This document discusses complex line integrals and Cauchy's integral theorem. It begins by defining a complex line integral as integrating a function along a curve with a parametric representation in the complex plane. It notes that complex line integrals depend on both the endpoints and path of integration. The document then introduces Cauchy's integral theorem, which states that the integral of an analytic function around a closed curve is zero. It provides several examples of evaluating complex line integrals and applying Cauchy's integral theorem and formula. It concludes by discussing how Cauchy's integral formula can be used to find derivatives of analytic functions.

1. Complex Integration
Yoses Manuel
2112033

2. 14.1 Line Integral in Complex Plane
• A complex definite integrals are called complex line integrals
• It denotes as
• Here, the integrand is integrated over a given curve .
• The curve is a path of integration with parametric
representation
• Examples
( )
f z
C

3. Line Integral in Complex Plane (cont.)
• We assume the curve is smooth, thus it has continuous and
non-zero derivatives
• The definition of the derivatives

4. Definition of Complex Line Integral
• Suppose that the time in parametric representation is divided
into
• The is indeed partitioned into
where
C

5. Definition of Complex Line Integral (cont.)
• On each subdivision of we choose any point
• The we can form the sum
• The limit of the sum is termed as line integration of over
the path of integration of
• Important definition
C i

( )
f z
C

6. Basic Properties
• Linearity
• Sense of reversal
• Partitioning path

7. Evaluation of Complex Integral
Proof :
We have z x iy
 

8. Step to Evaluate

9. Examples

10. Examples (cont.)

11. Examples (cont.)

12. Dependence on Path
• If we integrate from to along different
paths, then the integral in general will have
different values.
• A complex line integral depends not only on
the endpoints of the path, but also on the
path itself.
( )
f z 0
z 1
z

13. Example

14. Example (cont.)

15. 14.2 Cauchy’s Integral Theorem

19. Contoh 1

20. Contoh 2

21. Contoh 3

22. Contoh 4

23. Contoh 5

24. Contoh 6

26. Contoh 7

30. 14.3 Cauchy’s Integral Formula

32. Contoh 1

33. Contoh 2

34. 14.4 Derivatives of Analytic Functions

35. Contoh 1

36. Contoh 2

37. Contoh 3