Stirling's approximation

Stirling's Approximation

1. Stirling’s Approximation Dr. Anjali Devi J S Guest Faculty School of Chemical Sciences M G University

2. James Stirling and Stirling’s Approximation James Stirling (Scottish Mathematician ) Born May 1692 , Garden,  Stirlingshire Died 5 December 1770 (aged 78) Edinburgh ,  Scotland Resting place Greyfriars Kirkyard Nationality Scottish Known for Stirling's approximation Stirling numbers Scientific career Fields Mathematics

3. Why Stirling’s Approximation? Factorials are HUGE!! Find 10! Ans : 3628 In Statistics, We have to deal with N A ! Avogadro Number N A = 6.023 X10 23 Find 10 23 ! To solve large, we use Stirling’s Approximation

4. Derivation of Stirling’s Approximation N!= 1x2x3x… xN HUGE NUMBER Taking Natural Logarithm ln (N!)= ln (1)+ ln (2)+ ln (3).…+ ln (N) ln (N!)=   If N is large (say 10 23 ) ln (N!)=  

5. Derivation of Stirling’s Approximation ln (N!)= =   1 st function 2nd function by parts     = ln (N) N- ln (1) 1-N-1 =N ln (N)-N+1 =N ln (N) -N ln (N!)≈   This is Stirling’s Approximation

6. Accurate form of Stirling’s Approximation A more accurate form of Stirling’s approximation is  

7. Question ln (N!)≈   Calculate the following using Stirling’s approximation: (a) 10! (b) 50! N! Calculated value ln N! (by calculation) ln N! (by Stirling’s Approximation) Error (%) 10! 3.63 x10 6 15.1 13.02 13.8 N! Calculated value ln N! (by calculation) ln N! (by Stirling’s Approximation) Error (%) 50! 3.04 x10 64 148.4 145.6 1.88

8. N! Calculated value ln N! (by calculator) ln N! (by Stirling’s Approximation) Error (%) 100! 9.33 x10 157 363.7 360.5 0.88% Question ln (N!)≈   Calculate the following using Stirling’s approximation: (c) 100! (d) 150! N! Calculated value ln N! (by calculation) ln N! (by Stirling’s Approximation) Error (%) 150! 5.71 x10 262 605.0 601.6 0.56%