Page 906 - Advanced_Engineering_Mathematics o'neil
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886 Index
Polynomial coefficients (Continued) Principal axis theorem, 291–292
complex roots, 271–272 Projections, 158–159, 178–180,
differential equations with, 112–114 762–763
eigenvalues (λ) as, 269–273 dot products for vectors, 158–159
Laplace transform and, 112–117 orthogonal, 158, 178–180
repeated roots, 273 stereographic, 762–763
Position function y(x,t), 565–567 Proper node, 333–334
Position vector, 345–346 Pulses, Heaviside formula (H) and,
Potential equation, 641–666 87–89
Dirichlet problems, 641–655 Punctured disk, 725
Green’s first identity, 659 Pure imaginary numbers, 670
Laplace’s equation, 641–642 Pursuit problem, 35–37
Neumann problems, 659–665 Pythagorean theorem, 156–157
Poisson’s integral formula, 648–649
steady-state equation, 655–658
Q
Potential function ϕ, 22–25, 380–381
Quadratic forms of matrices, 290–293
exact first-order equations, 22–25
defined, 290
vector fields, 380–381
mixed product terms, 291
Potential theory, 380–387, 410–411
principal axis theorem, 291–292
conservative vector field test,
real, 290
383–387, 410–411
standard, 291–293
Green’s theorem for, 380–387
Quadrilateral vector, 152
independence of the path and,
380–387
Stoke’s theorem for, 410–411 R
3-space, 410–411 Radiating ends, heat equation for,
Power series, 121–126, 715–724 615–617
antiderivative, existence of, 721–722 Radius of convergence, 717–718
complex numbers, 715–716 Random walks in crystals, matrix
convergence of, 716–718 application of, 194–197
defined, 716 Rational functions, 740–745
differentiation of, 718 defined, 740
integration of, 718 of sine or cosine, 742–743
isolated zeros, 722–724 residue theorem integral evaluation
recurrence relations, 123–126 using, 740–745
sequences, 715–716 times sine or cosine, 742–743
solutions, 121–126 Rational powers, 692
Taylor expansion, 718–722 Real axis, 670
Powers of complex numbers, 690–692 Real distinct roots, linear second-order
nth roots 690–691 equations for, 51
rational, 692 Rectangles, 642–644, 660–662
Prey/predator model, phase portraits Dirichlet problem for, 642–644
applied to, 338–340 Neumann problem for, 660–662
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