Built on Facts

Sunday Function

The Sine Function. Calm and dignified, it sits among the royal court of the Elementary Functions, presiding with undulating grace over the trigonometric functions, partnered with the Exponential Function, and showing forth his power over the realms of physics and mathematics.

On one of the less subtle TV networks, this installment might be called When Good Functions Go Bad.

The sine function is ubiquitous in physics. Figuring out vector components, solving differential equations in E&M and quantum mechanics, decomposing Fourier series, you name it. It’s about as well-behaved as functions get. At least along the real line, it’s periodic, finite, bounded, continuous, and continuously differentiable with derivatives of all orders. It’s hard to ask for a cleaner function. But just for fun, let’s replace the argument x with 1/x and see what happens.

i-68d99e78d230c147ce86d86e5c68b525-sine.png

i-8d7070ce28f0af28047c7a5fb8246d53-4.png

It’s continuous and differentiable everywhere except the origin. Discontinuities are not exactly rare in the world of functions, but this one is of a type not often seen in physics. It’s not a point discontinuity, or an asymptote, or a jump discontinuity. Instead, the function oscillates infinitely often near the origin and so the limit completely fails to exist from either side.

When these types of functions turn up in physics, we tend to wave our hands and declare them “unphysical”. As indeed they are. What this means is usually that whatever theory has resulted in something like this is simply not valid near the point of discontinuity and needs to be extended.

Probably the most famous of this type of scenario was the ultraviolet catastrophe. Back in the early 20th century, there was a theory due to Rayleigh and Jeans that described the blackbody spectrum of objects very well for long wavelengths. But as wavelength approached 0, the emitted intensity approached infinity. Obviously this was impossible, and the theory which ended up describing the situation correctly was quantum mechanics.

Nonphysical equations popping up in physical theories are quite common even today, and remain both aggravating problems and tremendous opportunities to extend our understanding with better theories.

Comments

  1. #1 Blake Stacey
    August 3, 2008

    Out of curiosity, what are you using to create the PNG images for your equations?

  2. #2 Egaeus
    August 3, 2008

    Not to answer for him, but the borders around the function don’t look uniform, so probably Microsoft’s Equation Editor in Word, or if he’s a true geek, LaTeX. Then I would print to or save as a PDF, from there save as a png, copy the important part, and you have an image of the function that’s not brower-dependent.

  3. #3 Ron
    August 3, 2008

    sin(1/z) over the complex plane is also interesting with its essential singularity.

  4. #4 Blake Stacey
    August 3, 2008

    The Movable Type version of the LaTeXRender plugin (which automatically converts TeX formulas in blog posts to PNG images) only works on MT 4, while ScienceBlogs is still hosted on MT 3, so I figured the conversion has to be done manually. . . but maybe our host knows something I don’t! :-)

  5. #5 Philip R
    August 3, 2008

    Linked here (rather, to your front page) from Pharyngula. Then I saw you’re at A&M. WHOOP! So, I wanted to say Howdy! It makes me pretty proud to see an Aggie on ScienceBlogs. Gig’Em!

    Also, Physics ROCKS!

  6. #6 Matt
    August 3, 2008

    I write the equations in LaTeX and use one of the various online LaTeX to PNG converters you can find around the web.

    Eventually I’m going to start agitating for this site to implement LaTeX directly (including in the comments), but I don’t want to get a reputation as a pest so early!

    Incidentally, the graph itself was done in Mathematica and from there edited in Photoshop as something web-usable.

    Oh one more thing about the essential singularity: I wrote a post not too long ago about Exp[-1/x^2], which also has an essential singularity at the origin and thus has a very interesting issue in its Taylor series.

  7. #7 Taylor
    August 3, 2008

    In the off chance you are interested, I have made a video of the sin(1/x) function as you zoom into the origin which can be found here:

    http://happyquark.com/2008/08/03/sin1x-and-its-interesting-discontinuity/

    Thought this might make for a good visual aid.

  8. #8 Uncle Al
    August 3, 2008

    About the video: why are there f(x) maxima and minima that are not -1 or +1? One naively expects the plot to be self-similar with rescaled abscissa as x = 0 is approached by magnification. It’s more aesthetically pleasing that way.

  9. #9 Ron Avitzur
    August 3, 2008

    The asymptotes in the movie above are an artifact of Mathematica’s plotting algorithm. It naively evaluates the function at some number of points in the domain and connects the dots by lines. Since the function is varying faster than the frequency of the sampling, the value of the function is some not quite random number between +1 and -1. In general, plotting programs which sample a function can be fooled if the function behaves differently between the sampled points. It is possible to ask a different question, given a closed form function: what is the range of values it takes in the domain between x and x+dx and use that information to construct a more robust graph. Fateman’s paper Honest Plotting, Global Extrema, and Interval Arithmetic describes this in detail.

    I posted a movie zooming in on sin(1/x) using this technique here.

  10. #10 Bob Brashear
    August 3, 2008

    First time on this blog. All I can say is “Thank you”.

  11. #11 DG
    August 4, 2008

    this blog is awesome

  12. #12 FO
    August 4, 2008

    Awesome! Glad to see a physics blog that’s not reluctant to include equations (and LaTeX-produced ones too, at that). I hope the Seed Overlords can find a way to implement LaTeX directly into the comments section — that would make for some very interesting comment threads!

    Best of luck, and I hope to read more awesome stuff from you!

  13. #13 Chris Nowak
    August 4, 2008

    Whoa that’s cool. I’m looking forward to reading more posts on this blog in the future! I know very little about physics involving actual equations but i’ve understood all your posts so far.

  14. #14 Matti Sironen
    August 5, 2008

    A more familiar example nonphysical/mathematical artifacts might be space-time singularities such as the center of a black hole. The remedy should come from the same source as with the ultraviolet catastrophe, namely quantum mechanics, but a properly quantized theory of gravity seems to have proven itself one of the greatest hurdles in physics yet.

  15. #15 Edén
    March 10, 2012

    I’m currently solving a physical problem where this equation arises, so… it isn’t so unphysical as it might seem. :-)

    Thanks for the info.

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