1. The Cauchy–Riemann (CR) Equations

2. Introduction The Cauchy–Riemann (CR) equations is one of the most fundamental in complex function analysis. This provides analyticity of a complex function. In real function analysis, analyticity of a function depends on the smoothness of the function on But for a complex function, this is no longer the case as the limit can be defined many direction

3. The Cauchy–Riemann (CR) Equations A complex function can be written as It is analytic iff the first derivatives and satisfy two CR equations D

4. The Cauchy–Riemann (CR) Equations (2)

5. The Cauchy–Riemann (CR) Equations (3) Theorem 1 says that If is continuous, then obey CR equations While theorem 2 states the converse i.e. if are continuous (obey CR equation) then is analytic

6. Proof of Theorem 1 D The may approach the z from all direction We may choose path I and II, and equate them

7. Proof of Theorem 1 (2) g ff

8. Proof of Theorem 1 (3) F h

9. Example

10. Example (2)

11. Exponential Function It is denoted as or exp It may also be expressed as The derivatives is

12. Properties D F G D H F d

13. Example

14. Trigonometric Function Using Euler formula Then we obtain trigonometry identity in complex Furthermore The derivatives Euler formula for complex

15. Trigonometric Function (2) F f

16. Hyperbolic Function F Derivatives Furthermore Complex trigonometric and hyperbolic function is related by

17. Logarithm It is expressed as The principal argument Since the argument of is multiplication of And

18. Examples

19. General power G f

20. Examples

21. Homework Problem set 13.4 1, 2, 4, 10. Problem set 13.5 no 2, 9, 15. Problem set 13.6 no 7 & 11. Problem set 13.7 no 5, 10, 22.