Complex Analysis
Course Objectives: This course aims to introduce the basic ideas of analysis for complex
functions in complex variables with visualization through relevant Practicals. Particular
emphasis has been laid on Cauchy’s theorems, series expansions and calculation of residues.
Course Learning Outcomes: The completion of the course will enable the students to:
i) Understand the significance of differentiability of complex functions leading to the
understanding of Cauchy-Riemann equations.
ii) Evaluate the contour integrals and understand the role of Cauchy-Goursat theorem
and the Cauchy integral formula.
iii) Expand some simple functions as their Taylor and Laurent series, classify the nature
of singularities, find residues and apply Cauchy Residue theorem to evaluate
integrals.
iv) Represent functions as Taylor, power and Laurent series, classify singularities and
poles, find residues and evaluate complex integrals using the residue theorem.
Course Objectives: This course aims to introduce the basic ideas of analysis for complex
functions in complex variables with visualization through relevant Practicals. Particular
emphasis has been laid on Cauchy’s theorems, series expansions and calculation of residues.
Course Learning Outcomes: The completion of the course will enable the students to:
i) Understand the significance of differentiability of complex functions leading to the
understanding of Cauchy-Riemann equations.
ii) Evaluate the contour integrals and understand the role of Cauchy-Goursat theorem
and the Cauchy integral formula.
iii) Expand some simple functions as their Taylor and Laurent series, classify the nature
of singularities, find residues and apply Cauchy Residue theorem to evaluate
integrals.
iv) Represent functions as Taylor, power and Laurent series, classify singularities and
poles, find residues and evaluate complex integrals using the residue theorem.
Course Contents: Unit 1: Analytic functions [09 Lectures] 1.1 Functions of a Complex Variables 1.2 Limits, Theorems on limits (Without Proof), Limitsinvolving the point at infinity, Continuity, Derivatives, Differentiation formulas (WithoutProof) 1.3 Cauchy- Riemann Equations, Sufficient Conditions for differentiability (OnlyStatement and Examples) 1.4 Polar coordinates, Analytic functions, Harmonic functions. Unit 2: Elementary Functions [07 Lectures] 2.1 The Exponential functions 2.2 The Logarithmic function, Branches and derivatives oflogarithms, Some identities involvinglogarithms 2.3 Complex exponents, Trigonometricfunctions. Unit 3. Integrals [11 Lectures] 3.1 Derivatives of functions, Definite integrals of functions 3.2 Contours, Contour integral,Examples 3.3 Upper bounds for Moduli of contour integrals, Anti-derivatives (Only Examples) 3.4 Cauchy-GoursatTheorem (without proof), Simply and multiply Connected domains.Cauchy integral formula, Derivatives of analytic functions. Liouville’s Theorem andFundamental Theorem of Algebra (Without Proof). Unit 4. Series [04 Lectures] 4.1 Convergence of sequences and series (Theorems without proof) 4.2 Taylor’s series (without proof), Laurent series (withoutproof), examples only. Unit 5. Residues and Poles [05 Lectures] 5.1 Isolated singular points, Residues 5.2 Cauchy residue theorem (Without Proof), residue atinfinity, types of isolated singular points,residues at poles 5.3 Zeros of analytic functions, zerosand poles.