UNIT - 01 FUNDAMENTAL THEOREMS - LINE INTEGRALS - RECTIFIABLE ARCS - LINE INTEGRALS AS FUNCTIONS OF ARCS. UNIT - 02 CAVHY`S THEOREM FOR A RECTANGLE AND CAVCHY`S THEOREM FOR A CIRCULAR DISK - CAVCHY`S INTEGRAL FORMULA - THE INDEX OF A POINT WITH RESPECT TO A CLOSED CURVE - THE INTEGRAL FORMULA AND HIGHER DERIVATIVES. UNIT - 03 LOCAL PROPERTIES OF ANALYTIC FUNCTIONS - REMOVABLE SINGULARITIES - TAYLORS THERMO - ZEROS AND POLES - THE LOCAL MAPPING AND THE MAXIMUM PRINCIPLES. UNIT - 04 THE GENERAL FORM OF CAVCHY`S THEOREM - CHAINS AND CYCLES - SIMPLE CONNECTIVITY - EXACT DIFFERENTIALS IN SIMPLY CONNECTED REGIONS AND MULTIPLY CONNECTED REGIONS - THE CALCULUS OF RESIDUES - THE RESIDUE THEOREM, THE ARGUMENT PRINCIPLE AND THE EVALUATION OF DEFINITE INTEGRALS. UNIT - 05 HARMONIC FUNCTIONS - DEFINITIONS AND BASIC PROPERTIES THE MEAN-VALUE PROPERTY - POISSION`S FORMULA SCHWARY THEOREM AND THE REFLECTION PRINCIPLE. UNIT - 06 POWER SERIES EXPANSIONS - WEIERSTRASS`S THEOREM - THE TAYLOR SERIES AND THE LAURENT SERIES - PARTIAL FRACTIO NS AND FACTORIZATION - PARTIAL FRACTIONS. UNIT - 07 INFINITE RODUCTS AND CANONICAL PRODUCTS - ENTIRE FUNCTIONS - JENSEN`S FORMULA AND HADAMARD`S THEOREM. UNIT - 08 NORMAL FAMILIES - EQICONTINUITY - NORMALLY AND COMPACTNESS - ARZELA`S THEOREM - FAMILIES OF ANALYTIC FUNCTIONS AND THE CLASSICAL DEFINITION. UNIT - 09 THE RIEMANN MAPPING THEOREM - STATEMENT AND THE PROOF A CLOSER LOOK AT HARMONIC FUNCTIONS - FUNCTIONS WITH THE MEAN VALUE PROPERTY AND HARNACK`S PRINCIPLE. UNIT - 10 ELLIPTIC FUNCTION - SIMPLY PERIODIC FUNCTIONS AND DOUBLY PERIODIC FUNCTIONS - THE PERIODIC MODULE - UNOMIDULAR TRANSFORMATIONS. UNIT - 11 Vinayaka Missions University,Directorate of Distance Education Salem India MASTER OF SCIENCE IN MATHEMATICS 2 Yr. COMPLEX ANALYSIS(2030510) THE CANONICAL BASIS AND THE GENERAL PROPERTIES OF ELLIPTIC FUNCTIONS. UNIT - 12 THE WEIERSTRASS THEORY - THE WEIERSTRASS - FUNCTION, THE FUNCTION? (Z) AND? (Z) AND THE DIFFERENTIAL EQUATION.
UNIT - 01
FUNDAMENTAL THEOREMS - LINE INTEGRALS - RECTIFIABLE ARCS - LINE INTEGRALS AS
FUNCTIONS OF ARCS.
UNIT - 02
CAVHY`S THEOREM FOR A RECTANGLE AND CAVCHY`S THEOREM FOR A CIRCULAR
DISK - CAVCHY`S INTEGRAL FORMULA - THE INDEX OF A POINT WITH RESPECT TO A
CLOSED CURVE - THE INTEGRAL FORMULA AND HIGHER DERIVATIVES.
UNIT - 03
LOCAL PROPERTIES OF ANALYTIC FUNCTIONS - REMOVABLE SINGULARITIES - TAYLORS
THERMO - ZEROS AND POLES - THE LOCAL MAPPING AND THE MAXIMUM PRINCIPLES.
UNIT - 04
THE GENERAL FORM OF CAVCHY`S THEOREM - CHAINS AND CYCLES - SIMPLE
CONNECTIVITY - EXACT DIFFERENTIALS IN SIMPLY CONNECTED REGIONS AND
MULTIPLY CONNECTED REGIONS - THE CALCULUS OF RESIDUES - THE RESIDUE
THEOREM, THE ARGUMENT PRINCIPLE AND THE EVALUATION OF DEFINITE INTEGRALS.
UNIT - 05
HARMONIC FUNCTIONS - DEFINITIONS AND BASIC PROPERTIES THE MEAN-VALUE
PROPERTY - POISSION`S FORMULA SCHWARY THEOREM AND THE REFLECTION
PRINCIPLE.
UNIT - 06
POWER SERIES EXPANSIONS - WEIERSTRASS`S THEOREM - THE TAYLOR SERIES AND
THE LAURENT SERIES - PARTIAL FRACTIO NS AND FACTORIZATION - PARTIAL
FRACTIONS.
UNIT - 07
INFINITE RODUCTS AND CANONICAL PRODUCTS - ENTIRE FUNCTIONS - JENSEN`S
FORMULA AND HADAMARD`S THEOREM.
UNIT - 08
NORMAL FAMILIES - EQICONTINUITY - NORMALLY AND COMPACTNESS - ARZELA`S
THEOREM - FAMILIES OF ANALYTIC FUNCTIONS AND THE CLASSICAL DEFINITION.
UNIT - 09
THE RIEMANN MAPPING THEOREM - STATEMENT AND THE PROOF A CLOSER LOOK AT
HARMONIC FUNCTIONS - FUNCTIONS WITH THE MEAN VALUE PROPERTY AND
HARNACK`S PRINCIPLE.
UNIT - 10
ELLIPTIC FUNCTION - SIMPLY PERIODIC FUNCTIONS AND DOUBLY PERIODIC FUNCTIONS
- THE PERIODIC MODULE - UNOMIDULAR TRANSFORMATIONS.
UNIT - 11
Vinayaka Missions University,Directorate of Distance Education
Salem India
MASTER OF SCIENCE IN MATHEMATICS
2 Yr.
COMPLEX ANALYSIS(2030510)
THE CANONICAL BASIS AND THE GENERAL PROPERTIES OF ELLIPTIC FUNCTIONS.
UNIT - 12
THE WEIERSTRASS THEORY - THE WEIERSTRASS - FUNCTION, THE FUNCTION? (Z) AND?
(Z) AND THE DIFFERENTIAL EQUATION.
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