Ordinary Differential Equations DEFINITIONS LINEAR DIFFERENTIAL EQUATIONS OF FIRST ORDER LINEAR INDEPENDENCE AND THE WRONSKIAN LINEAR HOMOGENEOUS DIFFERENTIAL EQUATION OF ORDER N WITH CONSTANT COEFFICIENTS EULER''S EQUATION PARTICULAR SOLUTIONS BY METHOD OF UNDETERMINED COEFFICIENTS PARTICULAR SOLUTIONS BY THE METHOD OF VARIATIONS OF PARAMETERS ABEL''S FORMULA FOR THE WRONSKIAN INITIAL VALUE PROBLEMS Series Solutions of Ordinary Differential Equations INTRODUCTION POWER SERIES SOLUTIONS CLASSIFICATION OF SINGULARITIES FROBENIUS SOLUTION Special Functions BESSEL FUNCTIONS BESSEL FUNCTION OF ORDER ZERO BESSEL FUNCTION OF AN INTEGER ORDER N RECURRENCE RELATIONS FOR BESSEL FUNCTIONS BESSEL FUNCTIONS OF HALF ORDERS SPHERICAL BESSEL FUNCTIONS HANKEL FUNCTIONS MODIFIED BESSEL FUNCTIONS GENERALIZED EQUATIONS LEADING TO SOLUTIONS IN TERMS OF BESSEL FUNCTIONS BESSEL COEFFICIENTS INTEGRAL REPRESENTATION OF BESSEL FUNCTIONS ASYMPTOTIC APPROXIMATIONS OF BESSEL FUNCTIONS FOR SMALL ARGUMENTS ASYMPTOTIC APPROXIMATIONS OF BESSEL FUNCTIONS FOR LARGE ARGUMENTS INTEGRALS OF BESSEL FUNCTIONS ZEROES OF BESSEL FUNCTIONS LEGENDRE FUNCTIONS LEGENDRE COEFFICIENTS RECURRENCE FORMULAE FOR LEGENDRE POLYNOMIALS INTEGRAL REPRESENTATION FOR LEGENDRE POLYNOMIALS INTEGRALS OF LEGENDRE POLYNOMIALS EXPANSIONS OF FUNCTIONS IN TERMS OF LEGENDRE POLYNOMIALS LEGENDRE FUNCTION OF THE SECOND KIND QN(X) ASSOCIATED LEGENDRE FUNCTIONS GENERATING FUNCTION FOR ASSOCIATED LEGENDRE FUNCTIONS RECURRENCE FORMULAE FOR Pnm INTEGRALS OF ASSOCIATED LEGENDRE FUNCTIONS ASSOCIATED LEGENDRE FUNCTION OF THE SECOND KIND Qnm Boundary Value Problems and Eigenvalue Problems INTRODUCTION VIBRATION, WAVE PROPAGATION OR WHIRLING OF STRETCHED STRINGS LONGITUDINAL VIBRATION AND WAVE PROPAGATION IN ELASTIC BARS VIBRATION, WAVE PROPAGATION AND WHIRLING OF BEAMS WAVES IN ACOUSTIC HORNS STABILITY OF COMPRESSED COLUMNS IDEAL TRANSMISSION LINES (TELEGRAPH EQUATION) TORSIONAL VIBRATION OF CIRCULAR BARS ORTHOGONALITY AND ORTHOGONAL SETS OF FUNCTIONS GENERALIZED FOURIER SERIES ADJOINT SYSTEMS BOUNDARY VALUE PROBLEMS EIGENVALUE PROBLEMS PROPERTIES OF EIGENFUNCTIONS OF SELF-ADJOINT SYSTEMS STURM-LIOUVILLE SYSTEM STURM-LIOUVILLE SYSTEM FOR FOURTH-ORDER EQUATIONS SOLUTION OF NON-HOMOGENEOUS EIGENVALUE PROBLEMS FOURIER SINE SERIES FOURIER COSINE SERIES COMPLETE FOURIER SERIES FOURIER-BESSEL SERIES FOURIER-LEGENDRE SERIES Functions of a Complex Variable COMPLEX NUMBERS ANALYTIC FUNCTIONS ELEMENTARY FUNCTIONS INTEGRATION IN THE COMPLEX PLANE CAUCHY''S INTEGRAL THEOREM CAUCHY''S INTEGRAL FORMULA INFINITE SERIES TAYLOR''S EXPANSION THEOREM LAURENT''S SERIES CLASSIFICATION OF SINGULARITIES RESIDUES AND RESIDUE THEOREM INTEGRALS OF PERIODIC FUNCTIONS IMPROPER REAL INTEGRALS IMPROPER REAL INTEGRAL INVOLVING CIRCULAR FUNCTIONS IMPROPER REAL INTEGRALS OF FUNCTIONS HAVING SINGULARITIES ON THE REAL AXIS THEOREMS ON LIMITING CONTOURS INTEGRALS OF EVEN FUNCTIONS INVOLVING LOG X INTEGRALS OF FUNCTIONS INVOLVING Xa INTEGRALS OF ODD OR ASYMMETRIC FUNCTIONS INTEGRALS OF ODD OR ASYMMETRIC FUNCTIONS INVOLVING LOG X INVERSE LAPLACE TRANSFORMS Partial Differential Equations of Mathematical Physics INTRODUCTION THE DIFFUSION EQUATION THE VIBRATION EQUATION THE WAVE EQUATION HELMHOLTZ EQUATION POISSON AND LAPLACE EQUATIONS CLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONS UNIQUENESS OF SOLUTIONS THE LAPLACE EQUATION THE POISSON EQUATION THE HELMHOLTZ EQUATION THE DIFFUSION EQUATION THE VIBRATION EQUATION THE WAVE EQUATION Integral Transforms FOURIER INTEGRAL THEOREM FOURIER COSINE TRANSFORM FOURIER SINE TRANSFORM COMPLEX FOURIER TRANSFORM MULTIPLE FOURIER TRANSFORM HANKEL TRANSFORM OF ORDER ZERO HANKEL TRANSFORM OF ORDER ν GENERAL REMARKS ABOUT TRANSFORMS DERIVED FROM THE FOURIER INTEGRAL THEOREM GENERALIZED FOURIER TRANSFORM TWO-SIDED LAPLACE TRANSFORM ONE-SIDED GENERALIZED FOURIER TRANSFORM LAPLACE TRANSFORM MELLIN TRANSFORM OPERATIONAL CALCULUS WITH LAPLACE TRANSFORMS SOLUTION OF ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS BY LAPLACE TRANSFORMS OPERATIONAL CALCULUS WITH FOURIER COSINE TRANSFORM OPERATIONAL CALCULUS WITH FOURIER SINE TRANSFORM OPERATIONAL CALCULUS WITH COMPLEX FOURIER TRANSFORM OPERATIONAL CALCULUS WITH MULTIPLE FOURIER TRANSFORM OPERATIONAL CALCULUS WITH HANKEL TRANSFORM Green''s Functions INTRODUCTION GREEN''S FUNCTION FOR ORDINARY DIFFERENTIAL BOUNDARY VALUE PROBLEM GREEN''S FUNCTION FOR AN ADJOINT SYSTEM SYMMETRY OF THE GREEN''S FUNCTIONS AND RECIPROCITY GREEN''S FUNCTION FOR EQUATIONS WITH CONSTANT COEFFICIENTS GREEN''S FUNCTIONS FOR HIGHER ORDERED SOURCES GREEN''S FUNCTION FOR EIGENVALUE PROBLEMS GREEN''S FUNCTION FOR SEMI-INFINITE ONE DIMENSIONAL MEDIA GREEN''S FUNCTION FOR INFINITE ONE-DIMENSIONAL MEDIA GREEN''S FUNCTION FOR PARTIAL DIFFERENTIAL EQUATIONS GREEN''S IDENTITIES FOR THE LAPLACIAN OPERATOR GREEN''S IDENTITY FOR THE HELMHOLTZ OPERATOR GREEN''S IDENTITY FOR BI-LAPLACIAN OPERATOR GREEN''S IDENTITY FOR THE DIFFUSION OPERATOR GREEN''S IDENTITY FOR THE WAVE OPERATOR GREEN''S FUNCTION FOR UNBOUNDED MEDIA--FUNDAMENTAL SOLUTION FUNDAMENTAL SOLUTION FOR THE LAPLACIAN FUNDAMENTAL SOLUTION FOR THE BI-LAPLACIAN FUNDAMENTAL SOLUTION FOR THE HELMHOLTZ OPERATOR FUNDAMENTAL SOLUTION FOR THE OPERATOR, - ∇2+ μ2 CAUSAL FUNDAMENTAL SOLUTION FOR THE DIFFUSION OPERATOR CAUSAL FUNDAMENTAL SOLUTION FOR THE WAVE OPERATOR FUNDAMENTAL SOLUTIONS FOR THE BI-LAPLACIAN HELMHOLTZ OPERATOR GREEN''S FUNCTION FOR THE LAPLACIAN OPERATOR FOR BOUNDED MEDIA CONSTRUCTION OF THE AUXILIARY FUNCTION-METHOD OF IMAGES GREEN''S FUNCTION FOR THE LAPLACIAN FOR HALF-SPACE GREEN''S FUNCTION FOR THE LAPLACIAN BY EIGENFUNCTION EXPANSION FOR BOUNDED MEDIA GREEN''S FUNCTION FOR A CIRCULAR AREA FOR THE LAPLACIAN GREEN''S FUNCTION FOR SPHERICAL GEOMETRY FOR THE LAPLACIAN GREEN''S FUNCTION FOR THE HELMHOLTZ OPERATOR FOR BOUNDED MEDIA GREEN''S FUNCTION FOR THE HELMHOLTZ OPERATOR FOR HALF-SPACE GREEN''S FUNCTION FOR A HELMHOLTZ OPERATOR IN QUARTER-SPACE CAUSAL GREEN''S FUNCTION FOR THE WAVE OPERATOR IN BOUNDED MEDIA CAUSAL GREEN''S FUNCTION FOR THE DIFFUSION OPERATOR FOR BOUNDED MEDIA METHOD OF SUMMATION OF SERIES SOLUTIONS IN TWO DIMENSIONAL MEDIA Asymptotic Methods INTRODUCTION METHOD OF INTEGRATION BY PARTS LAPLACE''S INTEGRAL STEEPEST DESCENT METHOD DEBYE''S FIRST ORDER APPROXIMATION ASYMPTOTIC SERIES APPROXIMATION METHOD OF STATIONARY PHASE STEEPEST DESCENT METHOD IN TWO DIMENSIONS MODIFIED SADDLE POINT METHOD: SUBTRACTION OF A SIMPLE POLE MODIFIED SADDLE POINT METHOD: SUBTRACTION OF POLE OF ORDER N SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS FOR LARGE ARGUMENTS CLASSIFICATION OF POINTS AT INFINITY SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS WITH REGULAR SINGULAR POINTS ASYMPTOTIC SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS WITH IRREGULAR SINGULAR POINTS OF RANK ONE THE PHASE INTEGRAL AND WKBJ METHOD FOR AN IRREGULAR SINGULAR POINT OF RANK ONE ASYMPTOTIC SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS WITH IRREGULAR SINGULAR POINTS OF RANK HIGHER THAN ONE ASYMPTOTIC SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS WITH LARGE PARAMETERS Numerical Methods INTRODUCTION ROOTS OF NON-LINEAR EQUATIONS ROOTS OF A SYSTEM OF NON-LINEAR EQUATION FINITE DIFFERENCES NUMERICAL DIFFERENTIATION NUMERICAL INTEGRATION ORDINARY DIFFERENTIAL EQUATIONS: INITIAL VALUE PROBLEMS ORDINARY DIFFERENTIAL EQUATIONS: BOUNDARY VALUE PROBLEMS ORDINARY DIFFERENTIAL EQUATIONS: EIGENVALUE PROBLEMS PARTIAL DIFFERENTIAL EQUATIONS Appendix A: Infinite Series Appendix B: Special Functions Appendix C: Orthogonal Coordinate Systems Appendix D: Dirac Delta Functions Appendix E: Plots of Special Functions Appendix F: Vector Analysis Appendix G: Matrix Algebra References Answers Index Problems appear at the end of each chapter. ELATIONS FOR BESSEL FUNCTIONS BESSEL FUNCTIONS OF HALF ORDERS SPHERICAL BESSEL FUNCTIONS HANKEL FUNCTIONS MODIFIED BESSEL FUNCTIONS GENERALIZED EQUATIONS LEADING TO SOLUTIONS IN TERMS OF BESSEL FUNCTIONS BESSEL COEFFICIENTS INTEGRAL REPRESENTATION OF BESSEL FUNCTIONS ASYMPTOTIC APPROXIMATIONS OF BESSEL FUNCTIONS FOR SMALL ARGUMENTS ASYMPTOTIC APPROXIMATIONS OF BESSEL FUNCTIONS FOR LARGE ARGUMENTS INTEGRALS OF BESSEL FUNCTIONS ZEROES OF BESSEL FUNCTIONS LEGENDRE FUNCTIONS LEGENDRE COEFFICIENTS RECURRENCE FORMULAE FOR LEGENDRE POLYNOMIALS INTEGRAL REPRESENTATION FOR LEGENDRE POLYNOMIALS INTEGRALS OF LEGENDRE POLYNOMIALS EXPANSIONS OF FUNCTIONS IN TERMS OF LEGENDRE POLYNOMIALS LEGENDRE FUNCTION OF THE SECOND KIND QN(X) ASSOCIATED LEGENDRE FUNCTIONS GENERATING FUNCTION FOR ASSOCIATED LEGENDRE FUNCTIONS RECURRENCE FORMULAE FOR Pnm INTEGRALS OF ASSOCIATED LEGENDRE FUNCTIONS ASSOCIATED LEGENDRE FUNCTION OF THE SECOND KIN.