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 x
• 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 f ( z ) can be written as
• It is analytic iff the first derivatives u and v
satisfy two CR equations
• D
The Cauchy–Riemann (CR) Equations (2)
The Cauchy–Riemann (CR) Equations (3)
• Theorem 1 says that If f ( z ) is continuous, then
u and v obey CR equations
• While theorem 2 states the converse i.e. if u and v
are continuous (obey CR equation) then f ( z ) is
analytic
Proof of Theorem 1
• D
• The z 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 e z or exp z
• 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 z is multiplication of 2
• And
Examples
General power
• G
• f
Examples
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.