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03-5715131#35401 mengfen@mx.nthu.edu.tw 吳小姐

Title

Video

Syllabus

章節大綱

L7A
路徑積分例題：計算1/(5＋3cos(x))在[-pi,pi]上的積分
0:00 Contour Integrals 路徑積分
1:31 Example: Integral of 1/(5＋3cos(x)) on [-pi,pi]
3:05 View the integral as a contour integral on the unit circle 將此積分化為單位圓上的路徑積分
8:45 Find the poles of the integrand 求出被積函數的極點
10:09 Compute the contour integral by applying the residue theorem 利用留數定理計算路徑積分
11:13 Calculate the residues at poles 計算留數
14:32 Correction 修正筆誤
17:27 Integral of P(x)/Q(x) on R 形如P(x)/Q(x)的積分
19:22 Example: Integral of 1/(1＋x^4) on R

L7B
路徑積分例題：計算1/(1＋x^4)在R上的積分
此題有2種算法
方法一
0:00 Choose a contour 選擇適當路徑
0:34 Find the poles of the integrand 求出被積函數的極點
3:25 Compute the contour integral by applying the residue theorem 利用留數定理計算路徑積分
4:19 Calculate the residues at poles 計算留數
11:45 Compute the contour integral by direct computation 直接參數化計算路徑積分
13:03 Estimate the contour integral on the upper semicircle when R tends to infinity
估計上半圓上的路徑積分
17:33 Let R tend to infinity 令R趨近於無限大
方法二
18:08 Choose another contour 選擇適當路徑
19:58 Compute the contour integral by applying the residue theorem 利用留數定理計算路徑積分
20:23 Calculate the residues at the pole 計算留數

L7C
路徑積分例題：計算1/(x^2＋2x＋2)在R上的積分
方法二(續)
0:00 Compute the contour integral by direct computation 直接參數化計算路徑積分
2:17 Estimate the contour integral on the quarter circle when R tends to infinity
估計四分之一圓上的路徑積分
3:52 Let R tend to infinity 令R趨近於無限大
6:23 Example: Integral of 1/(x^2＋2x＋2) on R
7:39 Choose another contour 選擇另一條路徑
8:07 Find the poles of the integrand 求出被積函數的極點
9:33 Compute by direct computation and by applying the residue theorem
利用留數定理和參數化計算路徑積分
10:42 Calculate the residues at the pole 計算留數
12:49 Estimate the contour integral on the upper semicircle when R tends to infinity
估計上半圓上的路徑積分
16:37 Let R tend to infinity 令R趨近於無限大
18:41 Cauchy Principal Value 柯西主值

L7D
柯西主值
0:00 Cauchy Principal Value (cont.) 柯西主值 (續)
10:41 Example: The principal value of the integral of 1/((2-x)(x^2＋4)) on R
12:10 Choose contours 選擇路徑(不只一種選擇)

課程資訊

• 程守慶 教授

• 上架日期: 2023-07-19
• 點閱次數: 5069