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Mathematics · 300 Level
This is the complete lecture note for MTH 305 Complex Analysis II, 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 Series 1 Laurent Expansions, Isolated Singularities, and Residues Series Outline Part 1.1: Laurent Expansions Definition of Laurent series Derivation and examples Region of convergence Part 1.2: Isolated Singularities Definition and classification (removable, pole, essential) Behaviour near singularities Worked examples Part 1.3: Residues Definition of residue Methods of calculating residues (direct computation, limit method, coefficient extraction) Illustrative examples Part 1.4: Applications of Residues in Complex Functions Connection between residues and singularities Role of residues in contour integration Preliminary applications leading into the residue theorem Introduction In everyday mathematics, we often encounter functions that behave smoothly across their domains. However, in complex analysis, certain functions exhibit points where they fail to be analytic, known as singularities. To study such behaviour, mathematicians developed the Laurent expansion, a powerful tool that allows us to represent functions near these points using series that include both positive and negative powers. This expansion provides a way to analyse and classify singularities, and to compute important quantities called residues, which play a central role in evaluating complex integrals.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS The aim of this Series is to introduce you to Laurent expansions, isolated singularities, and residues, and to equip you with the skills needed to compute and interpret them in the study of complex functions. We shall commence this Series by examining the concept of Laurent expansions, including their definition, derivation, and regions of convergence. Next, we will explore isolated singularities, considering their classification into removable singularities, poles, and essential singularities. Following that, we will study residues, focusing on their definition and methods of calculation. Finally, we will conclude by discussing the applications of residues in complex functions, preparing the ground for the residue theorem in the next Series. Series Learning Outcomes Upon completing this Series, you should be able to: SLO 1.1 define the Laurent expansion and explain how it represents complex functions near singularities, SLO 1.2 analyse the region of convergence of a Laurent series and demonstrate its application with examples, SLO 1.3 classify isolated singularities into removable singularities, poles, and essential singularities, and explain their behaviour, SLO 1.4 define the concept of a residue and demonstrate methods of calculating residues using direct computation, limits, and coefficient extraction, SLO 1.5 evaluate the role of residues in complex functions and apply them to preliminary problems involving contour integration. 1.1 Laurent Expansions The Laurent expansion is a representation of a complex function as a series that includes both positive and negative powers of the variable. Unlike the Taylor series, which only involves non-negative powers, the Laurent series allows us to study functions near points where they are not analytic, known as singularities. This makes it a powerful tool in complex analysis. Fill in the blank: A Laurent series differs from a Taylor series because it includes __________ powers of the variable. 1.1.1 Definition of Laurent series A Laurent series is defined as an infinite series of the form:
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS f(z)=n=-an(z-z0)n where z0is the centre of expansion and the coefficients anare complex numbers. The terms with negative powers are called the principal part, while those with non-negative powers form the regular part. Structure of a Laurent series expansion Fill in the blank: The terms with negative powers in a Laurent series are called the __________ part.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS 1.1.2 Derivation and examples Laurent expansions can be derived using contour integration or by manipulating known series expansions. For example, the function f(z)=1z(1-z) can be expanded around z=0as: f(z)=1z+1+z+z2+z3+⋯ Here, the term 1zbelongs to the principal part, while the remaining terms form the regular part. Example of Laurent expansion around z = 0
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Fill in the blank: In the expansion of f(z)=1z(1-z), the term 1zbelongs to the __________ part. 1.1.3 Region of convergence The region of convergence of a Laurent series is an annulus (ring-shaped region) around the point of expansion. Within this region, the series converges absolutely and uniformly. The inner radius corresponds to the nearest singularity inside, and the outer radius corresponds to the nearest singularity outside. Region of convergence of a Laurent series
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Fill in the blank: The region of convergence of a Laurent series is shaped like an __________ around the point of expansion. Pilot questions 1.1 A Laurent series differs from a Taylor series because it includes: A. Only positive powers B. Negative powers C. No powers D. None of the above (Linked to SLO 1.1) The terms with negative powers in a Laurent series are called the: A. Regular part B. Principal part C. Convergent part D. None of the above (Linked to SLO 1.1) In the expansion of f(z)=1z(1-z), the term 1zbelongs to the: A. Regular part B. Principal part C. Convergent part D. None of the above (Linked to SLO 1.2) The region of convergence of a Laurent series is shaped like: A. A circle B. An annulus C. A rectangle D. None of the above (Linked to SLO 1.2) The coefficients of a Laurent series are: A. Real numbers only B. Complex numbers C. Integers D. None of the above (Linked to SLO 1.1) 1.2 Isolated Singularities In complex analysis, certain points exist where a function ceases to be analytic. These points are called isolated singularities, defined as points where a function is not analytic but remains analytic in some punctured neighbourhood around them. Studying isolated singularities is crucial because they determine the behaviour of functions near problematic points and form the basis for residue theory. Fill in the blank: An isolated singularity is a point where a function is not analytic but remains analytic in a __________ neighbourhood around it. 1.2.1 Definition and classification There are three main types of isolated singularities: Removable singularity: A point where the function is undefined, but can be redefined to make it analytic. Pole: A point where the function tends to infinity as the variable approaches it. Poles are classified by order, depending on how rapidly the function diverges.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Essential singularity: A point where the function exhibits chaotic behaviour, with values clustering densely around the complex plane near the singularity. Classification of isolated singularities Fill in the blank: A point where a function tends to infinity as the variable approaches it is called a __________.
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