By Walter E. Thirring

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F (x) = 2 + 3x + 4x2 + 5x3 + · · · 2—Infinite Series 46 If I multiply this by x I get 2x + 3x2 + 4x3 + 5x4 + · · · and that starts to look like a derivative. xf (x) = 2x + 3x2 + 4x3 + 5x4 + · · · = d 2 x + x3 + x4 + · · · dx Again, the geometric series pops up, though missing a couple of terms. 8 Diffraction When light passes through a very small opening it will be diffracted so that it will spread out in a characteristic pattern of higher and lower intensity. The analysis of the result uses many of the tools that you’ve looked at in the first two chapters, so it’s worth showing the derivation first.

381 You can see that the ratio of the exact to the approximate result is approaching one even though the difference is getting very large. This is not a handicap, as there are many circumstances for which this is all you need. I derived this assuming that n is large, but notice that the result is not too bad even for modest values. The error is less than 2% for n = 5. There are even some applications, especially in statistical mechanics, in which you can √ make a still cruder approximation and drop the factor 2πn.

2πN N N e−N N! (N − k)! 2πk k k e−k 2π(N − k) (N − k)N −k e−(N −k) 1 = ak bN −k √ 2π N NN k(N − k) k k (N − k)N −k (17) The complicated parts to manipulate are the factors with all the exponentials of k in them. Pull them out from the denominator for separate handling. k k (N − k)N −k a−k b−(N −k) The next trick is to take a logarithm and to do all the manipulations on it. ln → k ln k + (N − k) ln(N − k) − k ln a − (N − k) ln b = f (k) (18) 2—Infinite Series 44 The original function is a maximum when this denominator is a minimum.