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# Study BSc - Complex analysis

UNIT I Complex number system, Complex number – Field of Complex numbers – Conjugation – Absolute value – Arguments Simple Mappings. i) w= z+a ii) w= az iii) w=1/z invariance of cross-ratio under bilincar transformation-Definition of extended complex planc – Stereographic projection. Complex functions : Limit of a function – continuity – differentiability- Analytical function defined in a region – necessary conditions for differentiability – sufficient conditions for differentiability – Cauchy- Riemann equation in polar coordinates- Definition of entire function. UNIT II Power Series : Absolute convergence – circle of convergence – Analyticity of the sum of power series in the Circle of convergence ( term by term differentiation of a series) Elementary functions : Exponential, Logarithmic, Trigonometric and Hyperbolic functions. Anx.18 D - B.Sc Maths (SDE) 2007-08 Page 10 of 26 Conjugate horonic functions : Definition and determination, Conformal Mappings : Isogonal mapping – Conformal mapping- Mapping z --- f (z),where f is analystic, particularly the mappings. w=ez ; w =z ½ ; w=sin z ; w=1/2 (z+1/z) UNIT III Complex Integration : Simply and multiply connected regions in the complex plane. Integration of f(z) from definition along a curve joining Z1 and Z2 . Proff of Cauchy’s Theorem ( using Goursat’s lemma for a simply connected region). Cauchy’s integral formula for higher derivatives ( statement only) – Morera’s theorem. Results based on Cauchy’s theorem (I) : Zero-Cauchy’s Inequality – Lioville’s theorem – Fundamental theorem of algebra Maximum modulus theorem Gauss mean value theorem Gauss mean value theorem for a harmonic function on a circle. UNIT IV Results based on Cauchy’s theorem (II) – Taylor’s series-Laurent’s series. Singularities and Residues: Isolated singularities ( Removable Singularity, pole and essential singularity ) Residues Residue theorem. UNIT V Real definite integrals : Evaluation using the calculus of residues- Integration on the unit circle- Integral with - and as lower and upper ,limits with the following integrals: I. P(x) / Q(x) where the degree of Q(x) exceeds that of P(x) at least. II. (sin ax ).f(x),(cos ax).f(x), where a>0 and f(z) ---- 0 as z ----- and f(z) does not have a pole on the real axis. III. f(x) where f(z) has a finite number of poles on the real axis. Integral of the type Meromorphic functions : Theorem on number of zeros minus number of poles – Principle of argument : Rouche’s throrem – theorem that a function which is mcromorphic in the extended plane is a rational function.

## BSc Subjects

 Applied Electronics Atomic Physics and Nuclear Physics Classical Algebra and Calculus CLIENT SERVER COMPUTING Climatology and Oceanography Complex Analysis DATA COMMUNICATION AND NETWORKS DATA STRUCTURES AND C PROGRAMMING Differential Equation Lap lace Transforms DIGITAL FUNDAMENTALS AND ARCHITECTURE Economic geography Geography of India Geomorphology Heat Thermodynamics Mechanics Properties of Matter and Sound Human geography INTERNET and JAVA PROGRAMMING MATHEMATICAL FOUNDATION FOR COMPUTER Mathematical Physics Mechanics Modern Algebra OBJECT ORIENTED PROGRAMMING WITH C++ Optics Electricity and Magnetism Principles of cartography Quantum Mechanics and Relativity Real Analysis SOFTWARE ENGINEERING Solid State Physics Statistics for geographers Statistics for Mathematics SYSTEM SOFTWARE AND OPERATING SYSTEM Trigonometry Vector Calculus and Analytical Geometry VISUAL PROGRAMMING WEB TECHNOLOGY Business Economics Business Management Business Organisation and Office Management Business Statistics Commercial Law and Company Law Company law and secretarial practice I Company law and Secretarial practice II Corporate Accounting Cost and Management Accounting Executive Communication Financial Accounting Industrial law Life Office Management Managerial Economics Mathematics for Business Principles of Accountancy Principles of Life Insurance Principles of Management Principles of Marketing Priniciples of Reinsurance Statistics

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