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1 Fourier series


Piecewise Continuous Functions

Fourier Cosine Series 4

Examples 6
1 Fourier series Piecewise Continuous Functions Fourier Cosine Series 4 Examples 6 Fourier Sine Series 9 Examples 10 Fourier Series 14 3 Partial Differential Equations of Physics Linear Boundary Value Problems 60 One-Dimensional Heat Equation 62 Related Equations 65 Laplacian in Cylindrical and Spherical Coordinates 67 Derivations 70 Boundary Conditions 73 Duhamel's Principle 75 A Vibrating String 80 Vibrations of Bars and Membranes 83 General Solution of a Wave Equation 87 Types of Equations and Boundary Conditions 92 4 The Fourier method Linear Operators Principle of Superposition 97 Examples 99 Eigenvalues and Eigenfunctions 102 A Temperature Problem 104 A Vibrating String Problem 107 Historical Development 111 5 Boundary value Problems A Slab with Faces at Prescribed Temperatures 114 Related Temperature Problems 118 Temperatures in a Sphere 122 A Slab with Internally Generated Heat 125 Steady Temperatures in Rectangular Coordinates 131 Steady Temperatures in Cylindrical Coordinates 136 A String with Prescribed Initial Conditions 140 Resonance 146 An elastic Bar 149 Double fourier series 152 Periodic Boundary Conditions 155 Examples 16 Adaptations to Other Intervals 20 2 Convergence of Fourier Series One-Sided Derivatives 25 A Property of Fourier Coefficients 28 Two Lemmas 31 A Fourier Theorem 35 A Related Fourier Theorem 38 Examples 39 Convergence on Other Intervals 43 A Lemma 47 Absolute and Uniform Convergence of Fourier Series 49 The Gibbs phenomenon 51 Differentiation of Fourier Series 54 Integration of Fourier Series 55 6 Fourier Integrals and Applications The Fourier Integral Formula 161 Dirichlet's Integral 163 Two Lemmas 165 A Fourier Integral Theorem 169 The Cosine and Sine Integrals 173 Some eigenvalue problems on Unbounded Intervals 174 More on Superposition of Solutions 177 Steady Temperatures in a Semi-Infinite Strip 179 Temperatures in a semi-Infinite Solid 182 Temperatures in an Unlimited Medium 187 7 Orthonormal sets Inner Products and Orthonormal sets 189 Examples 191 Generalized Fourier Series 195 Examples 196 Best Approximation in the Mean 200 Bessel's Inequality and Parseval,s Equation 203 Applications to Fourier Series 205 8 Sturm-Liouville Problems and Applications Regular Sturm-Liouville Problems 210 Modifications 212 Orthogonal Eigenfunctions and Real Eigenvalues 213 Real-Valued Eigenfunctions 218 Nonnegative Eigenvalues 220 Methods of solution 221 Examples of Eigenfunction Expansions 227 A Temperature Problem in Rectangular Coordinates 234 Steady Temperatures 238 Other Coordinates 242 A Modification of the Method 245 Another modification 249 A Vertically Hung Elastic Bar 252 9 Bessel Functions and Applications The Gamma Function 261 Bessel Functions Jv(x) 263 Solutions when v=0.1.2 Recurrence Relations 273 Bessel's Integral Form 277 Some Consequences of the Integral Forms 279 The Zeros of Jn(x) 283 Zeros of related Functions 286 Orthogonal Sets of Bessel Functions 288 Proof of the Theorems 290 Two Lemmas 295 Fourier-Bessel Series 298 Examples 300 Temperatures in a Long Cylinder 305 A Temperature Problem in Shrunken Fittings 312 Internally Generated Heat 314 Temperatures in a Long Cylindrical Wedge 319 Vibration of a Circular Membrane 321 10 Legendre Polynomials and Applications Solutions of Legendre's Equation 326 Legendre Polynomials 328 Rodrigues' Formula 333 Laplace's Integral Form 336 Some Consequences of the Integral Form 338 Orthogonality of Legendre Polynomials 341 Normalized Legendre Polynomials 343 Legendre Series 345 The Eigenfunctions Pn(cos 8) 350 Dirichlet Problems in Spherical Regions 352 Steady Temperatures in a Hemisphere 356 11 Verification of Solutions and Uniqueness Abel's Test for Uniform Convergence 362 Verification of Solution of Temperature Problem 365 Uniqueness of Solutions of the Heat Equation 368 Verification of Solution of Vibrating String Problem 372 Uniqueness of Solutions of the Wave Equation 374 Appendixes Bibliography 377 Some Fourier Series Expansions 381 Solutions of Some Regular Sturm-Liouville Problems 383 Some Fourier-Bessel Series Expansions 387 Index ±à¼­ÊÖ¼Ç

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