A reader asked me about the hyperbolic trig functions, sinh(x) and cosh(x). What are they for, and do they have an intuitive interpretation in physics?
That's a pretty good question. After all, most of the time you first meet the hyperbolic trig functions in intro calculus, where their rather odd definitions are presented and then used as test beds for blindly applying newly-learned differentiation rules. Ok, great. But what are they really?
To answer the question, we should start off with Euler's identity, which relates the exponential function with the regular trig functions. Proving this identity would take us a little far afield for this post, so for now we'll just take it for granted:
Now, replace x with -x and write down the equation again. But remember that cos(-x) = cos(x) and sin(-x) = -sin(x) because of the even/odd properties of those functions. With that in mind, Euler's identity is just as well written:
Now we have two equations, and by adding the first equation to the second we can cancel out the sin(x) terms. Or by subtracting the second equation from the first we can cancel out the cos(x) terms. We might as well do both, an we end up with:
and
This is kind of neat - we've taken functions that have their origins in ancient people studying triangles and we've written them in terms of the modern language of exponential functions and imaginary numbers. Pythagoras and crew would have no idea at all what something like cos(iπ) would be, but now we're in position to answer those sorts of questions. In the equations we've just derived, substitute ix in place of just x. Since i*i = -1 by definition, the complex exponentials become purely real, like this:
and
Thus if we want to know what cos(iπ) is, we just plug in (1/2)*(e^π + e^-π) into our calculators, and it turns out to be about 11.592.
Which brings us to the hyperbolic trig functions. Instead of the strange ex cathedra definition in intro calc books, we see that they're simply defined as the regular old trig functions when you plug imaginary numbers into them:
and
Which is actually sort of a nice little connection. For completeness, here's their graphs:
sinh(x):
cosh(x):
Now what's the physical interpretation of sinh and cosh? To be honest there isn't much of one - really they're just sort of a change of basis, and anything you can write in terms of sinh and cosh can usually be written more clearly in terms of exponential or ordinary trig functions. They do crop up in various differential equations, including the Laplace equation in classical E&M, and potential steps in the Schrodinger equation. But at least now we know where they come from.
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Actually 1/cosh is a function that some would say controls the rise and fall of civilization. It looks like a Gaussian but its growth and subsequent decay is merely exponential. More details here
http://sep.stanford.edu/sep/jon/hubbert.pdf
Enjoy!
The Tanh function has quite a bit of use in material physics, especially in continuum theories. It is used to describe the change through an interface.
The transition solution to the famous Cahn-Hilliard (http://en.wikipedia.org/wiki/Cahn%E2%80%93Hilliard_equation) equation is a Tanh profile.
In simpler terms, whenever you want to describe a system that has a bulk value in one domain, and another bulk value else where, chances are a Tanh like interface is involved.
Now what's the physical interpretation of sinh and cosh? To be honest there isn't much of one - really they're just sort of a change of basis, and anything you can write in terms of sinh and cosh can usually be written more clearly in terms of exponential or ordinary trig functions. They do crop up in various differential equations, including the Laplace equation in classical E&M, and potential steps in the Schrodinger equation. But at least now we know where they come from.
They have one very common application that you didn't mention: the catenary curve of a hanging chain or wire. If you live somewhere where you regularly see power or telephone cables on poles, the shape of the wire between the two poles is the same as the hyperbolic cosine.
The hyperbolic trig functions are still about triangles.
Just like ( sin θ , cos θ, 1 ) describe the sides of a right triangle with a constant hypotenuse, so do the hyperbolic functions ( 1 , sinh Ï, cosh Ï ) describe a right triangle with a constant leg.
Special Relativity says for a massive particle with invariant mass of m, the following is also a right triangle with constant leg ( mc^2, |p|c, E ). The substitution Ï = tanh^(-1) (v/c) relates |p|c = mc^2 sinh Ï and E = mc^2 cosh mc^2.
Similarly, the Lorentz transform, when parametrized by Ï, is just
( cosh Ï, sinh Ï \\ sinh Ï, cosh Ï ) which preserves c²(Ît)²â(Îx)² while a rotation is a trasform in the form ( cos θ, - sin θ \\ sin θ, cos θ ) which preserves (Îx)²+(Îy)². This hyperbolic analog of rotation is why the geometry of Minkowski (or Lorentzian) space-time is called hyperbolic as opposed to the Euclidean plane.
just as trig functions parameterize ellipses, so hyperbolic functions parameterize hyperbolas.
Some excellent replies so far. All I have to add is that sinh is the odd part of exp(x) while cosh is the odd part - dividing a function into even and odd parts is frequently useful when symmetries are involved.
So if cosh(x) = cos(ix), then what does a 3D plot of cos(x+iy) look like? sin(x+iy)? tan(x+iy)? I don't have anything handy to plot that.
@Colin: Plugging complex numbers into the trig functions will generally yield complex results, so there is no straightforward way to plot that.
As BlackGriffen pointed out, all above replies are excellent. I wish I could add another example.
@rpenner: +100 for that comment!
@Colin: Using the trig addition formula on
cos(x + i y) = cos(x)cosh(y) + i sin(x)sinh(y)
This makes it easier to imagine the structure on the complex plane.
Type cos(x+I*y) into http://www.wolframalpha.com/ and it will give you a 3D plot.
Adding to what bill said at #5, the parametrizations of the circle x^2+y^2=1 and the hyperbola x^2-y^2=1 by trigonometric and hyperbolic trigonometric functions are not arbitrary; they are related to the area of a region bounded by the circle or hyperbola.
For positive s, let P(s)=(x,y) be either (cos(s),sin(s)) or (cosh(s),sinh(s)), and let R(s) be the region bounded by the line segment from the origin to (1,0), the line segment from the origin to P(s), and either the circle or hyperbola (respectively). Then, in both cases, the area of R(s) is s/2. (In the case of the circle, if s is greater than pi, then you need to be careful how you interpret "area".)
Great replies all around, for sure.
@Colin: However you plot cos(z) in the complex plane, rotate the resulting plot 90 degrees counter-clockwise and it'll be the plot of cosh(z).
At some level the really striking thing here is that cos(ix) actually gives you real numbers for all real x. Once you've gone to the complex plane, there's no reason to think that any particular value will be real. But in this case we get that they all are. This means that cos(z) is real on both the real and imaginary axis. This is in fact arising from the fact that the Taylor series for cos z has only even degree terms. Similarly, we need that -i term in front of sin(ix) to get a real value because the Taylor series for sin(z) has only odd degree terms.
Graphs of cosh(x+iy) and sinh(x+iy):
http://www.wolframalpha.com/input/?i=cosh%28x%2Biy%29
http://www.wolframalpha.com/input/?i=sinh%28x%2Biy%29
Wolfram Alpha is pretty much my favorite.
These come in hyperbolic trigonometry, see for example the law of sines in various geometries:
http://en.wikipedia.org/wiki/Law_of_sines
One of the inventors of hyperbolic functions, Lambert tried to prove (at least examined) Euclid's fifth postulate and ended up proving what turned out to be theorems in hyperbolic geometry. Seems to be a curious connection.
Tanh(x) is widely used in neural networks as a versatile non-linear transfer function (although other functions such as sigmoid will also work just as well).
If you know the identities for trigonometric functions, then these can easily be generalized to hyperbolic functions using Osborne's rule (http://mathworld.wolfram.com/OsbornesRule.html), which follows from their definitions in terms of exponential functions.
I will try to explain the analogy between the circular and
hyperbolic functions and derive the hyperbolic functions
from first principles.
The function which takes a real number theta to the rotation of angle theta is a continuous homomorphism from
the additive group of real numbers onto the connected
component of the automorphism group of the quadratic metric
x^2 + y^2 ( a.k.a SO(R,2) ).
The matrix of a rotation of angle theta relative to the
standard basis of R2 is
cos(theta) -sin(theta)
sin(theta) cos(theta)
This mapping reduces the study of Euclidean plane geometry
to the study of the familiar abelian groups R, Z, and Z2(orientation).
Any other continuous ( or even measurable ) homomorphism of the additive group R onto SO(R,2) is given by replacing
theta with (a theta) where a is a non-zero real number.
Now consider the hyperbolic plane with quadratic metric
x^2 - y^2. We wish to do something similar. First we de-
termine the automorphism of the form x^2 - y^2. To do so it is convenient to make a change of variables. Introducing
x + y and x - y as new variables we transform this form to
the form xy.
Note that -
1) the standard basis vectors e1 & e2 are null vectors
2) the null vectors of the form xy are precisely the
multiples of e1 & e2
3) the scalar product of e1 & e2 is 1
From this we see that T is an automorphism of xy if and only if it's matrix relative to the basis e1, e2 has columns
null vectors with a scalar product of 1.
Thus the columns of such a matrix must be multiples of e1 & e2 with scalar product 1. Thus the columns must be ae1 and
(1/a)e2 where a<>0.
Now staying the connected componenet of the identity, a must
be positive and the multiple of e1 must come first. So the
automorphisms of the form xy in the connected component of
the identity have matrices
a 0
0 1/a
where a > 0.
Now we verify that the function taking a positive number a
to the above matrix is a continuous isomorphism of the
multiplicative group of real numbers onto the connected
componenet of the automorphism group of the form xy.
Now the real exponential gives an isomorphism of the additive group of all real numbers onto the multiplicative group of positive numbers. Composing with the above we
get an isomorphism sending a real number x to
exp(x) 0
0 exp(-x)
This is an isoomorphism of the additive group of real numbers with the connected component of the automorphism
group of the form xy.
Now transforming back to the form x^2 - y^2 we get the
formulas for cosh & sinh.
The function taking a real number x to
cosh(x) sinh(x)
sinh(x) cosh(x)
is an isomorphism of the additive group of real numbers with
the connected component of the automorphism group of plane hyperbolic geometry. Any such (continuous or even measurable ) isomorphism is obtained by replacing x with ax
where a is a non-zero real number.
Thus as with Euclidean plane geometry the geometry of the
hyperbolic plane is reduced to the study of familiar abelian groups.
In higher dimensions such a reduction is not possible. Indeed then the automorphisms groups are not solvable.
However any higher dimensional indefinite geometry contains subspaces isometric to the hyperbolic plan so the hyperbolic functions are relevant to any such geometry including for example Minkowski geometry.
After complexification the forms x^2 + y^2 and x^2 - y^2
are linearly equivalent. This is reflected in the circular and hyperbolic functions being linearly equivalent after
complexification.
Hyperbolic trig functions represent the motion of particles near unstable equilibria the same way circular trig functions represent the motion of particles near stable equilibria.
If you have a spring with negative spring constant -k, there differential equation is d^2x/dt^2 = kx, as opposed to the usual d^2x/dt^2 = -kx. The general solution is a sum of hyperbolic sines and cosines. Thus, a pendulum nearly vertically up or a ball at the top of a hill have motions described by hyperbolic trig (for small angles).
To Annonymous | August 2, 2011 9:12 AM
Is there a dictionary to convert plane trigonometry formaulae to those in plane hyperbolic trigonometry? Long ago i taught hyperbolic geometry (as a part of trying to understand William Thurston's work) and even though I could see the analogy, I could not find a dictionary for conversion. I am retired now but would like to understand a bit better if I can. Thanks.
gaddeswarup
1) Excuse me for not responding sooner. I had minor
knee surgery yesterday.
2) I'm afraid I don't where to find the information
you are looking for. You might try Thurston's
work "Three-Dimensional Geometry and Topology". I
have only browsed through it not read it. The biblio-
graphy lists Fricke/Klein. Presumably anything can
be found in that august work if you can get access
to it. I've never seen a copy.
3) My use of the term "hyperbolic plane" is in the context
of metric affine geometry. This is Artin's teminology
in his "Geometric Algebra".
rpenner -
If one considers a right angle triangle with constant
leg (1,y,x) then 1 = x^2 - y^2 and we are back to the
form x^2 - y^2. Euclidean plane geometry and hyperbolic
plane geometry are closely related. There is only some
fussing with signs. Complexification obliterates the
difference between positive and negative (so orientation
is not relevant in complex geometry).
Over the real field the hyperbolic function plays the same
role with respect to the form x^2 - y^2 as the circular
functions play with respect to the form x^2 + y^2. They
reduce the study of the automorphism group of the form to
the additive group of real numbers and some other other
familiar abelian groups. This is a great miracle. In higher
dimensions one must deal with non-solvable groups.
After complexification the two forms are linearly equivalent
and so the circular and hyperbolic functions merge.
If one stays in the real field it is rather easier to treat
the form xy which is linearly equivalent to x^2 - y^2. Then
the functions exp(x) & exp(-x) on the real line play the role played by the circular and hyperbolic functions. The
linear transformation between the forms x^2 - y^2 and the
form xy also transforms exp(x) & exp(-x) to the hyperbolic
functions.
It is interesting to note that the exponential function and
sin x & cos x, probably the most important functions in
mathematics (after linear functiuons ) are both group homo-
morphisms. For sin x & cos x this is somewhat indirect. It
is rather the function sending theta to a rotation of angle
theta which is a group homomorphism of the additive group
of real numbers onto SO(R,2) the connected component of the
automorphism group of the form x^2 + y^2. Looking at the
matrix multiplication of the rotation matrices what the
addition laws for the sine and cosine say is that a rotation
of angle theta1 composed with a rotation of angle2 is a rotation of angle theta1 + theta2. This immediately implies
that SO(R,2) is divisible as well as giving the torsion of
SO(R,2). Neither of these things are at all obvious from the
definition of SO(R,2). One then has the keys to plane Euclidean geometry.
Although the difference between Euclidean plane geometry
and hyperbolic plane geometry at the algebraic level seems
only a matter of some signs there is another important
difference. The automorphism group of the form x^2 + y^2
is compact while that of the form x^2 - y^2 is non-compact.
Well this is where projective geometry comes in.
My comment that complexification obliterates the difference
between positive and negative should have been expressed a
little more precisely. Obviously it doesn't obliterate
the difference between z and -z. It takes characteristic 2
to do that but complexification does obliterate the difference between the set of positive numbers and the set
of negative numbers because zero no longer disconnects the
the complex field. So orientation goes away in complex
geometry and complex varieties cannot be disconnected by
lower dimensional subvarieties.
Daha basit terimlerle, toplu deÄeri bir etki alanında olan bir sistem açıklamak istediÄiniz zaman ve nerede, Åans are Tanh arayüzü gibi baÅka bir yıÄın deÄeri baÅka ilgilenmektedir.