the hyperbolic-circular function
"sector area" relationship, which many books include
(sometimes sketched out in a problem).
There's a nice discussion of this in Sections 267-269
(pp. 307-310) of Hobson's "A Treatise on Plane
Trigonometry", which is on-line at
http://books.google.com/books?id=_ktLAAAAMAAJ
Hobson uses u and theta for the hyperbolic and trig.
parameters involved in this way of relating the hyperbolic
functions to the trig. functions. As one might imagine,
for something so well known and entrenched in the
curriculum, there must be other connections as well,
and there are, such as cosh(ix) = cos(x) and
sinh(ix) = i*sin(x) [these are immediate from Euler's
identity for e^(ix)], but let's stay with the sector
area version for now. Hobson continues, in Section 270
on p. 310, and studies the relation between the
hyperbolic parameter 'u' and the trig. parameter 'theta',
which is
tan(theta) = sinh(u)
If we solve for theta, we get what is called the
Gudermannian function. (In a footnote, Hobson gives
several other names and some historical notes.)
At this point, let's dispense with 'theta' and 'u'
and just use 'x' and 'y'.
The Gudermannian function, denoted by gd(x), is defined by
gd(x) = arctan(sinh x).
Curiously, this is also equal to arcsin(tanh x), and
I've sometimes asked students to prove this equality
on take-home assignments.
The Gudermannian function shows up in a lot of diverse
ways in mathematics and the physical sciences. Without
trying all that hard, here are some things I googled up
that are on-line:
http://mathworld.wolfram.com/Gudermannian.html
http://en.wikipedia.org/wiki/Gudermannian_function
See p. 2 of "A New Formula Describing the Scaffold
Structure of Spiral Galaxies" by Harry I. Ringermacher
and Lawrence R. Mead
http://arxiv.org/ftp/arxiv/papers/0908/0908.0892.pdf
http://en.wikipedia.org/wiki/Angle_of_parallelism
The Gudermannian and hyperbolic functions show up in
special relativity:
http://hubpages.com/hub/The-x-y-Minkowski-diagram
The Gudermannian makes an appearance in a paper in an
aerospace engineering journal:
https://netfiles.uiuc.edu/prussing/www/AIAA-Prussing-alpha-beta.pdf
See p. 7 of Explicit Formulas for Bernoulli and Euler
Numbers by David C. Vella
http://www.emis.de/journals/INTEGERS/papers/i1/i1.pdf [see p. 7]
The next two documents give lots of formulas involving
trig. functions, hyperbolic functions, and the
Gudermannian function.
http://gottschalksgestalts.org/pdf/GG88.pdf
http://www.kevincarmody.com/math/beautrig.pdf
The next paper isn't freely available on-line,
but I'm including it because it gives some geometrical
connections with the Gudermannian function and it
isn't mentioned in most of the other places I've cited.
J. M. H. Peters, "The Gudermannian", Mathematical Gazette
68 #445 (October 1984), 192-196.
O-K, so what's this got to do with the antiderivative
of the secant?
Let's see if we can get a formula for the
function-inverse of
gd(x) = arctan(sinh x).
To do this, we need to solve for y in
x = arctan(sinh y).
Taking the tangent of both sides gives
tan(x) = sinh(y) = [e^y - e^(-y)]/2
Multiplying both sides by 2e^y gives
(2e^y)(tan x) = e^(2y) - 1
or
e^(2y) - (2e^y)(tan x) - 1 = 0.
This is quadratic in e^y, so we get
e^y = {2(tan x) +/- sqrt[4(tan x)^2 + 4]} / 2
e^y = (tan x) +/- sqrt[(tan x)^2 + 1]
e^y = (tan x) +/- sqrt[(sec x)^2]
e^y = (tan x) +/- |sec x|
Since x is the arctangent of something (recall from
above that x = arctan(sinh y)), it follows that x
lies between -pi/2 and pi/2. Thus, the cosine of x
is positive, hence the secant of x is positive,
and thus |sec x| = sec(x), which gives
e^y = tan(x) +/- sec(x)
Also, since tan(x) < sec(x) for x between -pi/2
and pi/2, and e^y can't be negative, the solution
with the negative sign is extraneous. Thus,
e^y = tan(x) + sec(x)
or
y = ln[tan(x) + sec(x)]
Whoo-hoo, lookie at what we got -- the antiderivative
of secant!
Since one of the ways to get the antiderivative of secant
is to use the u = tan(x/2) substitution (which I mentioned
in [2]), it should come as no surprise that there are some
connections between this transformation and the Gudermannian
function:
http://en.wikipedia.org/wiki/Tangent_half-angle_formula
[2] http://mathforum.org/kb/message.jspa?messageID=6832887
Also, since the function-inverse of the Gudermannian
function is the antiderivative of secant, what could
be more satisfying than to have the Gudermannian
function be the antiderivative of hyperbolic secant?
gd(x) = integral from t=0 to t=x of sech(t)
I'll end with some papers related to these issues.
Note that the last one is freely available on-line.
V. Frederick Rickey and Philip M. Tuchinsky, "An application
of geography to mathematics: History of the integral of
the secant", Mathematics Magazine 53 #3 (May 1980) 162?166.
Janet Heine Barnett, "Enter, stage center: The early drama
of the hyperbolic functions", Mathematics Magazine 77 #1
(February 2004), 15-30.
Stephen Fulling, "How to avoid the inverse secant (and even
the secant itself)", College Mathematics Journal 36 #5
(November 2005), 381-387.
http://www.math.tamu.edu/~fulling/secant_reprint.pdf
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Dave L. Renfro
Gerry Myerson
<d1883595-351b-4f74...@o36g2000vbl.googlegroups.com>,
"Dave L. Renfro" <renf...@cmich.edu> wrote:
> This post, whose main body is a post I made on
> 11 September 2009 in another group, is intended
> in the same spirit as the post I made yesterday,
> which is at
>
> http://groups.google.com/group/sci.math/msg/1472c60d9d643639
>
> ***********************************************************
> ***********************************************************
>
> [URL for archived version of the original post]
> http://mathforum.org/kb/message.jspa?messageID=6841108
>
> On Sept. 2 Jack Bross wrote (in part):
>
> http://mathforum.org/kb/message.jspa?messageID=6832566
>
> > If anyone has a way of deriving this [antiderivative of
> > secant] that DOESN'T seem like pulling a rabbit out of
> > a hat, I'd be interested to hear it.
>
> On Sept. 9 Don Katze wrote:
>
> http://mathforum.org/kb/message.jspa?messageID=6838361
>
> > Can anyone give me the basis for the definition
> > of sinh x to be (e^x - e^-x) / 2 ?
>
> I'm surprised that no one has yet put these two
> together and mentioned the Gudermannian function,
Which satisfies the functional equation,
f(a f(x / a)) = x for a = -i.
John Brillhart has asked whether there are any interesting functions
satisfying that functional equation for other values of a.
--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)