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Operational Mathematics / Edition 3

Operational Mathematics / Edition 3

by Ruel V. Churchill

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ISBN-10: 0070108706

ISBN-13: 9780070108707

Pub. Date: 06/01/1971

Publisher: McGraw-Hill Higher Education

Product Details

McGraw-Hill Higher Education
Publication date:
Edition description:
Product dimensions:
6.35(w) x 9.26(h) x 1.27(d)

Table of Contents

Chapter 1.The Laplace Transformation1
2.Definition of the Laplace Transformation3
3.Sectionally Continuous Functions. Exponential Order5
4.Transforms of Derivatives7
5.Examples. The Gamma Function10
6.The Inverse Transform14
7.A Theorem on Substitution16
8.The Use of Partial Fractions (Table 1)17
9.The Solution of Simple Differential Equations20
10.Generation of the Transformation24
Chapter 2.Further Properties of the Transformation27
11.Translation of F(t)27
12.Step Functions29
13.The Impulse Symbol [delta](t - t[subscript 0])33
14.Integrals Containing a Parameter39
15.Improper Integrals41
17.Properties of Convolution46
18.Differential and Integral Equations50
19.Derivatives of Transforms54
20.Series of Transforms57
21.Differential Equations with Variable Coefficients61
22.Integration of Transforms65
23.Periodic Functions66
24.Partial Fractions70
25.Repeated Linear Factors73
26.Quadratic Factors75
27.Tables of Operations and Transforms78
Chapter 3.Elementary Applications85
28.Free Vibrations of a Mass on a Spring85
29.Forced Vibrations without Damping88
31.Forced Vibrations with Damping95
32.A Vibration Absorber96
33.Electric Circuits102
34.Evaluation of Integrals106
35.Exponential- and Cosine-integral Functions110
36.Static Deflection of Beams113
37.The Tautochrone115
39.Mortality of Equipment119
Chapter 4.Problems in Partial Differential Equations123
40.The Wave Equation123
41.Displacements in a Long String126
42.A Long String under Its Weight130
43.The Long String Initially Displaced133
44.A Bar with a Prescribed Force on One End135
45.Equations of Diffusion143
46.Temperatures in a Semi-infinite Solid145
47.Prescribed Surface Temperature146
48.Temperatures in a Slab151
49.A Bar with Variable End Temperature153
50.A Cooling Fin or Evaporation Plate153
51.Temperatures in a Composite Solid155
52.Observations on the Method159
Chapter 5.Functions of a Complex Variable162
53.Complex Numbers162
54.Analytic Functions163
55.Exponential and Trigonometric Functions166
56.Contour Integrals168
57.Integral Theorems170
58.Power Series171
59.Singular Points and Residues173
60.Branches of Multiple-valued Functions177
61.Analytic Continuation179
62.Improper Cauchy Integrals181
Chapter 6.The Inversion Integral186
63.Analytic Transforms186
64.Permanence of Forms188
65.Order Properties of Transforms189
66.The Inversion Integral193
67.Conditions on f(s)195
68.Conditions on F(t)198
69.Uniqueness of Inverse Transforms201
70.Derivatives of the Inversion Integral202
71.Representation by Series of Residues206
72.Residues at Poles208
73.Validity of the Representation by Series210
74.Alterations of the Inversion Integral213
Chapter 7.Problems in Heat Conduction219
75.Temperatures in a Bar with Ends at Fixed Temperatures220
76.The Solution Established222
77.The Series Form Established224
78.Properties of the Temperature Function226
79.Uniqueness of the Solution228
80.Arbitrary End Temperatures230
81.Special End Temperatures232
82.Arbitrary Initial Temperatures237
83.Temperatures in a Cylinder240
84.Evaporation from a Thick Slab245
85.Duhamel's Formula247
Chapter 8.Problems in Mechanical Vibrations253
86.A Bar with a Constant Force on One End253
87.Another Form of the Solution256
88.Resonance in the Bar with a Fixed End258
89.Verification of Solutions259
90.Free Vibrations of a String264
91.Resonance in a Bar with a Mass Attached266
92.Transverse Vibrations of Beams268
93.Duhamel's Formula for Vibration Problems270
Chapter 9.Generalized Fourier Series276
94.Self-adjoint Differential Equations277
95.Green's Functions278
96.Construction of Green's Function281
97.Orthogonal Sets of Functions283
98.Eigenvalue Problems288
99.A Representation Theorem291
100.The Reduced Sturm-Liouville System292
101.A Related Boundary Value Problem293
102.The Transform y(x,s)294
103.Existence of Eigenvalues296
104.The Generalized Fourier Series299
105.Steady Temperatures in a Wall305
106.Verification of the Solution308
107.Singular Eigenvalue Problems309
Chapter 10.General Integral Transforms317
108.Linear Integral Transformations317
109.Kernel-product Convolution Properties319
111.Sturm-Liouville Transforms325
112.Inverse Transforms327
113.Further Properties329
114.Transforms of Certain Functions332
115.Example of Sturm-Liouville Transformations333
116.Singular Cases336
117.A Problem in Steady Temperatures341
118.Other Boundary Value Problems343
Chapter 11.Finite Fourier Transforms348
119.Finite Fourier Sine Transforms348
120.Other Properties of S[subscript n]350
121.Finite Cosine Transforms354
122.Tables of Finite Fourier Transforms356
123.Joint Properties of C[subscript n] and S[subscript n]357
124.Potential in a Slot360
125.Successive Transformations362
126.A Modified Sine Transformation368
127.Generalized Cosine Transforms370
128.A Generalized Sine Transform372
129.Finite Exponential Transforms E[subscript n]{F}376
130.Other Properties of E[subscript n]378
Chapter 12.Exponential Fourier Transforms383
131.The Transformation E[subscript alpha]{F}383
132.The Inverse Transformation385
133.Other Properties of E[subscript alpha]388
134.The Convolution Integral for E[subscript alpha]391
135.Convolution Theorem393
136.Tables of Transforms396
137.Boundary Value Problems397
Chapter 13.Fourier Transforms on the Half Line401
138.Fourier Sine Transforms f[subscript s]([alpha])401
139.Fourier Cosine Transforms f[subscript c]([alpha])404
140.Further Properties of S[subscript alpha] and C[subscript alpha]405
141.Convolution Properties406
142.Tables of Sine and Cosine Transforms407
143.Steady Temperatures in a Quadrant408
144.Deflections in an Elastic Plate410
145.A Modified Fourier Transformation T[subscript alpha]414
146.Convolution for T[subscript alpha]416
147.Surface Heat Transfer418
Chapter 14.Hankel Transforms420
149.Finite Hankel Transformations421
150.Inversion of H[subscript nj]423
151.Modified Finite Transformations H[subscript nh]424
152.A Boundary Value Problem426
153.Nonsingular Hankel Transformations431
154.Hankel Transformations H[subscript n alpha] on the Half Line (x [less than sign] 0)432
155.Further Properties of H[subscript n alpha]434
156.Tables of Transforms H[subscript n alpha]{F}436
157.Axially Symmetric Heat Source437
Chapter 15.Legendre and Other Integral Transforms441
158.The Legendre Transformation T[subscript n] on the Interval (-1,1)442
159.Further Properties of T[subscript n]444
160.Legendre Transforms on the Interval (0,1)446
161.Dirichlet Problems for the Sphere448
162.Laguerre Transforms452
163.Mellin Transforms453
Appendix ATables of Laplace Transforms458
Table A.1Operations458
Table A.2Laplace Transforms459
Appendix BTables of Finite Fourier Transforms467
Table B.1Finite Sine Transforms467
Table B.2Finite Cosine Transforms469
Appendix CTable of Exponential Fourier Transforms471
Appendix DTables of Fourier Sine and Cosine Transforms473
Table D.1Sine Transforms on the Half Line473
Table D.2Cosine Transforms on the Half Line475

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