The Cauchy–Riemann (CR) Equations

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

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

The Cauchy–Riemann (CR) Equations (2)

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

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

Proof of Theorem 1 (2) g ff

Proof of Theorem 1 (3) F h

Example

Example (2)

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

Properties D F G D H F d

Example

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

Trigonometric Function (2) F f

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

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

Examples

General power G f

Examples

Homework Problem set , 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.