Looking For a little History on the Hyperbolic Functions

[Saladsamurai]

I was just browsing through my textbook in the section on hyperbolic trig functions. It defines sinhx to be $$\frac{e^x-e^{-x}}{2}$$, which comes from breaking the function $$f(x)=e^x$$ into two functions, the other of which forms coshx.

Oddly enough, this is one of the only sections in the text that does not include a brief history of the topic at hand.

I came across one site that said that a Lambert discovered (or created, I don't know which) the hyperbolic functions.

Does anyone know of any good sources where I could get the rundown on the history of these things.

I am just curious as to why someone would have wanted to break $$e^x$$ into parts in the first place.

I know that the hyperbolic functions will serve some purposes in integration, but I would assume that that was not their original intent.

Any insight would be appreciated,
Casey

 

[Hurkyl]

As their name suggests, they are useful for hyperbolic trigonometry. For example, the unit hyperbola defined by

x^2 - y^2 = 1​

is parametrized by

$$(x, y) = (\pm \cosh u, \sinh u)$$.​

I don't remember the details, but this is very closely related to hyperbolic geometry -- the non-Euclidean geometry that Lambert was studying. (Of course, he was trying to find a contradiction, but still, he laid the foundations for this particular subject)

 

[Saladsamurai]

As their name suggests, they are useful for hyperbolic trigonometry. For example, the unit hyperbola defined by

x^2 - y^2 = 1​

is parametrized by

$$(x, y) = (\pm \cosh u, \sinh u)$$.​

I don't remember the details, but this is very closely related to hyperbolic geometry -- the non-Euclidean geometry that Lambert was studying. (Of course, he was trying to find a contradiction, but still, he laid the foundations for this particular subject)

Thanks Hurkly. Then, I guess I was looking for something more along the lines of what is hyperbolic trig.

What made somebody say to themselves, "Hey, I think I'll break the function e^x up into two ridiculous looking fractions that when summed equal just e^x again after breakfast today..."

Know what I mean? Seems like it probably had an application or some purpose...

Casey

 

[Hurkyl]

Oh phooey, I confused Lambert with Saccheri. (But I think Lambert did some work along those lines too)


Anyways, as I was trying to imply, hyperbolic trig functions are to (rectangular) hyperbolas as circular trig functions are to circles. So anytime a hyperbola can be made interesting to study, the hyperbolic trig functions will probably come into play.


One major application is in Minkowski geometry (the space-time of special relativity); squared distances in the Minkowski plane are given by $\Delta(ct)^2 - \Delta x^2$, so the hyperbola plays the same role in Minkowski geometry as the circle does in Euclidean geometry. (it's the locus of all points a fixed distance from a given point)

 

[Saladsamurai]

Oh phooey, I confused Lambert with Saccheri. (But I think Lambert did some work along those lines too)


Anyways, as I was trying to imply, hyperbolic trig functions are to (rectangular) hyperbolas as circular trig functions are to circles. So anytime a hyperbola can be made interesting to study, the hyperbolic trig functions will probably come into play.


One major application is in Minkowski geometry (the space-time of special relativity); squared distances in the Minkowski plane are given by $\Delta(ct)^2 - \Delta x^2$, so the hyperbola plays the same role in Minkowski geometry as the circle does in Euclidean geometry. (it's the locus of all points a fixed distance from a given point)

Word. I'll search this Minkowski geometry a little. So I guess my question really should have been, why study hyperbolas. And this you have answered.

Thanks Hurkyl,
Casey

 

[Hurkyl]

Hyperbolas are one of the conic sections: after lines, they are the simplest of all shapes, and were known even to Euclid.

Because of their simplicity, they tend to crop up frequently, just like their cousins: circles, ellipses, and parabolas.

In fact, in projective geometry, circles, ellipses, parabolas, and hyperbolas are all the same thing. Their apparent difference is an artifact of perspective: a hyperbola has two points at infinity, a parabola 1, and an ellipse none.

 

[neutrino]

http://www-history.mcs.st-andrews.ac.uk/Search/historysearch.cgi?TOPICS=1&WORD=hyperbolic+function is usually a good site on the history of mathematics, but in the case of hyper-trig funcions, it just seems to have two sentences, one in each article(one by trig. functions and the other, the biography of Lambert.), something along the lines of Lambert made important discoveries... .

 

[Saladsamurai]

Thanks neutrino. I had run into that site earlier from a google search. I just was not sure what kind of website it was. Is it a school?

Casey

 

[LukeD]

Thanks Hurkly. Then, I guess I was looking for something more along the lines of what is hyperbolic trig.

What made somebody say to themselves, "Hey, I think I'll break the function e^x up into two ridiculous looking fractions that when summed equal just e^x again after breakfast today..."

Know what I mean? Seems like it probably had an application or some purpose...

Casey

As you're probably aware though, cosine and sine come from break up e^x in a different way, using complex numbers which is that cos(x) = (e^(ix)+e^(-ix))/2 and sin(x) = (e^(ix)-e^(-ix))/(2i)

From that actually you find that cosh(x) = cos(ix) and sinhy(x) = -i*sin(ix)

 

[neutrino]

Thanks neutrino. I had run into that site earlier from a google search. I just was not sure what kind of website it was. Is it a school?

Casey

It's a site on history of mathematics maintained by the maths dept.(or the school of maths and stat.) of the Univ. of St.Andrews, Scotland.

 

[eeuler]

Hyperbolic functions stem back to ancient times i believe. Hypatia, a female mathematician/astronomer from Alexandria contributed to the development, mainly through conic sections, which i think she elaborated on. Well i do know that her contributions lead to the development of hyperbolic functions.

 

[neutrino]

Hyperbolic functions stem back to ancient times i believe. Hypatia, a female mathematician/astronomer from Alexandria contributed to the development, mainly through conic sections, which i think she elaborated on. Well i do know that her contributions lead to the development of hyperbolic functions.

You could ask her if you want to. She is a regular, here, at PF. :tongue2:

 

[eeuler]

You could ask her if you want to. She is a regular, here, at PF. :tongue2:

Oh wow, really!;) hehe what a coincidence i mentioned her and she ends up being a member here:)

 

[Gib Z]

I would imagine that when Euler found his identity $e^{ix} = \cos x + i \sin x$ and rearranged the series of cos and sin to derive that, he would have had to check if he was allowed to rearranged the series in the way he did. To be able to arrange the terms as he did, he would have to to prove that the series for sin and cos, which have alternating signs, also converge absolutely. As you should know, the absolute series for sin and cos and sinh and cosh respectively.

 

[neutrino]

Lots of links regarding Euler's publication of this identity are available here.