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KTU B.Tech S3 Lecture Note – LINEAR ALGEBRA AND COMPLEX ANALYSIS

S3 Lecture Note  LINEAR ALGEBRA  COMPLEX ANALYSIS

MODULE I

  • Complex differentiation

 Limit, continuity and derivative of complex functions

Analytic Functions

Cauchy–Riemann Equation

Harmonic functions, Harmonic Conjugate

MODULE II

Geometry of Analytic functions Conformal Mapping,

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Circles and straight lines, extended complex plane, fixed points Special linear fractional Transformations,

Cross Ratio, Cross Ratio property-Mapping of disks and half planes Conformal mapping by w

Conformal mapping by w=  sinz & w = cos z

 

MODULE III

  • Complex Integration

Cauchy’s Integral Theorem

Cauchy’s Integral Formula- Derivatives of Analytic Functions

Taylor and Maclaurin series

Laurent’s series 

MODULE IV

  • Residue Integration

Singularities, Zeros, Poles, Essential singularity, Zeros of analytic functions.

Residue Integration Method, Formulas for Residues.

Several singularities inside the contour Residue  Theorem.

MODULE V

  • Linear system of Equations

Linear systems of Equations,

Coefficient Matrix, Augmented Matrix Gauss Elimination and back substitution,  Elementary row operations.

 Linear independence-rank of a matrix Vector Space-Dimension-basis-vector spaceR 3

Solution of linear systems, Fundamental theorem of non homogeneous linear s/m ,Homogeneous linear s/m

MODULE VI

  • Matrix Eigen value Problem

Determination of Eigen values and Eigen vectors-Eigen space

Symmetric, Skew Symmetric and Orthogonal matrices –simple properties

Basis of Eigen vectors- Similar matrices Diagonalization of a matrixQuadratic forms- Principal axis theorem

 

S3 Lecture Note  LINEAR ALGEBRA  COMPLEX ANALYSIS

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