Core-13 Metric Spaces and Complex Analysis ,Bsc 6th sem Mathematics Dibrugarh University CBCS ,Syllabus
C6.1
Metric Spaces and Complex Analysis
Total Marks: 100, Theory: 80, IA: 20,
Credit: 5+1=6;
(L=5, P=0, T=1)
Objectives: After going through this course the students will be
able to describe
• various properties of
metrics paces
• complex number system, its
differentiation and integration.
Unit-1
Marks: 20, Contact hrs:20
Metric spaces: definition and examples. Sequences in metric
spaces, Cauchy sequences. Complete Metric Spaces.
Open and closed balls, neighbourhood, open set, interior of a
set. Limit point of a set, closed set, diameter of a set,
Cantor’s theorem. Subspaces,
dense sets, separable spaces.
Unit-2
Marks:
15, Contact hrs:15
ontinuous mappings, sequential criterion and other
characterizations of continuity. Uniform continuity.
Homeomorphism, Contraction mappings, compactness Banach Fixed
point Theorem. Connectedness, connected subsets of R.
Unit-3
Marks: 15, Contact hrs: 20
Limits, Limits involving the point at infinity, continuity. Properties
of complex numbers, regions in the complex
plane, functions of complex variable, mappings. Derivatives,
differentiation formulas, Cauchy-Riemann equations,
sufficient conditions for differentiability.
Unit-4 Marks:
12, Contact hrs:15
Analytic functions, examples of analytic functions, exponential
function, Logarithmic function, trigonometric
function, derivatives of functions, definite integrals of functions.
Contours, Contour integrals and its examples,upper bounds for moduli of contour
integrals. Cauchy- Goursat theorem, Cauchy integralformula
.
Unit-5 Marks: 10,
Contact
hrs:10
Liouville’s theorem and the
fundamental theorem of algebra. Convergence of sequences and series, Taylor
series and its examples.
Unit-6 Marks:
08,
Contact hrs:10
Laurent series and its examples, absolute and uniform
convergence of power series.
Text Books:
1. S. Kumaresan, Topology of Metric
Spaces, 2nd Ed., Narosa
Publishing House,2011.
2. G.F. Simmons, Introduction to
Topology and Modern Analysis, McGraw-Hill,2004.
3. J. W. Brown and R. V.
Churchill, Complex Variables
and Applications, 8th Ed., McGraw – Hill
International Edition,2009.
Reference Books:
1. S. Shirali and H. L.
Vasudeva, Metric Spaces, Springer Verlag,
London, 2006.
2. J. Bak and D. J. Newman, Complex Analysis, 2nd Ed., Undergraduate
Texts in Mathematics, Springer-Verlag
RECOMENDED YOUR TEXT BOOK IF ANY BUY
New York, Inc., New York,1997.
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