Limit, Continuity
(Complex Functions)
MATHEMATICS-II (20MA103T)
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Limit and Continuity of Complex Functions, Lecture notes of Mathematics
An introduction to the concepts of limit and continuity of complex functions. It covers the basics of complex numbers, including their arithmetic, magnitude, and polar form. The document then defines the limit and continuity of complex functions, and provides several examples to illustrate these concepts. Likely part of a mathematics course, specifically a course on complex analysis or advanced calculus. It could be useful for university students studying complex functions, as well as high school students interested in advanced mathematics topics. Important concepts that are foundational for further study in complex analysis, and could be used as study notes, lecture notes, or for exam preparation.
Typology: Lecture notes
2022/2023
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Limit, Continuity
(Complex Functions)
MATHEMATICS-II (20MA103T)
PRELIMINARIES
Motivation : If one wants to solve this type of equation , then we
have to extend the set of Real numbers to set of Complex
numbers.
Complex Numbers : or , where and are real numbers and
A point in the plane
(a,b) a b x y Real axis Imaginary axis
PRELIMINARIES (Cont…) (^) Magnitude: Magnitude of is denoted by and is defined as (a,b) Distance x y O P
PRELIMINARIES (Cont…)
Polar Form of Complex Numbers: (^) If z has polar coordinate then (^) The angle (OP makes with the positive ) is called the argument of. So, and (^) Then x y O P r θ a b
COMPLEX FUNCTIONS
- (^) , where and are some sets of complex numbers.
Limit (Definition):
Let be defined and single valued in a neighbourhood of Let be a complex number then, if and only if for given , there exists a positive number such that whenever We call the limit of as approaches. OR if the difference in the absolute value between and can be made arbitrarily small by choosing close enough to.
LIMIT(Cont…)
Example-1:
Example-2: (Hint: means and )
CONTINUITY
is continuous at if
- is defined at i.e. exists
- , i.e. the exists NOTE: If exists but not equal to , we call removable discontinuity since by redefining to be the same as the function becomes continuous.
CONTINUITY (Cont…)
- (^) Example-1: Test the continuity of the function at , if
- (^) Solution: Given and
Hence is continuous at.
CONTINUITY (Cont…)
Example-3: Test the continuity of the function at , if Solution: Given and , Put , (Limit does not exist) Hence is not continuous at.
CONTINUITY (Cont…)
Example-4: Test the continuity of the function at , if Solution: Given and , Put , (Limit does not exist) Hence is not continuous at.
Unsolved Examples: (Continuity)
Test the continuity of the following functions at , if
Ex-1: (Ans: continuous)