1. 講者：許永昌 老師 1

2. Contents I 2

3. Contents II 3

4. Contents III 4

5. Matrices Basic Operations 2 matrices: A+B,A-B,A*B 1 matrix: A †, A T, tr(A), |A|, A -1. How about A=BC for these operators? E.g. tr(BC)=tr(CB). 這些請務必要知道他們 element 形式下的定義與操作。 E.g. (A T ) ij =A ji. 5

6. Matrices 6

7. Series Convergent {u n }: If for every  >0 there is a positive number N such that |u n  u| N. Cauchy’s criterion Tests for absolute convergent series {s || n } Properties of Conditionally and Absolutely convergent series. s || n  s ||, we can get s n  s. S a+b n  S a +S b.  u n =  u p(n) for absolutely convergent series.  u n v m =  c l if one of them is absolutely convergent series. Leibniz criterion for  (-1) n a n. 7

8. Series 8

9. Functions of Complex variables Complex algebra: Representations of z: z=x+iy  f(z)=u+iv. z=|z|e i . Complex algebra: 1 variable: z*, arg(z), z -1. 2 variables: z 1 +z 2, z 1 z 2. 9

10. Functions of Complex variables Complex algebra 10 Differentiable Analytic Cauchy-Riemann condition. f x & f y are continuous. Cauchy-Riemann condition. f x & f y are continuous. Cauchy’s integral theorem Cauchy’s integral Formula Morera’s theorem. Laurent expansion Taylor expansion Residue Theorem

11. Product theorem: |AB|=|A||B| 11

12. Tr(AB)=Tr(BA), A † B † =(BA) †,A -1 B -1 =(BA) -1 12

13. Normal matrix has complete orthogonal eigenvectors 13

14. A matrix has complete orthogonal eigenvectors is a Normal matrix 14

15. Leibniz criterion 15

16. Weierstrass M test ( 可用在複數 ) 16

17. 17

18. Abel’s test 18 1. 骨幹已給，細節可以像 Weierstrass 的證明那樣補齊 2. 此證明的 b n 改成 f n (x) 3. 沒法用在複變

19. Remainder of Taylor Expansion (real number) 19 Proof: ( 用 mathematical induction) 1. n=1 時  LHS:f(x)-f(a), RHS:R 1 (x)=f(x)-f(a): 相等。 2. 設 n=m 時對 F(x)=f (1) (x) 是對的，證明 n=m+1 對 f(x) 也是對的。 3. 由於 n=1 以證明是對的，根據 (2) 得到對所有 n 都是 對的。

20. Uniform convergence (f n (z)  f(z) ) 1. If f n (z)  f(z) in z  D,  c f n (z)dz   c f(z)dz. 2. If f n (z)  f(z) and f ’ n (z)  g(z) in z  D f ’ n (z)  f ’ (z) 3. If f n (z)  f(z) and f n (z) is continuous in z  D 1. f(z) is continuous 20

21. Uniform convergence (3) 21

22. Uniform Convergence (1) 22

23. Uniform convergence (2) 23

24. Cauchy-Riemann Condition 24 z0z0 For eq. (1)

25. Cauchy’s Integral Theorem 25

26. Cauchy’s integral formula 26

27. General Cauchy’s integral Formula Mathematical Induction 27

28. Morera’s Theorem for every closed contour C within R f(z) is continuous in a simply connected region R f(z) is analytic throughout R. Cauchy’s Integral Theorem 28

29. Taylor’s expansion uniqueness 與 均勻收斂 的問題在下 一頁 29

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31. Laurent Expansion 證明法與 Taylor‘s expansion 很像， 一樣可以證明唯一性與均勻收斂。 一樣當 z 0 變時， {a n } 會跟著變。 不同處在於： a n  f (n) /n! 用來證明的 contour 變成兩個圓。 r<|z-z 0 |<R {a n } 還會針對不同收斂區域而改變。 若不明講範圍，則收斂區域對應的是 z 0 附近的圓 31

32. Conformal mapping 32

33. Residue Theorem 33

34. Summary 34