1. Cauchy’s theory for complex integrals
MTH 324
Lecture # 19
Cauchy’s theory for complex integrals

2. Previous Lecture’s Review
Contours
Complex integrals
ML-inequality 2

3. Simply and multiply connected domains Cauchy’s theorem
Lecture’s Outline
Simply and multiply connected domains
Cauchy’s theorem
Cauchy’s Goursat theorem 3

4. Connected set

5. Simply connected domain:5

7. Green’s theorem: 7

8. Cauchy’s theorem:
Proof: 8

9. 9

10. 10

11. Cauchy-Goursat theorem:
Example:
Solution: 11

12. 12

13. Remark:
Example:

14. 14

15. Cauchy-Goursat theorem for doubly connected domain:
Proof: 15

16. 16

17. 17

18. Example:
Solution:

20. 20

21. Example:
Solution:

23. Cauchy-Goursat theorem for multiply connected domain:23

24. Example:
Solution: 24

27. 27

28. References
A First Course in Complex Analysis with Applications by Dennis G. Zill and Patrick D. Shanahan.
Complex variables and applications by James Brown and Ruel Churchill