Nair S. Advanced Topics in Applied Mathematics: For Engineering and the Physical Sciences
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Author Index
Abramowitz, M., 47, 92
Andrews, L. C., 165, 212
Barber, J. R., 92
Beck, J., 47
Bender, C. M., 212
Berkooz, G., 93
Brebbia, C. A., 92
Brillouin, L., 212
Chatterjee, A., 93
Cole, K., 47
Cooley, J. W., 165
Courant, R., 47, 93
Davies, B., 165, 212
Ewing, W. M., 202, 212
Feshbach, H., 48, 166
Fung, Y. C., 202, 212
Gladwell, G. M. L., 165
Haji-Sheikh, A., 47
Hartmann, F., 93
Heaviside, O., 174
Hilbert, D., 47, 93
Hildebrand, F. B., 47, 93
Holmes, P., 93
Jardetzky, W. S., 202, 212
King, F., 166
Litkouhi, B., 47
Lumley, J. L., 93
Miles, J. W., 212
Milne-Thomson, L. M., 166
Morse, P. M., 48, 166
Noble, B., 166
Orzag, S. A., 212
Pozrikides, C., 93
Press, F., 202, 212
Rahman, M., 93
Shivamoggi, B. K., 165, 212
Sneddon, I. N., 93, 166, 212
Stakgold, I., 48
Stegun, I., 47, 92
Tricomi, F. G., 93
Tukey, J. W., 165
219
Subject Index
Abel
constant, 22
equation, 197
identity, 21, 23
integral equation, 82
absolutely integrable, 101, 152
Adjoint
boundary conditions, 17
Green’s function, 18
of a matrix, 17
operator, 16, 17
airfoil theory, 146
analytic
continuation, 160, 162
domain, 152, 153
functions, 146
signal, 144
angular frequency, 211
anti-derivative, 123
approximate kernel, 74
asymptotic forms, 52
Bessel
function, 20, 34
inequality, 73
bi-linear, 44
concomitant, 18
bi-orthogonal, 27
Boundary Element Method, 84, 88
boundary integral method, 84
branch cut, 186
Brilluoin approximation, 210
Bromwich contour, 176, 183
Cagniard–De Hoop method, 198
calculus of variations, 70
Cauchy
integral, 39, 145
principal value, 88, 129, 144, 147
Cauchy’s theorem, 143
causal, 142
characteristic
equation, 63
line, 196
polynomial, 188
variables, 196
circulation, 146
collocation, 87
complex conjugate, 100
conformal mapping, 36
convolution
integral, 119, 155, 180
operator, 119
sum, 164
theorem, 123, 128
crystal acoustics, 210
D’Alembert solution, 194
degenerate kernel, 74
delta sequence, 115
determinant, 63
difference equation, 205
differential
equation, 10
operator, 10, 62
diffusion equation, 134
diffusivity, 130
Dirac delta function, 3, 104
220
Subject Index 221
direct integration, 13
dispersion relation, 212
distribution, 1, 7, 8
dual integral equations, 80
dynamical systems, 88
Eddington’s Method, 140
eigenfunction, 26, 62
elastic media, 202
evaluation of integrals, 140
expansion theorems, 71
Feynman diagrams, 47
Finite Difference Method, 88
Finite Element Method, 88
first variation, 70, 90
forcing function, 155
Fourier
Cosine transform, 106
integral theorem, 100, 104
inverse transform, 116
multiple transform, 124
series, 98, 99
Sine transform, 106
transform, 99
transform, complex, 151
transform, discrete, 162
transform, Fast, 165
transform, properties, 108
Fredholm integral equation, 87
frequency domain, 120
Fubini’s theorem, 14
function
absolutely integrable, 101
piecewise continuous, 101
Gaussian function, 113
Goursat’s theorem, 156
gradient operator, 29
Green’s, 84
function, 1, 12, 14–16, 19, 20, 22–26,
58, 128, 202
function, generalized, 44
operator, 12
Haar function, 2
Hankel
functions, 34, 52
transform, 80
harmonic motion, 208
heat
conduction, 32, 186, 191
source, 32
Heaviside step function, 1, 175
Helmholtz equation, 34, 52, 210
hereditary kernels, 196
Hilbert transform, 142
finite, 146
Hilbert-Schmidt theory, 65, 91
hyperbola, 201
image processing, 146
incompressible, 124
infinite domain, 34
initial condition, 134
inner product, 17, 38, 120
integral equation, 56
first kind, 56, 78
Fredholm type, 57
second kind, 57
singular, 58
Volterra type, 57
inverse transform, 101
irrotational, 124
iterated kernel, 67
Jacobian, 151
Jordan’s lemma, 182
jump, 16, 21
condition, 16, 21
Karhunen-Loève transformation, 88
Kellogg’s method, 73
kernel, 57
kinetic energy, 82
Kramers-Krönig relations, 143
Kronecker delta, 12
Laplace
equation, 31, 32, 34, 36, 37, 39, 51, 53
inversion formula, 176
operator, 52
transform, 47, 174
transform, properties, 176
222 Subject Index
Legendre function, 20
Linear Differential Operator, 10
linear functional, 7
localization lemma, 101, 103
Macaulay brackets, 6
Mercer’s theorem, 71, 92
method of
images, 34
variable constants, 22
mixed trigonometric transform, 122
mode shape, 209
modulated, 144
Neumann functions, 34
Neumann series, 68
Newton’s law of cooling, 122
non-self-adjoint problem, 24
norm, 11
normal derivative, 35
Nyquist theorem, 144
observation point, 22
ortho-normal, 11
orthogonal, 11, 61
basis, 99
functions, 62, 98
Parseval’s formula, 165
Parseval’s relation, 120
partial fraction, 190
Petrov-Galerkin method, 75
phonons, 210
piecewise continuous, 101
Plemelj formulas, 149
Poisson’s equation, 32
positive definite kernel, 67
potential energy, 82
power spectrum, 120
principal component analysis, 88
probability distribution, 113
proper orthogonal decomposition, 88
quadratic form, 70
radiation condition, 122
residue theorem, 158, 182
resolvent kernel, 68
Riemann-Lebesgue lemma, 101
Schwartz-Christoffel transform, 37
self-adjoint
problem, 23
self-reciprocal, 113
separable kernel, 60
Simpson’s rule, 75
singular value decomposition, 88
sink, 35
slit, 151
Sommerfeld radiation conditions, 53
source, 8, 32
source point, 22
spatial basis functions, 89
spectral lines, 117
string, 208
Sturm-Liouville
equation, 29, 31
operator, 20, 44
Tauber’s theorems, 186
Taylor series, 205
temperature, 130, 192
test functions, 7
theta function, 168
transform operator, 107
transient heat conduction, 130
transpose, 17
trapezoidal rule, 75, 97, 162
Tricomi’s notation, 144
turbulence, 88
unit circle, 151
vibrations, 189
Volterra equation, 79
wave propagation, 193, 202
wavelet, 2
weight coefficients, 75
Wiener-Hopf method, 155, 162
Z-transform, 203