Message from Herb Wilf About Chandrasekhar and Discrete Analysis, Dated March 11, 2007

Back to "Real" Analysis is a Degenerate Case of Discrete Analysis

From wilf at math dot upenn dot edu  Sat Mar 10 21:25:35 2007

I was reading something of yours about how analysis is a degenerate 
case of discrete mathematics, and I was reminded of one of the most 
beautiful pieces of science that I have ever seen, due to the late, 
great Indian astrophysicist Subrahmanyan Chandrasekhar. In his 
elegant book "Radiative Transfer", there is one chapter in which the 
following sequence of events takes place:

1. He formulates some astrophysical problem as the solution  to an 
integro-differential equation.
2. Despairing of solving that by any standard method, he replaces the 
integral sign by a numerical approximation, maybe trapezoidal rule 
with mesh size h, or something of that kind.
3. The resulting system of approximate equations can be solved 
analytically, by standard linear algebra methods.
4. Now the fun part. In the solution, which is the exact solution of 
the approximate equations, there is the mesh size h, and he takes the 
limit as h->0. Successfully. The result is that he has found the 
*exact* solution of his original integro-differential equation.

So what he did was this. Confronted with a tough problem in 
continuous mathematics, he approximated it by a problem in discrete 
mathematics, found the exact solution of the approximate problem, and 
took the limit to make the approximate solution exact. Continuous 
mathematics was too feeble to solve the problem, so he stood on the 
shoulders of discrete folks to do it.


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