answers-02-15-2011 - Answers to Physics 176 One-Minute...

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Unformatted text preview: Answers to Physics 176 One-Minute Questionnaires Lecture date: February 15, 2011 What is the historical reason that there is an offset of 1 in the definition of the Gamma function, Γ( x + 1) ? I don’t know. If you somehow find out, please let me know. By the way, if you like mathematics enough to enjoy reading mathematics books, I would recommend the book Gamma: Exploring Euler’s Constant by Julian Havil. It is one of the few books that I know of that gives an exciting sense of the many interesting ideas and discoveries that Euler and his contemporaries explored. The book includes a discussion of the Gamma function and also several other famous functions like the zeta function that relates to the theory of prime numbers. It is not technical, nearly everything can be understood with a freshman level of mathematics (some calculus and a little number theory). What is the error in Stirling’s approximation? The error is often written as an infinite series involving powers of the small quantity 1 /n where n 1 is the integer for which you are trying to esti- mate n !. Problem B.13 on page 391 of Schroeder gives one way to calculate the errors associated with Stirling’s approximation but the recommended heavy- duty way to calculate the error systematically is a valuable result of clas- sical applied mathematics called the Euler-Maclaurin formula. An explicit expression for the error is given in the Wikipedia article “Euler-Maclaurin formula”.) If you ever take a course in computational or numerical mathematics, you will see frequent use of the Euler-Maclaurin formula since it provides a deep understanding of how well one can approximate the value of an integral numerically by a finite sum of values of the integrand. Can you point me in the direction of a more formal discussion of the fact that f ( x ) N ≈ Gaussian for large N ?...
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This note was uploaded on 10/20/2011 for the course PHYSICS 176 taught by Professor Behringer during the Spring '08 term at Duke.

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answers-02-15-2011 - Answers to Physics 176 One-Minute...

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