Book Description

Unit-1 1. Analytic Functions, Cauchy-Riemann Equations, Harmonic Functions 1-40 Complex Number System 1; Complex Numbers as Ordered Pairs 1; The Polar Form 1; Function of a Complex Variable 2; Single Valued Function(or Uniform Function) 2; Multiple-Valued Function(or Many-Valued Function) 3; Limit of a Function 3; Theorems on Limits 3; Continuity 3; Fundamental Operations as Applied to Continuous Function 4; Continuity in Terms of Real and Imaginary Parts of f(z) 4; Uniform Continuity 4; Differentiability of a Complex Function 5; Geometric Interpretation of the Derivative 5; Partial Derivative 6; Analytic Function 6; The Necessary Conditions for f(z) to be Analytic [(Cauchy-Riemann Equations (C-R Equations)] 6; The Sufficient Condition for f(z) to be Analytic 8; Polar Form of Cauchy-Riemann Equations 9; Derivative of w in Polar Form 11; Functions of a Function 12; Derivative of a Function of a Function 12; Inverse Function 12; Laplace Equation 13; Harmonic Function 13; Theorem 13; Conjugate Harmonic Functions 14; Theorem 14; Determination of the Conjugate Function 14; To Construct a Function f(z) when One Conjugate Function is Given 15; Orthogonal System 16; Theorem 16. 2. Mobius Transformation, Cross Ratio 41-68 Elementary Functions 41; Mapping or Transformation 43; Definition of Mapping 43; Mobius Transformation or Bilinear Transformation or Fractional Transformation 43; Inverse Transformation 44; Critical Points and Critical Mapping 44; Resultant or Product of Two Mobius Transformations (Group Property) 45; Some Theorems 46; Fixed Points (or Invariant Points) of Mobius Transformation 47; Theorem 48; Cross Ratio 48; Some Theorems 49; The Circle 54; Inverse Points with Respect to a Circle 55; To find the Relation between the Inverse Points with Respect to the Circle 55; Nature of Transformations (Elliptic, Hyperbolic and Parabolic Transformations) 56; Some Theorems 58. 3. Vector Space 69-123 Vector Space 69; Various Notations 70; General Properties (Elementary Properties) of Vector Spaces 70; Vector Subspace 77; Union and Intersection of Subspaces 83; Sum of Subspaces 85; Some Definitions 91; Basis of a Vector Space 91; Dimension of a Vector Space 92; Finite Dimensional Vector Space 102; Some Theorems on Finite Dimensional Spaces 102; Quotient Space 115. 4. Linear Transformation 124-158 Definition 124; Purpose 124; Image 124; Existence and Uniqueness 124; Types of Linear Transformation 127; Determining whether a Mapping is Linear Transformation or Not 127; Isomorphism of Vector Spaces 133; Theorems on Isomorphism 134; Kernel of Linear Transformation T or Kernel of a Homomorphism T 142; Theorem 142; Range of a Linear Transformation 143; Theorem 143; Lemma 144; Sylvester Law of Nullity [Rank-Nullity Theorem] 144; Fundamental Theorem of Vector Space Homomorphism 146. 5. Inner Product Spaces 159-200 Inner Product 159; Usual or Standard Inner Product 159; Inner Product Space 162; Theorems 162; Some Important Terms about Vectors 168; Norm or Length of a Vector a in an Inner Product Space 168; Theorems 169; Orthogonality 170; Orthogonal Complement 171; Orthogonal Basis and Orthonormal Basis 172; Gram-Schmidt Orthogonalization Process 172; Theorems on Orthogonal/ Orthonormal Bases 173; Cauchy-Schwarz’s Inequality or Schwarz’s Inequality 186; Bessel’s Inequality 188; Normed Vector Space or Normal Vector Space 194; Distance in an Inner Product Space 195.