Section 2.2 Limits and Continuity MAT 3730 Complex Variables Section 2.2 Limits and Continuity http://myhome.spu.edu/lauw

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)

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

Recall (Real Variables) Functions Limits Continuity Derivatives

Complex Variables Functions Limits Continuity Derivatives Analyticity

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

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

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.

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

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.

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

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)

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

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)

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)

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)

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

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

Theorem

Theorem

Common Continuous Functions

Example 1

MAT 3730 Complex Variables Section 2.3 Analyticity http://myhome.spu.edu/lauw

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

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

Example 2

(The Usual…) Rules of Differentiation

Example 3 (a)

Example 3 (b)

(The Usual…) The Chain Rule

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

Example 4

Example 4

Next Class Section 2.4