1. Section 2.2 Limits and Continuity
MAT 3730 Complex Variables
Section 2.2
Limits and Continuity

2. Preview
We will follow a similar path of real variables to define the derivative of complex functions
We will look at the conceptual idea of taking limits without going into the precise definition
(MAT 3749 Intro Analysis)

3. Default All coefficients are complex numbers
All functions are complex-valued functions

4. Recall (Real Variables)
Functions
Limits
Continuity
Derivatives

5. Complex Variables
Functions
Limits
Continuity
Derivatives
Analyticity

6. Recall: Limits of Real Functions
Left hand Limit
Right Hand Limit
Limit

7. Left-Hand Limit (Real Variables)y
The left-hand limit is 2 when x approaches 3
Notation:
Independent of f(3) 2
y=f(x) x 3

8. Left-Hand Limit (Real Variables)
We write
and say “the left-hand limit of f(x), as x approaches a, equals L”
if we can make the values of f(x) arbitrarily close to L (as close as we like) by taking x to be sufficiently close to a and x less than a.

9. Right-Hand Limit (Real Variables)y
The right-hand limit is 4 when x approaches 2
Notation:
Independent of f(2) 4
y=f(x) x 2

10. Right-Hand Limit (Real Variables)
We write
and say “the right-hand limit of f(x), as x approaches a, equals L”
if we can make the values of f(x) arbitrarily close to L (as close as we like) by taking x to be sufficiently close to a and x greater than a.

11. Limit of a Function (Real Variables)
if and only if
and
Independent of f(a)

12. Observations
The limit exists if the limits along all possible paths exist and equal
The existence of the limit is independent of the existence and value of f(a)

13. Definition (Conceptual)
w0 is the limit of f(z) as z approaches z0 ,
if f(z) stays arbitrarily close to w0 whenever z is sufficiently near z0

14. Observations (Complex Limits)
The limit exists if the limits along all possible paths exist and equal
There are infinitely many paths
2 paths with 2 different limits prove non-existence
Use the precise definition to prove the existence
The existence of the limit is independent of the existence and value of f(z0)

15. Observations (Complex Limits)
The limit exists if the limits along all possible paths exist and equal
There are infinitely many paths
2 paths with 2 different limiting values prove non-existence
Use the precise definition to prove the existence
The existence of the limit is independent of the existence and value of f(z0)

16. Observations (Complex Limits)
The limit exists if the limits along all possible paths exist and equal
There are infinitely many paths
2 paths with 2 different limiting values prove non-existence
Use the precise definition to prove the existence
The existence of the limit is independent of the existence and value of f(z0)

17. Definition
Let f be a function defined in a nhood of z0. Then f is continuous at z0 if

18. The Usual….
We are going to state the following standard results without going into the details.

19. Theorem

20. Theorem

21. Common Continuous Functions

22. Example 1

23. MAT 3730 Complex Variables Section 2.3 Analyticity

24. Preview We are going to look at Definition of Derivatives
Rules of Differentiation
Definition of Analytic Functions

25. Definition
Let f be a function defined on a nhood of z0. Then the derivative of f at z0 is given by
provided this limit exists.
f is said to be differentiable at z0

26. Example 2

27. (The Usual…) Rules of Differentiation

28. Example 3 (a)

29. Example 3 (b)

30. (The Usual…) The Chain Rule

31. Definition
A function f is said to be analytic on an open set G if it has a derivative at every point of G
f is analytic at a point z0 means
f is analytic on an nhood of a point z0

32. Example 4

33. Example 4

34. Next Class
Section 2.4