Built on Facts

Sunday Function

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:

i-d4bb9fe09d6b78eb832985821d8a307f-1.png

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:

i-88516ca505d84f969b7ab15a965c3fb3-2.png

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:

i-d18063683dcc0d42b9be45451a84d1e3-3.png

and

i-8d7070ce28f0af28047c7a5fb8246d53-4.png

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:

i-bb5c2c6b0452df43a61e3974bd9b473f-5.png

and

i-80f9713f6d11461837a9f9b540684e36-6.png

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:

i-9ef27a331268eac0961c3fd4d1a55446-7.png

and

i-2a911bc4888cdd6dfd35cad351fab7d5-8.png

Which is actually sort of a nice little connection. For completeness, here’s their graphs:

sinh(x):
i-a17c0f9310f609158c7c4a708e0d4c48-sinh.png

cosh(x):
i-05f3715cad79a58519cc032de9db88ec-cosh.png

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.

Comments

  1. #1 Jon Claerbout
    August 1, 2011

    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!

  2. #2 Ari Adland
    August 1, 2011

    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.

  3. #3 Chad Orzel
    August 1, 2011

    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.

  4. #4 rpenner
    August 1, 2011

    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 φ, cosh φ ) 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 φ = tanh^(-1) (v/c) relates |p|c = mc^2 sinh φ and E = mc^2 cosh mc^2.

    Similarly, the Lorentz transform, when parametrized by φ, is just
    ( cosh φ, sinh φ \\ sinh φ, cosh φ ) which preserves c²(Δt)²−(Δx)² while a rotation is a trasform in the form ( cos θ, – sin θ \\ sin θ, cos θ ) which preserves (Δx)²+(Δy)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.

  5. #5 --bill
    August 1, 2011

    just as trig functions parameterize ellipses, so hyperbolic functions parameterize hyperbolas.

  6. #6 BlackGriffen
    August 1, 2011

    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.

  7. #7 Colin
    August 1, 2011

    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.

  8. #8 Odysseus
    August 1, 2011

    @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.

  9. #9 Simon
    August 1, 2011

    @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.

  10. #10 Kevin
    August 1, 2011

    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”.)

  11. #11 Matt Springer
    August 1, 2011

    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).

  12. #12 Joshua Zelinsky
    August 1, 2011

    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.

  13. #13 Tyler Breisacher
    August 2, 2011

    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.

  14. #14 gaddeswarup
    August 2, 2011

    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.

  15. #15 lordaxil
    August 2, 2011

    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.

  16. #16 Annonymous
    August 2, 2011

    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.

  17. #17 Mark Eichenlaub
    August 2, 2011

    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).

  18. #18 gaddeswarup
    August 2, 2011

    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.

  19. #19 Annonymous
    August 4, 2011

    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”.

  20. #20 Annonymous
    August 4, 2011

    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.

  21. #21 Annonymous
    August 4, 2011

    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.

  22. #22 Sesli Chat
    February 7, 2012

    Daha basit terimlerle, toplu değeri bir etki alanında olan bir sistem açıklamak istediğiniz zaman ve nerede, şans are Tanh arayüzü gibi başka bir yığın değeri başka ilgilenmektedir.

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