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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
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