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

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### 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 |

431 | |

433 | |