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Mathematics · 300 Level
This is the complete lecture note for MTH 304 Complex Analysis I, prepared for Mathematics students at 300 Level. You can read it online below, open it in the full viewer, or download the PDF for offline study. It pairs with the course's video lectures, so you can read and watch together.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Outline for Series 1 Part 1.1: Functions of a Complex Variable • Definition of a complex variable • Functions of a complex variable • Examples of simple complex functions Part 1.2: Limits and Continuity of Complex Functions • Definition of limits in the complex plane • Continuity of complex functions • Comparison with real functions Part 1.3: Analytic Functions • Definition of analytic functions • Properties of analytic functions • Examples of analytic and non-analytic functions Part 1.4: Introduction to Differentiability in Complex Analysis • Concept of differentiability in the complex plane • Relationship between differentiability and analyticity • Simple examples Introduction Complex numbers extend the real number system and open up a rich field of study known as complex analysis. In this Series, we begin our journey into the subject by focusing on the fundamental building blocks: functions of a complex variable, their limits, continuity, and the concept of analyticity. These ideas are crucial because they form the foundation upon which more advanced topics such as conformal mapping, contour integration, and series expansions are built. Understanding them will enable you to appreciate how complex functions behave and why they are so powerful in mathematics and applied sciences. The aim of this Series is to introduce you to the basic concepts of functions of a complex variable, to explain how limits and continuity are defined in the complex plane, and to establish the meaning and importance of analytic functions.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS We shall commence this Series by defining functions of a complex variable and examining simple examples. Next, we will explore the notions of limits and continuity in the complex plane, comparing them with their real counterparts. Following that, we will study analytic functions, highlighting their properties and significance. Finally, we will wrap up by considering differentiability in complex analysis and its relationship to analyticity, thereby rounding off the Series with a clear understanding of these foundational concepts. Series Learning Outcomes Upon completing this Series, you should be able to: • SLO 1.1 define functions of a complex variable and illustrate them with examples, • SLO 1.2 explain the concept of limits and continuity for complex functions, • SLO 1.3 describe analytic functions and identify examples of analytic and non-analytic functions, and • SLO 1.4 analyse the relationship between differentiability and analyticity in complex analysis. 1.1 Functions of a Complex Variable A complex variable is a variable that can take values in the complex plane, expressed as z=x+iy, where xand yare real numbers and iis the imaginary unit defined by i2=-1. A function of a complex variable is a rule that assigns to each complex number zanother complex number w. For example, f(z)=z2is a function that maps each complex number to its square. Fill in the blank: A complex variable is expressed as z = x + \, __________. 1.1.1 Definition of functions of a complex variable Formally, if Dis a subset of the complex plane, then a function fdefined on Dis a mapping: f:D→C,z↦f(z) This means that for each z∈D, there is a unique complex number f(z). Key term: A domain is a set of complex numbers on which a function is defined. 1.1.2 Examples of simple complex functions • Polynomial functions: f(z)=z2+2z+1
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS • Exponential functions: f(z)=ez • Trigonometric functions: f(z)=sin(z), f(z)=cos(z) • Rational functions: f(z)=1z These examples illustrate that many familiar real functions extend naturally to the complex plane. Fill in the blank: The function f(z)=1zis an example of a __________ function. 1.1.3 Graphical representation Functions of a complex variable can be visualised by mapping points in the complex plane to other points. For instance, the function f(z)=z2maps circles centred at the origin to other circles, but with radii squared. Mapping of circles under the function f(z)=z2. Pilot questions 1.1 1. A complex variable is expressed as: A. z=x+iyB. z=x-yC. z=x+yD. None of the above (SLO 1.1)
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS 2. A function of a complex variable is a mapping from: A. Real numbers to real numbers B. Complex numbers to complex numbers C. Real numbers to complex numbers D. None of the above (SLO 1.1) 3. The function f(z)=z2+2z+1is an example of: A. Polynomial function B. Rational function C. Trigonometric function D. None of the above (SLO 1.1) 4. The function f(z)=1zis an example of: A. Rational function B. Polynomial function C. Exponential function D. None of the above (SLO 1.1) 5. The imaginary unit iis defined by: A. i2=-1B. i2=1C. i2=0D. None of the above (SLO 1.1) 1.2 Limits and Continuity of Complex Functions The concepts of limits and continuity in complex analysis are extensions of those in real analysis, but with important differences due to the two-dimensional nature of the complex plane. A limit describes the behaviour of a function as the input approaches a particular point, while continuity ensures that the function does not have sudden jumps or breaks at that point. Fill in the blank: A limit describes the behaviour of a function as the input approaches a particular __________. 1.2.1 Definition of limits in the complex plane For a function f(z), we say that the limit of f(z)as zapproaches z0is Lif, for every ϵ>0, there exists a δ>0such that whenever ∣z-z0∣<δ, it follows that ∣f(z)-L∣<ϵ. Key term: A limit in complex analysis is the value that a function approaches as the input variable tends to a specific point in the complex plane. Fill in the blank: In the definition of a limit, the condition ∣z-z0∣<δensures that zis sufficiently __________ to z0. 1.2.2 Continuity of complex functions A function f(z)is said to be continuous at a point z0if: limz→z0f(z)=f(z0) This means the value of the function at z0matches the limit as zapproaches z0. Key term: Continuity ensures that a function has no sudden jumps or breaks at a given point.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS 1.2.3 Comparison with real functions In real analysis, limits are approached from two directions (left and right). In complex analysis, however, limits must be the same regardless of the path taken in the complex plane. This makes continuity and limits in complex analysis more restrictive. Limits in complex analysis must be the same along all paths. Fill in the blank: In complex analysis, limits must be the same regardless of the __________ taken in the plane. Pilot questions 1.2 1. A limit describes the behaviour of a function as the input approaches: A. A particular point B. Infinity only C. Zero only D. None of the above (SLO 1.2) 2. In the definition of a limit, the condition ∣z-z0∣<δensures that: A. zis sufficiently close to z0B. zis far from z0C. zequals z0exactly D. None of the above (SLO 1.2)
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS 3. A function is continuous at z0if: A. limz→z0f(z)=f(z0)B. limz→z0f(z)=0C. limz→z0f(z)=D. None of the above (SLO 1.2) 4. In real analysis, limits are approached from: A. Two directions (left and right) B. Only one direction C. Infinite directions D. None of the above (SLO 1.2) 5. In complex analysis, limits must be the same regardless of the: A. Path taken in the plane B. Axis chosen C. Function chosen D. None of the above (SLO 1.2) 1.3 Analytic Functions An analytic function is a function of a complex variable that is differentiable at every point in its domain. Differentiability in the complex plane is a stronger condition than in the real case, because the derivative must exist regardless of the direction from which the point is approached. Analytic functions are central to complex analysis, as they exhibit remarkable properties such as infinite differentiability and representation by power series. Fill in the blank: An analytic function is one that is __________ at every point in its domain. 1.3.1 Definition of analytic functions A function f(z)is said to be analytic at a point z0if it is differentiable in some neighbourhood of z0. If it is analytic at every point in a domain D, then f(z)is called analytic in D. Key term: A neighbourhood of a point is a small region around that point within which certain properties of the function are considered. 1.3.2 Properties of analytic functions Analytic functions have several important properties: • They are infinitely differentiable within their domain. • They can be represented by a power series expansion. • They obey the Cauchy–Riemann equations, which are necessary conditions for differentiability. Fill in the blank: Analytic functions can be represented by a __________ series expansion.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS 1.3.3 Examples of analytic and non-analytic functions • Analytic functions: • f(z)=z2 • f(z)=ez • f(z)=sin(z), cos(z) • Non-analytic functions: • f(z)=z‾(the complex conjugate of z) • f(z)=∣z∣
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