Syllabus
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Course Code: MT-601 Course Name: (B) Complex Analysis |
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| MODULE NO / UNIT | COURSE SYLLABUS CONTENTS OF MODULE | NOTES |
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| 1 | Complex numbers and their representation, algebra of complex numbers; Complex plane, Open set, Domain and region in complex plane; Stereographic projection and Riemann sphere. De Moivre’s Theorem and its Applications. Expansion of trigonometrical functions. Direct circular and hyperbolic functions and their properties, Logarithm of a complex quantity, Summation of Trigonometric series. | |
| 2 | Analytic Functions and Cauchy-Riemann Equations Complex functions and their limits including limit at infinity; Continuity and differentiability of a complex valued function. Analytic functions; Cauchy-Riemann equations, Harmonic functions, necessary and sufficient conditions for differentiability. Analyticity and zeros of exponential, trigonometric and logarithmic functions. |
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| 3 | Line integral, Path independence. Branch cut and branch of multi-valued functions. Complex integration, Green’s theorem, Anti-derivative theorem, Cauchy-Goursat theorem, Cauchy integral formula, Cauchy’s inequality, Derivative of analytic function, Liouville’s theorem, Fundamental theorem of algebra, Maximum modulus theorem and its consequences. | |
| 4 | Sequences, series and their convergence, Taylor series and Laurent series of analytic functions, Power series, Radius of convergence, Integration and differentiation of power series, Absolute and uniform convergence of power series. | |
