Error Function

In the November 6th entry to this blog, I mentioned one special function introduced in the 4^th edition of Intermediate Physics for Medicine and Biology: the Bessel function. Another special function Russ Hobbie and I discuss briefly is the error function, which arises naturally when solving the one-dimensional cable equation (Chapters 6 and 7) or the diffusion equation (Chapter 4). The error function is the integral of the familiar Gaussian function, and has a sigmoidal shape, being minus one for large negative values of its argument and one for large positive values.

Handbook of Mathematical Functions
with Formulas, Graphs, and
Mathematical Tables,
by Abramowitz and Stegun.

To learn more about the error function, see the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables by Milton Abramowitz and Irene Stegun (1972). This classic math handbook is available online at http://www.math.ucla.edu/~cbm/aands//. Also, Wikipedia has a very thorough article about the error function, including beautiful plots of the error function in the complex plane.
I’m not sure how the error function got its name. Perhaps it has something to do with experimental errors often being Gaussianly distributed. If anyone knows, please let me know.


P.S. Speaking of errors: For any students or instructors preparing to use the 4^th edition of Intermediate Physics for Medicine and Biology next semester, I recommend you download the errata, which can be found at https://sites.google.com/view/hobbieroth. In it, Russ Hobbie and I list all known errors in our book. The number of errors has grown, and in particular some are present in homework problems. Generally I frown on writing in my books, but in this case do yourself a favor: download the errata and mark the corrections in your copy of the text. And as always, let us know if you find additional errors. The only thing worse than finding errors in a book you wrote is having errors in a book you wrote that you are not even aware of.

P.P.S. I have written in this blog about Steven Strogatz, a mathematician and author, and about Kleber’s law, which relates metabolic rate to body mass. Here is an article by Strogatz about Kleber’s law. It doesn’t get much better than that!