Scribd
Upload
387 views21 pages

Limits and Continuity

This document provides an overview of limits and continuity of complex functions. It defines the limit of a real function as values getting arbitrarily close to a real number L as the input gets closer to a point. For complex functions, the limit may not exist if the function approaches different values along different paths to the point. Continuity requires the limit to exist and equal the function value at the point. The document presents theorems on limits of real and imaginary parts, properties of limits and continuous functions, and examples demonstrating these concepts. It also discusses conditions for continuity and functions that are always continuous like polynomials.

Uploaded by

Abdullah Saeed
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
387 views21 pages

Limits and Continuity

This document provides an overview of limits and continuity of complex functions. It defines the limit of a real function as values getting arbitrarily close to a real number L as the input gets closer to a point. For complex functions, the limit may not exist if the function approaches different values along different paths to the point. Continuity requires the limit to exist and equal the function value at the point. The document presents theorems on limits of real and imaginary parts, properties of limits and continuous functions, and examples demonstrating these concepts. It also discusses conditions for continuity and functions that are always continuous like polynomials.

Uploaded by

Abdullah Saeed
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 21

Limits and

continuity

Dr. Hina Dutt


hina.dutt@seecs.edu.pk
SEECS-NUST
Advanced
Engineering
Mathematics (10th
• Chapter: 13
Edition) by Ervin • Sections: 13.3
Kreyszig

A First Course in
Complex Analysis
with Applications by
• Chapter: 2
Dennis G. Zill and • Section: 2.5
Patrick D. Shanahan.
Limit of a Real Function f(x)

The limit of a function 𝑓 given by


lim 𝑓(𝑥) = 𝐿
𝑥→𝑥0
means that values 𝑓(𝑥) of the function 𝑓 can be made arbitrarily
close to the real number 𝐿 if values of 𝑥 are chosen sufficiently close
to, but not equal to, the real number 𝑥0 .
Limit of a Real Function f(x)
Limit of a Complex Function
Nonexistence of a Real Limit
Criterion for the Nonexistence of a Limit
If 𝑓 approaches two complex numbers 𝐿1 ≠ 𝐿2
for two different curves or paths through 𝑧0 ,
then lim 𝑓(𝑧) does not exist.
𝑧→𝑧0

Note:
In general, computing values of lim 𝑓(𝑧) as 𝑧
𝑧→𝑧0
approaches 𝑧0 from different directions can
prove that a limit does not exist, but this
technique cannot be used to prove that a limit
does exist.
Example 1
𝑧
Show that lim ҧ does not exist.
𝑧→0 𝑧
Theorem 1: Real and Imaginary Parts of a Limit
Example 2

Compute the following limits:

a) lim (𝑧 2 + 𝑖)
𝑧→1+𝑖

Im(𝑧 2 )
b) lim
𝑧→3𝑖 𝑧+Re(𝑧)
Theorem 2: Properties of Complex Limits
Example 3
Compute the following limits:
3+𝑖 𝑧 4 −𝑧 2 +2𝑧
a) lim
𝑧+1
𝑧→𝑖

𝑧 2 −2𝑧+4
b) lim
𝑧→1+ 3𝑖 𝑧−1− 3𝑖

𝑖𝑧+1
c) lim
𝑧→∞ 2𝑧−𝑖
Continuity of a Function

A real valued function 𝑓 is continuous at a point 𝑥0 if


lim 𝑓(𝑥) = 𝑓 𝑥0 .
𝑥→𝑥0

A complex function 𝑓 is continuous at a point 𝑧0 if


lim 𝑓(𝑧) = 𝑓 𝑧0 .
𝑧→𝑧0
Criteria for Continuity at a Point
A complex function 𝑓 is continuous at a point 𝑧0 if each of the following
three conditions hold:
i. lim 𝑓(𝑧) exists,
𝑧→𝑧0
ii. 𝑓 is defined at 𝑧0 , and
iii. lim 𝑓(𝑧) = 𝑓 𝑧0 .
𝑧→𝑧0
Example 4
Theorem 3: Real and Imaginary Parts of a
Continuous Function

Suppose that 𝑓(𝑧) = 𝑢(𝑥, 𝑦) + 𝑖𝑣(𝑥, 𝑦) and 𝑧0 = 𝑥0 + 𝑖𝑦0 .


Then the complex function 𝑓 is continuous at the point 𝑧0 if and
only if both real functions 𝑢 and 𝑣 are continuous at the point
𝑥0 , 𝑦0 .
Example 5
Show that the function 𝑓 𝑧 = 𝑧ҧ is continuous on ℂ.
Theorem 4: Properties of Continuous
Functions
Continuity of Some Functions

Polynomial Functions:
Polynomial functions are continuous on the entire complex plane ℂ.

Rational functions:
Rational functions are continuous on their domains.
Continuous Function is Bounded

If a complex function 𝑓 is continuous on a closed and


bounded region 𝑅, then 𝑓 is bounded on 𝑅. That is, there
is a real constant 𝑀 > 0 such that
|𝑓(𝑧)| ≤ 𝑀 for all 𝑧 in 𝑅.
Practice Questions

A First Course in
Complex Analysis with • Chapter: 2
Applications by Dennis • Exercise: 2.6 Questions: 1-40
G. Zill and Patrick D.
Shanahan.

You might also like