
Infinite Series, Power Series 


1  (45) 


1  (3) 


4  (2) 


6  (1) 

Convergent and Divergent Series 


6  (3) 

Testing Series for Convergence; the Preliminary Test 


9  (1) 

Convergence Tests for Series of Positive Terms: Absolute Convergence 


10  (7) 


10  (1) 


11  (2) 


13  (2) 

A Special Comparison Test 


15  (2) 


17  (1) 

Conditionally Convergent Series 


18  (1) 

Useful Facts About Series 


19  (1) 

Power Series; Interval of Convergence 


20  (3) 

Theorems About Power Series 


23  (1) 

Expanding Functions in Power Series 


23  (2) 

Techniques for Obtaining Power Series Expansions 


25  (8) 

Multiplying a Series by a Polynomial or by Another Series 


26  (1) 

Division of Two Series or of a Series by a Polynomial 


27  (1) 


28  (1) 

Substitution of a Polynomial or a Series for the Variable in Another Series 


29  (1) 


30  (1) 

Taylor Series Using the Basic Maclaurin Series 


30  (1) 


31  (2) 

Accuracy of Series Approximations 


33  (3) 


36  (8) 


44  (2) 


46  (36) 


46  (1) 

Real and Imaginary Parts of a Complex Number 


47  (1) 


47  (2) 


49  (2) 


51  (5) 


51  (1) 

Complex Conjugate of a Complex Expression 


52  (1) 

Finding the Absolute Value of z 


53  (1) 


54  (1) 


54  (1) 


55  (1) 


56  (2) 

Complex Power Series; Disk of Convergence 


58  (2) 

Elementary Functions of Complex Numbers 


60  (1) 


61  (3) 

Powers and Roots of Complex Numbers 


64  (3) 

The Exponential and Trigonometric Functions 


67  (3) 


70  (2) 


72  (1) 


73  (1) 

Inverse Trigonometric and Hyperbolic Functions 


74  (2) 


76  (4) 


80  (2) 


82  (106) 


82  (1) 


83  (6) 

Determinants; Cramer's Rule 


89  (7) 


96  (10) 


106  (8) 


114  (10) 

Linear Combinations, Linear Functions, Linear Operators 


124  (8) 

Linear Dependence and Independence 


132  (5) 

Special Matrices and Formulas 


137  (5) 


142  (6) 

Eigenvalues and Eigenvectors; Diagonalizing Matrices 


148  (14) 

Applications of Diagonalization 


162  (10) 

A Brief Introduction to Groups 


172  (7) 


179  (5) 


184  (4) 


188  (53) 

Introduction and Notation 


188  (3) 

Power Series in Two Variables 


191  (2) 


193  (3) 

Approximations using Differentials 


196  (3) 

Chain Rule or Differentiating a Function of a Function 


199  (3) 


202  (1) 


203  (8) 

Application of Partial Differentiation to Maximum and Minimum Problems 


211  (3) 

Maximum and Minimum Problems with Constraints; Lagrange Multipliers 


214  (9) 

Endpoint or Boundary Point Problems 


223  (5) 


228  (5) 

Differentiation of Integrals; Leibniz' Rule 


233  (5) 


238  (3) 


241  (35) 


241  (1) 

Double and Triple Integrals 


242  (7) 

Applications of Integration; Single and Multiple Integrals 


249  (9) 

Change of Variables in Integrals; Jacobians 


258  (12) 


270  (3) 


273  (3) 


276  (64) 


276  (1) 

Applications of Vector Multiplication 


276  (2) 


278  (7) 

Differentiation of Vectors 


285  (4) 


289  (1) 

Directional Derivative; Gradient 


290  (6) 

Some Other Expressions Involving 


296  (3) 


299  (10) 

Green's Theorem in the Plane 


309  (5) 

The Divergence and the Divergence Theorem 


314  (10) 

The Curl and Stokes' Theorem 


324  (12) 


336  (4) 

Fourier Series and Transforms 


340  (50) 


340  (1) 

Simple Harmonic Motion and Wave Motion; Periodic Functions 


340  (5) 

Applications of Fourier Series 


345  (2) 

Average Value of a Function 


347  (3) 


350  (5) 


355  (3) 

Complex Form of Fourier Series 


358  (2) 


360  (4) 


364  (8) 


372  (3) 


375  (3) 


378  (8) 


386  (4) 

Ordinary Differential Equations 


390  (82) 


390  (5) 


395  (6) 

Linear FirstOrder Equations 


401  (3) 

Other Methods for FirstOrder Equations 


404  (4) 

SecondOrder Linear Equations with Constant Coefficients and Zero RightHand Side 


408  (9) 

SecondOrder Linear Equations with Constant Coefficients and RightHand Side Not Zero 


417  (13) 

Other SecondOrder Equations 


430  (7) 


437  (3) 

Solution of Differential Equations by Laplace Transforms 


440  (4) 


444  (5) 


449  (12) 

A Brief Introduction to Green Functions 


461  (5) 


466  (6) 


472  (24) 


472  (2) 


474  (4) 


478  (4) 

The Brachistochrone Problem; Cycloids 


482  (3) 

Several Dependent Variables; Lagrange's Equations 


485  (6) 


491  (2) 


493  (1) 


494  (2) 


496  (41) 


496  (2) 


498  (4) 

Tensor Notation and Operations 


502  (3) 


505  (3) 

Kronecker Delta and LeviCivita Symbol 


508  (6) 

Pseudovectors and Pseudotensors 


514  (4) 


518  (3) 


521  (4) 

Vector Operators in Orthogonal Curvilinear Coordinates 


525  (4) 


529  (6) 


535  (2) 


537  (25) 


537  (1) 


538  (1) 

Definition of the Gamma Function; Recursion Relation 


538  (2) 

The Gamma Function of Negative Numbers 


540  (1) 

Some Important Formulas Involving Gamma Functions 


541  (1) 


542  (1) 

Beta Functions in Terms of Gamma Functions 


543  (2) 


545  (2) 


547  (2) 


549  (3) 


552  (2) 

Elliptic Integrals and Functions 


554  (6) 


560  (2) 

Series Solutions of Differential Equations; Legendre, Bessel, Hermite, and Laguerre Functions 


562  (57) 


562  (2) 


564  (3) 

Leibniz' Rule for Differentiating Products 


567  (1) 


568  (1) 

Generating Function for Legendre Polynomials 


569  (6) 

Complete Sets of Orthogonal Functions 


575  (2) 

Orthogonality of the Legendre Polynomials 


577  (1) 

Normalization of the Legendre Polynomials 


578  (2) 


580  (3) 

The Associated Legendre Functions 


583  (2) 

Generalized Power Series or the Method of Frobenius 


585  (2) 


587  (3) 

The Second Solution of Bessel's Equation 


590  (1) 

Graphs and Zeros of Bessel Functions 


591  (1) 


592  (1) 

Differential Equations with Bessel Function Solutions 


593  (2) 

Other Kinds of Bessel Functions 


595  (3) 


598  (3) 

Orthogonality of Bessel Functions 


601  (3) 

Approximate Formulas for Bessel Functions 


604  (1) 

Series Solutions; Fuchs's Theorem 


605  (2) 

Hermite Functions; Laguerre Functions; Ladder Operators 


607  (8) 


615  (4) 

Partial Differential Equations 


619  (47) 


619  (2) 

Laplace's Equation; SteadyState Temperature in a Rectangular Plate 


621  (7) 

The Diffusion or Heat Flow Equation; the Schrodinger Equation 


628  (5) 

The Wave Equation; the Vibrating String 


633  (5) 

Steadystate Temperature in a Cylinder 


638  (6) 

Vibration of a Circular Membrane 


644  (3) 

Steadystate Temperature in a Sphere 


647  (5) 


652  (7) 

Integral Transform Solutions of Partial Differential Equations 


659  (4) 


663  (3) 

Functions of a Complex Variable 


666  (56) 


666  (1) 


667  (7) 


674  (4) 


678  (4) 


682  (1) 

Methods of Finding Residues 


683  (4) 

Evaluation of Definite Integrals by Use of the Residue Theorem 


687  (15) 

The Point at Infinity; Residues at Infinity 


702  (3) 


705  (5) 

Some Applications of Conformal Mapping 


710  (8) 


718  (4) 

Probability and Statistics 


722  (57) 


722  (2) 


724  (5) 


729  (7) 


736  (8) 


744  (6) 


750  (6) 


756  (5) 

The Normal or Gaussian Distribution 


761  (6) 


767  (3) 

Statistics and Experimental Measurements 


770  (6) 


776  (3) 
References 

779  (2) 
Answers to Selected Problems 

781  (30) 
Index 

811  