# Mathematical Methods for Engineers and Scientists 3: Fourier Analysis, Partial Differential Equations and Variational Methods

Springer Science & Business Media, 30 de nov de 2006 - 440 páginas

Pedagogical insights gained through 30 years of teaching applied mathematics led the author to write this set of student oriented books. Topics such as complex analysis, matrix theory, vector and tensor analysis, Fourier analysis, integral transforms, ordinary and partial differential equations are presented in a discursive style that is readable and easy to follow. Numerous clearly stated, completely worked out examples together with carefully selected problem sets with answers are used to enhance students' understanding and manipulative skill. The goal is to make students comfortable and confident in using advanced mathematical tools in junior, senior, and beginning graduate courses.

### O que estão dizendo -Escrever uma resenha

Não encontramos nenhuma resenha nos lugares comuns.

### Conteúdo

 Fourier Series 3 112 The Fourier Coefficients 5 113 Expansion of Functions in Fourier Series 6 12 Convergence of Fourier Series 9 122 Fourier Series and Delta Function 10 13 Fourier Series of Functions of any Period 13 132 Fourier Series of Even and Odd Functions 21 14 Fourier Series of Nonperiodic Functions in Limited Range 24
 473 Recurrence Relations 208 474 Orthogonality and Normalization of Legendre Polynomials 211 48 Associated Legendre Functions and Spherical Harmonics 212 482 Orthogonality and Normalization of Associated Legendre Functions 214 483 Spherical Harmonics 217 49 Resources on Special Functions 218 Exercises 219 Partial Differential Equations in Cartesian Coordinates 228

 15 Complex Fourier Series 29 16 The Method of Jumps 32 17 Properties of Fourier Series 37 172 Sums of Reciprocal Powers of Integers 39 173 Integration of Fourier Series 42 174 Differentiation of Fourier Series 43 18 Fourier Series and Differential Equations 45 182 Periodically Driven Oscillator 49 Exercises 52 Fourier Transforms 60 211 Fourier Cosine and Sine Integrals 65 212 Fourier Cosine and Sine Transforms 67 22 Tables of Transforms 72 24 Fourier Transform and Delta Function 79 242 Fourier Transforms Involving Delta Functions 80 243 ThreeDimensional Fourier Transform Pair 81 25 Some Important Transform Pairs 85 253 Exponentially Decaying Function 87 26 Properties of Fourier Transform 88 262 Linearity Shifting Scaling 89 263 Transform of Derivatives 91 264 Transform of Integral 92 27 Convolution 94 272 Convolution Theorems 96 28 Fourier Transform and Differential Equations 99 29 The Uncertainty of Waves 103 Exercises 105 Orthogonal Functions and SturmLiouville Problems 111 312 Inner Product and Orthogonality 113 313 Orthogonal Functions 116 32 Generalized Fourier Series 121 33 Hermitian Operators 123 332 Properties of Hermitian Operators 125 34 SturmLiouville Theory 130 342 Boundary Conditions of SturmLiouville Problems 132 343 Regular SturmLiouville Problems 133 344 Periodic SturmLiouville Problems 141 345 Singular SturmLiouville Problems 142 35 Greens Function 149 352 Greens Function and Delta Function 150 Exercises 157 Bessel and Legendre Functions 163 41 Frobenius Method of Differential Equations 164 412 Classifying Singular Points 166 413 Frobenius Series 167 42 Bessel Functions 171 421 Bessel Functions Jnx of Integer Order 172 422 Zeros of the Bessel Functions 174 423 Gamma Function 175 424 Bessel Function of Noninteger Order 177 425 Bessel Function of Negative Order 179 43 Properties of Bessel Function 182 432 Generating Function of Bessel Functions 185 433 Integral Representation 186 44 Bessel Functions as Eigenfunctions of SturmLiouville Problems 187 442 Orthogonality of Bessel Functions 188 443 Normalization of Bessel Functions 189 45 Other Kinds of Bessel Functions 191 452 Spherical Bessel Functions 192 46 Legendre Functions 196 462 Legendre Polynomials 200 463 Legendre Functions of the Second Kind 202 47 Properties of Legendre Polynomials 204 472 Generating Function of Legendre Polynomials 206
 51 OneDimensional Wave Equations 230 512 Separation of Variables 232 513 Standing Wave 238 514 Traveling Wave 242 515 Nonhomogeneous Wave Equations 248 516 DAlemberts Solution of Wave Equations 252 52 TwoDimensional Wave Equations 261 522 Vibration of a Rectangular Membrane 262 53 ThreeDimensional Wave Equations 267 531 Plane Wave 268 532 Particle Wave in a Rectangular Box 270 54 Equation of Heat Conduction 272 55 OneDimensional Diffusion Equations 274 551 Temperature Distributions with Specified Values at the Boundaries 275 552 Problems Involving Insulated Boundaries 278 553 Heat Exchange at the Boundary 280 Heat Transfer in a Rectangular Plate 284 57 Laplaces Equations 286 SteadyState Temperature in a Rectangular Plate 287 SteadyState Temperature in a Rectangular Parallelepiped 289 58 Helmholtzs Equations 291 Exercises 292 Partial Differential Equations with Curved Boundaries 301 61 The Laplacian 302 62 TwoDimensional Laplaces Equations 304 622 Poissons Integral Formula 312 63 TwoDimensional Helmholtzs Equations in Polar Coordinates 315 TwoDimensional Wave Equation in Polar Coordinates 316 TwoDimensional Diffusion Equation in Polar Coordinates 322 633 Laplaces Equations in Cylindrical Coordinates 326 634 Helmholtzs Equations in Cylindrical Coordinates 331 64 ThreeDimensional Laplacian in Spherical Coordinates 334 642 Helmholtzs Equations in Spherical Coordinates 345 643 Wave Equations in Spherical Coordinates 346 65 Poissons Equations 349 651 Poissons Equation and Greens Function 351 652 Greens Function for Boundary Value Problems 355 Exercises 359 Calculus of Variation 366 71 The EulerLagrange Equation 368 712 Fundamental Theorem of Variational Calculus 370 713 Variational Notation 372 714 Special Cases 373 72 Constrained Variation 377 73 Solutions to Some Famous Problems 380 732 Isoperimetric Problems 384 733 The Catenary 386 734 Minimum Surface of Revolution 391 735 Fermats Principle 394 74 Some Extensions 397 742 Several Dependent Variables 399 743 Several Independent Variables 401 75 SturmLiouville Problems and Variational Principles 403 752 Variational Calculations of Eigenvalues and Eigenfunctions 405 76 RayleighRitz Methods for Partial Differential Equations 410 761 Laplaces Equation 411 762 Poissons Equation 415 763 Helmholtzs Equation 417 77 Hamiltons Principle 420 Exercises 425 References 431 Index 433 Direitos autorais

### Sobre o autor (2006)

K.T. Tang received his B.S. in Engineering Physics and M.A. in Mathematics from University of Washington and his Ph.D. in Physics from Columbia University. He did postdoctoral studies in Chemistry at Berkeley and Harvard. He worked as an engineer at Collins Radio Company and Boeing Company. Dr. Tang regards teaching as his calling, although his research accomplishments are also considerable. He authored/co-authored over 130 research papers in professional journals and a monograph "Asymptotic Methods in Quantum Mechanics". He lectured widely in Asia, Europe, and North America. He had been a long-term visiting scientist at Max-Planck-Institut in Göttingen. He is a recipient of a Distinguished U.S. Senior Scientist Award from Alexander von Humboldt Foundation and a Faculty Excellence Award from Pacific Lutheran University where he is Professor of Physics.