Schaum's Outline of Advanced Mathematics for Engineers and Scientists

Contents..................................................................  7
Chapter 1 Review of Fundamental Concepts.................................. 11
 Real Numbers.......................................................... 11
 Rules of Algebra...................................................... 11
 Functions............................................................. 12
 Special Types of Functions............................................ 12
 Limits................................................................ 13
 Continuity............................................................ 14
 Derivatives........................................................... 14
 Differentiation Formulas.............................................. 14
 Integrals............................................................. 15
 Integration Formulas.................................................. 15
 Sequences and Series.................................................. 16
 Uniform Convergence................................................... 17
 Taylor Series......................................................... 18
 Functions of Two or More Variables.................................... 18
 Partial Derivatives................................................... 18
 Taylor Series for Functions of Two or More Variables.................. 19
 Linear Equations and Determinants..................................... 19
 Maxima and Minima..................................................... 21
 Method of Lagrange Multipliers........................................ 21
 Leibnitz\'s Rule for Differentiating an Integral....................... 21
 Multiple Integrals.................................................... 21
 Complex Numbers....................................................... 21
Chapter 2 Ordinary Differential Equations................................. 48
 Definition of a differential equation................................. 48
 Order of a differential equation...................................... 48
 Arbitrary Constants................................................... 48
 Solution of a Differential Equation................................... 48
 Differential Equation of a Family of Curves........................... 49
 Special First Order Equations and Solutions........................... 49
 Equations of Higher Order............................................. 51
 Existence and Uniqueness of Solutions................................. 51
 Applications of Differential Equations................................ 51
 Some Special Applications............................................. 52
  Mechanics......................................................... 52
  Electric Circuits................................................. 52
  Orthogonal Trajectories........................................... 53
  Deflection of Beams............................................... 53
  Miscellaneous Problems............................................ 53
 Numerical Methods for Solving Differential Equations.................. 53
Chapter 3 Linear Differential Equations................................... 81
 General Linear Differential Equation of Order n....................... 81
 Existence and Uniqueness Theorem...................................... 81
 Operator Notation..................................................... 81
 Linear Operators...................................................... 82
 Fundamental Theorem on Linear differential Equations.................. 82
 Linear Dependence and Wronskians...................................... 82
 Solutions of Linear Equations with Constant Coefficients.............. 83
 Non-Operator Techniques............................................... 83
  The Complementary or Homogeneous Solution......................... 83
  The Particular Solution........................................... 83
 Operator Techniques................................................... 85
  Method of Reduction of Order...................................... 85
  Method of Inverse Operators....................................... 85
 Linear Equations with Variable Coefficients........................... 86
 Simultaneous Differential Equations................................... 87
 Applications.......................................................... 87
Chapter 4 Laplace Transforms..............................................108
 Definition of a Laplace Transform.....................................108
 Laplace Transforms of Some Elementary Functions.......................108
 Sufficient Conditions for Existence of Laplace Transforms.............109
 Inverse Laplace Transforms............................................109
 Laplace Transforms of Derivatives.....................................110
 The Unit Step Function................................................110
 Some Special Theorems on Laplace Transforms...........................111
 Partial Fractions.....................................................112
 Solutions of Differential Equations by Laplace Transforms.............112
 Applications to Physical Problems.....................................112
 Laplace Inversion Formulas............................................112
Chapter 5 Vector Analysis.................................................131
 Vectors and Scalars...................................................131
 Vector Algebra........................................................131
 Laws of Vector Algebra................................................132
 Unit Vectors..........................................................132
 Rectangular Unit Vectors..............................................132
 Components of a Vector................................................133
 Dot or Scalar Product.................................................133
 Cross or Vector Product...............................................134
 Triple Products.......................................................134
 Vector Functions......................................................135
 Limits, Continuity and Derivatives of Vector Functions................135
 Geometric Interpretation of a Vector Derivative.......................136
 Gradient, Divergence and Curl.........................................136
 Formulas Involving V..................................................137
 Orthogonal Curvilinear Coordinates. Jacobians.........................137
 Gradient, Divergence, Curl and Laplacian in Orthogonal Curvilinear....138
 Special Curvilinear Coordinates.......................................139
Chapter 6 Multiple, Line and Surface Integrals and Integral Theorems......157
 Double Integrals......................................................157
 Iterated Integrals....................................................157
 Triple Integrals......................................................158
 Transformations of Multiple Integrals.................................158
 Line Integrals........................................................159
 Vector Notation for Line Integrals....................................160
 Evaluation of Line Integrals..........................................160
 Properties of Line Integrals..........................................161
 Simple Closed Curves. Simply and Multiply-Connected Regions...........161
 Green\'s Theorem in the Plane..........................................161
 Conditions for a Line Integral to be Independent of the Path..........162
 Surface Integrals.....................................................163
 The Divergence Theorem................................................164
 Stokes\' Theorem.......................................................164
Chapter 7 Fourier Series..................................................192
 Periodic Functions....................................................192
 Fourier Series........................................................192
 Dirichlet Conditions..................................................193
 Odd and Even Functions................................................193
 Half Range Fourier Sine or Cosine Series..............................193
 Parseval\'s Identity...................................................194
 Differentiation and Integration of Fourier Series.....................194
 Complex Notation for Fourier Series...................................194
 Orthogonal Functions..................................................194
Chapter 8 Fourier Integrals...............................................211
 The Fourier Integral..................................................211
 Equivalent forms of Fourier\'s Integral Theorem........................211
 Fourier Transforms....................................................212
 Parseval\'s Identities for Fourier Integrals...........................212
 The Convolution Theorem...............................................213
Chapter 9 Gamma, Beta and Other Special Functions.........................220
 The Gamma Function....................................................220
 Table of Values and Graph of the Gamma Function.......................220
 Asymptotic Formula for T(n)...........................................221
 Miscellaneous Results Involving the Gamma Function....................221
 The Beta Function.....................................................221
 Dirichlet Integrals...................................................222
 Other Special Functions...............................................222
  Error Function....................................................222
  Exponential Integral..............................................222
  Sine Integral.....................................................222
  Cosine Integral...................................................222
  Fresnel Sine Integral.............................................222
  Fresnel Cosine Integral...........................................222
 Asymptotic Series or Expansions.......................................222
Chapter 10 Bessel Functions...............................................234
 Bessel\'s Differential Equation........................................234
 Bessel Functions of the First Kind....................................234
 Bessel Functions of the Second Kind...................................235
 Generating Function for J[sub(n)](x)..................................235
 Recurrence Formulas...................................................235
 Functions Related to Bessel Functions.................................236
  Hankel Functions of First and Second Kinds........................236
  Modified Bessel Functions.........................................236
  Ber, Bei, Ker, Kei Functions......................................236
 Equations Transformed into Bessel\'s Equation..........................236
 Asymptotic Formulas for Bessel Functions..............................237
 Zeros of Bessel Functions.............................................237
 Orthogonality of Bessel Functions.....................................237
 Series of Bessel Functions............................................237
Chapter 11 Legendre Functions and Other Orthogonal Functions..............252
 Legendre\'s Differential Equation......................................252
 Legendre Polynomials..................................................252
 Generating Function for Legendre Polynomials..........................252
 Recurrence Formulas...................................................252
 Legendre Functions of the Second Kind.................................253
 Orthogonality of Legendre Polynomials.................................253
 Series of Legendre Polynomials........................................253
 Associated Legendre Functions.........................................253
 Other Special Functions...............................................254
  Hermite Polynomials...............................................254
  Laguerre Polynomials..............................................254
 Sturm-Liouville Systems...............................................255
Chapter 12 Partial Differential Equations.................................268
 Some Definitions Involving Partial Differential Equations.............268
 Linear Partial Differential Equations.................................268
 Some Important Partial Differential Equations.........................269
  Heat Conduction Equation..........................................269
  Vibrating String Equation.........................................269
  Laplace\'s Equation................................................269
  Longitudinal Vibrations of a Beam.................................269
  Transverse Vibrations of a Beam...................................270
 Methods of Solving Boundary-Value Problems............................270
  General Solutions.................................................270
  Separation of Variables...........................................270
  Laplace Transform Methods.........................................271
Chapter 13 Complex Variables and Conformal Mapping........................296
 Functions.............................................................296
 Limits and Continuity.................................................296
 Derivatives...........................................................296
 Cauchy-Riemann Equations..............................................297
 Integrals.............................................................297
 Cauchy\'s Theorem......................................................297
 Cauchy\'s Integral Formulas............................................298
 Taylor\'s Series.......................................................298
 Singular Points.......................................................298
 Poles.................................................................298
 Laurent\'s Series......................................................299
 Residues..............................................................299
 Residue Theorem.......................................................299
 Evaluation of Definite Integrals......................................300
 Conformai Mapping.....................................................301
 Riemann\'s Mapping Theorem.............................................301
 Some General Transformations..........................................302
 Mapping of a Half Plane on to a Circle................................302
 The Schwarz-Christoffel Transformation................................303
 Solutions of Laplace\'s Equation by Conformal Mapping..................303
Chapter 14 Complex Inversion Formula for Laplace Transforms...............334
 The Complex Inversion Formula.........................................334
 The Bromwich Contour..................................................334
 Use of Residue Theorem in Finding Inverse Laplace Transforms..........334
 A Sufficient Condition for the Integral Around T to Approach Zero.....335
 Modification of Bromwich Contour in Case of Branch Points.............335
 Case of Infinitely Many Singularities.................................335
 Applications to Boundary-Value Problems...............................335
Chapter 15 Matrices.......................................................352
 Definition of a Matrix................................................352
 Some Special Definitions and Operations Involving Matrices............352
 Determinants..........................................................354
 Theorems on Determinants..............................................355
 Inverse of a Matrix...................................................356
 Orthogonal and Unitary Matrices.......................................356
 Orthogonal Vectors....................................................356
 Systems of linear Equations...........................................357
 Systems of n Equations in n Unknowns. Cramer\'s Rule...................357
 Eigenvalues and Eigenvectors..........................................358
 Theorems on Eigenvalues and Eigenvectors..............................359
Chapter 16 Calculus of Variations.........................................385
 Maximum or Minimum of an Integral.....................................385
 Euler\'s Equation......................................................385
 Constraints...........................................................386
 The Variational Notation..............................................386
 Generalizations.......................................................387
 Hamilton\'s Principle..................................................387
 Lagrange\'s Equations..................................................388
 Sturm-Liouville Systems and Rayleigh-Ritz Methods.....................388
Index.....................................................................409
 A.....................................................................409
 B.....................................................................409
 C.....................................................................409
 D.....................................................................410
 E.....................................................................411
 F.....................................................................411
 G.....................................................................412
 H.....................................................................412
 I.....................................................................412
 J.....................................................................413
 K.....................................................................413
 L.....................................................................413
 M.....................................................................413
 N.....................................................................414
 O.....................................................................414
 P.....................................................................414
 Q.....................................................................415
 R.....................................................................415
 S.....................................................................415
 T.....................................................................416
 U.....................................................................417
 V.....................................................................417
 W.....................................................................417
 X.....................................................................417
 Y.....................................................................417
 Z.....................................................................417