講者： 許永昌 老師 1

Contents Bernoulli function Euler-Maclaurin Integration Formula Improvement of Convergence Asymptotic Series 2

Bernoulli Functions ( 請預讀 P305~P306) We use its properties to derive the Euler-Maclaurin Integration Formula. Bernoulli Functions: B n (s) Properties: Bernoulli numbers: B n =B n (0). B 0 (s)=1. B 2n+1 =0, n  1. 3

Bernoulli functions (continue) From the properties of Bernoulli functions, we get However, you will find that We can get B 2 by the relation B 3 (1)=(-1) 3 B 3 =0, i.e. 1-3/2+3B 2 =0. B 2 =1/6. 當然，您也可以用課本 Eq. (5.124) 來回答。 4

Euler-Maclaurin Integration Formula ( 請預讀 P306~P307) Euler-Maclaurin Integration Formula Hint: 5

Example  m 2 =? Hint: Based on the Euler-Maclaurin Formula f (p) (x)=0 when p>2. 6

Improvement of convergence ( 請 預讀 P307~P311) 7

Improvement of convergence (continue) 8

Homework

An Example of Asymptotic Series ( 請 預讀 P314~P316) 10

An example of Asymptotic series (continue) 11

An example of Asymptotic series (continue) Another way to get this result: 12

An example of Asymptotic series (continue) 13

Asymptotic Series 14

Homework

Nouns 16