Stirling's Approximation for Large Factorials - Prof. Stefan Franzen, Study notes of Physical Chemistry

Download Stirling's Approximation for Large Factorials - Prof. Stefan Franzen and more Study notes Physical Chemistry in PDF only on Docsity! Stirling’s Approximation In confronting statistical problems we often encounter factorials of very large numbers. The factorial N! is a product N(N-1)(N-2)..(2)(1). Therefore, ln N! is a sum where we have used the property of logarithms that log(abc) = log(a) + log(b) + log(c). The sum is shown in figure below. The sum of the area under the blue rectangles shown below up to N is ln N!. As you can see the rectangles begin to closely approximate the red curve as m gets larger. The area under the curve is given the integral of ln x. To solve the integral use integration by parts Here we let u = ln x and dv = dx. Then v = x and du = dx/x. Notice that x/x = 1 in the last integral and x ln x is 0 when evaluated at zero, so we have ln N! = ln mΣ m = 1 N ln N! = ln mΣ m = 1 N ≈ ln x dx 1 N u dv = uv – v du ln x dx 0 N = x ln x 0 N – x dxx 0 N