Hassani S. Mathematical Methods: For Students of Physics and Related Fields

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32.5 Problems 813

32.25. Using partial fractions write the integrand of Example 32.4.3 as

+1)[(u −v)

+1]

av + b

cv + d

(u −v)

Now write the right-hand side as a single fraction with the same denominator

as the left-hand side. Set the coeﬃcients of the powers of v in the numerator

equal to zero, except the constant which must be equal to 1. Find a, b, c,and

d. Show that the integral becomes

(b + d + cu)

∞

−∞

∞

−∞

32.26. Using the Stirling approximation and a procedure similar to the one

used for binomial distribution in the text, show that in the limit of large

n and λ, the Poisson distribution of Equation (32.43) becomes the normal

distribution.

32.27. Certain measurements are assumed to be normally distributed with

25 as the mean 25 and 0.5 as the standard deviation. What is the probability

that a measurement lies between 23 and 27?

Bibliography

[1] Alligood, K.T., Sauer, T.D., and Yorke, J.A. Chaos: An Introduction to

Dynamical Systems, Springer-Verlag, 1996.

[2] Auer, J.W. Linear Algebra with Applications, Prentice-Hall, 1991.

[3] Axler, S. Linear Algebra Done Right, Springer-Verlag, 1996.

[4] Baker, G.L. and Gollub, J.P. Chaotic Dynamics: An Introduction,Cam-

bridge University Press, 1990.

[5] Birkhoﬀ, G. and Rota, G.-C. Ordinary Diﬀerential Equations,3rded.,

Wiley, 1978.

[6] Churchill, R. and Verhey, R. Complex Variables and Applications,3rd

ed., McGraw-Hill, 1974.

[7] Edwards, C.H., Penney, D.E. Calculus and Analytic Geometry,Prentice

Hall, 1990.

[8] Finney, R.L., Thomas, G.B., Damana, F., and Waits, B.K. Calculus,

Addison-Wesley, 1994.

[9] Friedberg, S., Insel, A., and Spence, L. Linear Algebra, Prentice-Hall,

1997.

[10] Gamow, G., The Great Physicists: From Galileo to Einstein, Dover, 1961.

[11] Gelfand, I. M. and Fomin, S. V., Calculus of Variations, translated and

edited by Silverman, R. A., Dover, 1991.

[12] Goode, S.W. An Introduction to Diﬀerential Equations and Linear Alge-

bra, Prentice-Hall, 1991.

[13] Halmos, P. Finite Dimensional Vector Spaces, 2nd ed., Van Nostrand,

1958.

[14] Hartle, J. Gravity, Addison Wesley, 2003.

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816 BIBLIOGRAPHY

[16] Hilborn, R.C. Chaos and Nonlinear Dynamics: An Introduction for Sci-

entists and Engineers, Oxford University Press, 1994.

[17] Holmgren, R.A. A First Course in Discrete Dynamical Systems, Springer-

Verlag, 1996.

[18] Kaplan, W. Advanced Calculus, Addison-Wesley, 1991.

[19] Kline, M., Mathematical Thought: From Ancient to Modern Times,Vols.

1–3, Oxford University Press, 1972.

[20] Lang, S. Calculus of Several Variables, Springer-Verlag, 1988.

[21] Lang, S. Complex Analysis, 2nd ed., Springer-Verlag, 1985.

[22] Lorrain, P., Corson D., and Lorrain, F. Electromagnetic Fields and

Waves, 3rd ed., W. H. Freeman, 1988.

[23] Lovelock, D. and Rund, H. Tensors, Diﬀerential Forms, and Variational

Principles, Dover, 1989.

[24] Miller, R.K. Introduction to Diﬀerential Equations, Prentice-Hall, 1991.

[25] Pinsky, M.A. Partial Diﬀerential Equations and Boundary-Value Prob-

lems with Applications, McGraw-Hill, 1991.

[26] Rahman, M. Ordinary Diﬀerential Equations and Partial Diﬀerential

Equations, Computational Mechanics Publication, 1991.

[27] Schiﬀ, J. L. The Laplace Transform, Springer, 1991.

[28] Seydel, R. From Equilibrium to Chaos: Practical Bifurcation and Stability

Analysis, Elsevier, 1988.

[29] Simmons, G. Calculus Gems, McGraw-Hill, 1992.

[30] Stewart, J. Calculus, 4th ed., Brooks/Cole, 1999.

Index

Abel, 331, 503, 675

Abel, Niels Henrik

biography, 326

absolute diﬀerential, 463

Acceleration, 44

acceleration

components

spherical coordinates, 353

active transformation, 178

addition of velocities

relativistic law of, 246

adjoint DO, 571–573

advanced Green’s function, 712

aﬃne connection, 462–464, 470

algebra

fundamental theorem, 478, 511

amplitude

complex, 486

angle

as integral, 345

solid, 344–350

total, 349

total, 346

angular momentum, 28

conservation, 351

central force, 354

angular momentum operator, 412

spherical coordinates, 435

antisymmetric tensor, 452

Arago, 321

Archimedes, 47

Archimedes, of Syracuse

biography, 81

area element

primary, 60

astragalus, 781

astrophysics, 415

Poisson equation, 415

attractor, 766

strange, 778

autonomous, 767

average, 790, 801

azimuth, 12

azimuthal

angle, 12

symmetry, 607

azimuthal equation, 549

Barrow, 87, 96, 97, 481

Barrow, Isaac

biography, 47

basin of attraction, 766

basis, 16

for plane, 175

orthonormal, 186

standard, 216

Bayes’ theorem, 788

Beltrami identity, 731

Bernoulli, 272, 294, 320, 321, 326, 642

Bernoulli’s FODE, 560

Bessel, 322

Bessel diﬀerential equation, 548, 641

recursion relation, 644

Bessel equation

Liouville substitution, 589

Bessel function, 333–335, 644

addition theorem, 651

conﬂuent hypergeometric, 652

expansion in, 653–654

physical examples, 654–656

generating function, 651

integral representation, 652

Laplace’s equation, 642–654

order

negative integer, 646

orthogonality relation, 649

properties, 646–652

recurrence relation, 646

second kind, 645

Bessel’s integral, 652

818 INDEX

Bessel, Friedrich Wilhelm

biography, 641

beta function, 320

Bezout, 210

Bianchi identity, 470

bifurcation, 757

Hopf, 770

period-doubling, 757

binding energy, 582

binomial probability distribution,

792–797

binomial theorem, 265

biography

Abel, Niels Henrik, 326

Archimedes, of Syracuse, 81

Barrow, Isaac, 47

Bessel, Friedrich Wilhelm, 641

Biot, Jean-Baptiste, 115

Cauchy, Augustin-Louis, 503

Cavalieri, Bonaventura, 90

Cavendish,Henry,23

Cayley, Arthur, 192

Coulomb, Charles, 23

d’Alembert, Jean Le Rond, 548

Descartes, Rene, 15

Dirac, Paul Adrien Maurice, 151

Euler, Leonhard, 321

Fermat, Pierre de , 15

Fourier, Joseph, 304

Gauss, Johann Carl Friedrich,

330

Gibbs, Josiah Willard, 381

Hamilton, William R., 10

Heaviside, Oliver, 382

Hermite, Charles, 674

Jacobi, Carl Gustav Jacob, 326

Kepler, Johannes, 579

Laplace, Pierre Simon de, 593

Legendre, Adrien-Marie, 617

Leibniz, Gottfried Wilhelm, 103

Maxwell, James Clerk, 419

Newton, Isaac, 96

Savart, Felix, 115

Stokes, George Gabriel, 398

Sylvester, James Joseph, 210

Taylor, Brook, 294

Wallis, John, 90

Biot, Jean-Baptiste

biography, 115

Biot–Savart law, 30

circuit, 111

general, 110

bipolar coordinates, 73

three-dimensional, 74, 438

Boltzmann, 419

Bose-Einstein statistics, 792

boundary condition, 542

Dirichlet, 593

Neumann, 593

boundary conditions

periodic, 574

separated, 574

brachistochrone, 731

Bromwich contour, 722

butterﬂy eﬀect, 771

calculus

fundamental theorem, 87

calculus of residues, 525–536

canonical equations, 749

Cantor set, 776

Cardan, 481

Cartesian vector, 216

component, 216

Cauchy, 279, 326, 331, 594

Cauchy criterion, 261

Cauchy integral formula, 508–509

Cauchy, Augustin-Louis

biography, 503

Cauchy–Goursat theorem, 505

Cauchy–Riemann conditions, 500

Cauchy–Riemann theorem, 501

Cavalieri, Bonaventura

biography, 90

Cavendish, Henry

biography, 23

Cayley, 192, 211

Cayley, Arthur

biography, 192

center of mass, 21

central force, 354, 579–583

eccentricity, 581

central limit theorem, 809

centrifugal potential, 581

chain rule, 55–57

Champollion, 304

chaos, 753

theory

systems obeying DE, 770–773

INDEX 819

systems obeying iterated map, 763

universality, 773–778

coeﬃcient, 173

cofactor, 205

collision

relativistic, 250–253

column vector, 177

combination, 791

complement, 783

complex

conjugate, 479

function

analytic, 499

continuous, 499

regular point, 499

singular point, 499

singularity, 499

integral

positive sense, 507

number, 478

absolute value, 479

argument, 483

imaginary part, 478

real part, 478

plane, 478

complex amplitude, 486

complex function, 497–511

derivative, 499–503

derivative as integral, 509–511

integration, 503–508

complex number

Cartesian form, 478

Fourier series, 489

polar form, 482

roots, 486

complex numbers, 477–488

Cartesian form, 477–481

Fourier series, 488–491

polar form, 482–488

complex power series, 516

analyticity, 517

convergence circle, 517

diﬀerentiation, 517

integration, 517

uniform convergence, 517

complex series

absolute convergence, 516

component, 176

Compton wavelength, 253

conditional probability, 786–789

conducting cylindrical can, 655

conductor

electrical, 594

heat, 598

conﬂuent hypergeometric function, 332–

333

connection

aﬃne, 462–464, 470

metric, 465–468

constraints, 360, 738

continuity equation, 378–381

diﬀerential form, 380

integral form, 380

contour, 505

Bromwich, 722

simple closed, 505

contractible to zero, 400

contraction, 451

contravariant vector, 445–447

convergence

test, 267–272

convolution, 716

coordinate

generalized, 741

coordinate system, 11–15

bipolar, 73

three-dimensional, 74, 438

Cartesian, 11, 12

cylindrical, 12

elliptic, 73, 213

elliptic cylindrical, 73, 213, 436

parabolic, 73

paraboloidal, 74, 437

polar, 11

prolate spheroidal, 74, 213, 437

spherical, 12

toroidal, 74, 213, 437

unit vector, 31–36

vector, 16–31

coordinate time, 239–240

Copernicus, 97, 417, 580

correlation

probability, 803

cosine transform, 697

Coulomb, 744

Coulomb’s law, 22, 24

Coulomb, Charles

biography, 23

covariance

in probability, 803

820 INDEX

covariant derivative, 464–465

covariant diﬀerential, 462–464

covariant vector, 445–447

Cramer, 210

Crelle, 326

cross product, 7, 28–31

as a tensor, 447

Levi-Civita symbols, 458

parallelepiped volume, 10

parallelogram area, 9

curl

curvilinear coordinates, 431–435

vector ﬁeld, 391–398

current density, 379

and ﬂux, 379

curvature, 468–471

scalar, 470

curve

parametric equation, 61

primary, 59

curvilinear

vector analysis, 423–435

curvilinear coordinates

curl, 431–435

divergence, 427–431

gradient, 425–427

Laplacian, 435

cycloid, 732

d’Alembert, 273, 303

d’Alembert, Jean Le Rond

biography, 548

d´Alembert, 743

damping factor, 311

ﬁrst-order, 551–561

integrating factor, 553–555

linear, 556–561

second-order, 563–570

de Broglie, 666

de Moivre theorem, 485

del operator, 359

delta

Kronecker, 442

delta function

and Laplacian, 412

cylindrical, 160

derivative, 147, 159

Legendre expansion, 630

limit of sequence, 492

one-variable, 139–151

point sources, 144

polar, 156

representation, 491–492

spherical, 160

three-variable, 159–165

two-dimensional, 155

two-variable, 154–159

density, 45

current, 379

ﬂux, 371–381

of states, 677

probability, 801

density function

surface, 154

derivative, 44–46

covariant, 464–465

functional, 730

mixed, 52

normal, 593

partial, 47–59

time

vector, 350–355

total, 86

Descartes, 46, 97, 103, 215, 417, 481, 482

Descartes, Rene

biography, 15

determinant, 202–207, 222–227

parallelepiped volume, 10

diﬀerential, 53–54

absolute, 463

covariant, 462–464

exact, 553

diﬀerential equation

Bessel, 548, 641

recursion relation, 644

second solution, 645–646

solutions, 642–645

conﬂuent hypergeometric, 332

Hermite

recursion relation, 668

hypergeometric, 328

Legendre, 608

second solution, 617–619

order of, 556

ordinary, 542

partial, 542

second-order linear

adjoint, 572

integrating factor, 571

INDEX 821

diﬀerential operator, 217, 576

diﬀusion equation, 661

time-dependent, 663

dimension, 176

fractal, 775–778

dipole

approximation, 298

magnetic, 410

dipole moment, 298

dipole potential, 299

Dirac, 26

biography, 151

Dirac delta function

in variational problems, 730

step function, 153

Dirac, Paul Adrien Maurice

biography, 151

disjoint sets, 783

distance

spacetime, 240–242

distribution, 146

normal, 806–809

sum of two, 807

divergence, 371–381

curvilinear coordinates, 427–431

spherical coordinates, 430

theorem, 374–378

vector ﬁeld, 374

Doppler shift

relativistic, 255

dot product, 5, 21

double del operation, 407–412

double factorial, 319

dummy index, 262

dynamical system

autonomous, 767

nonautonomous, 767

eccentricity, 581

eigenvalue, 224

eigenvalue equation, 224

eigenvector, 224

Einstein, 215, 666

summation convention, 441

Einstein curvature tensor, 471

Einstein equation, 471

electric ﬁeld, 104

point charge, 25

electrical conductor, 594

electrodynamics

Lagrangian density, 745

tensors, 459–461

element

area, 59–68

Cartesian, 60–62

cylindrical, 65–68

length, 59–68

spherical, 62–64

volume, 59–68

elliptic coordinates, 73, 213

elliptic cylindrical coordinates, 73, 213,

436

elliptic functions, 322–326

elliptic integral

complete, 324

ﬁrst kind, 323

second kind, 323

empty set, 782

energy

relativistic, 249

zero mass particle, 250

energy momentum tensor, 471

equation

canonical, 749

Klein-Gordon, 747

error function, 322, 806

Euclid, 47, 80, 90

Euler, 272, 303, 326, 330, 503, 642, 743

Euler angles, 201

Euler equation, 483

Euler’s equation, 414

Euler, Leonhard

biography, 321

Euler-Lagrange equation, 729–731,

734–736, 738, 739

event, 784

compound, 784

elementary, 784

random, 781

exact diﬀerential, 553

expectation value, 790

extremum problem, 727

gradient, 359–361

factorial

double, 319

factorial function, 99, 318

Faraday, 26

Feigenbaum alpha, 774

Feigenbaum delta, 773

822 INDEX

Feigenbaum numbers, 773–775

Fermat, Pierre de

biography, 15

Fermi energy, 677

Fermi-Dirac statistics, 791

Feynman, 26

ﬁeld, 21–28, 343

electric, 104

scalar, 343

spinor, 343

tensor, 343

vector, 343

ﬁeld point, 25, 78

ﬁne structure constant, 679

ﬁnite constraint problem, 739

ﬁxed point

iterated map, 755

stable, 756

ﬂat space, 470

Florence Nightingale, 210

ﬂuid dynamics, 413–415

ﬂux, 365–369

density, 371–381

vector ﬁeld, 365–369

FODE, 551–561

Bernoulli, 560

homogeneous, 560

integrating factor, 553–555

Lagrange, 561

linear, 556–561

normal, 551

integral of, 552

FOLDE, 556–561

explicit solution, 557

force

central, 354

force density, 414

form factor, 702

four-acceleration, 248

four-momentum, 247–250

four-vector, 243

four-velocity, 247–250

Fourier, 115, 279, 322

Fourier series, 299–303

complex numbers, 488–491

to Fourier transform, 693–696

Fourier transform, 693–712

and derivatives, 702–703

and quark model, 702

application to DEs, 702–704

convolution theorem, 724

Coulomb potential

charge distribution, 701

point charge, 700

deﬁnition, 695

examples, 698–702

Gaussian, 699

Green’s functions, 705–712

heat equation

one-dimensional, 704

higher dimensions, 696

inverse, 695

of delta function, 698

properties, 696

Fourier, Joseph

biography, 304

Fourier-Bessel series, 655

fractal, 777

fractal dimension, 775–778

free index, 440

frequency

natural, 586

Frobenius method, 608–610, 693

function

analytic

isolated singularity, 525

principal part, 528

antiderivative, 87

as integral, 317–326

as power series, 327–335

Bessel, 333–335, 644

Laplace’s equation, 642–654

beta, 320

complex, 497–511

derivative, 499–503

residue, 526

complex hyperbolic, 502

complex trigonometric, 502

conﬂuent hypergeometric, 332

delta

point sources, 144

elliptic, 322–326

error, 322

even, 84

factorial, 318

gain, 586

gamma, 318–319

Stirling approximation, 319

harmonic, 501

homogeneous, 57–59

INDEX 823

hypergeometric, 328–330

integral representation, 329

iterated map, 755

linear density, 143

logistic map, 755

odd, 84

periodic, 299

piecewise continuous, 82

primitive, 87

rational

integral, 529–531

sequence, 274–279

series, 274–279

special, 550

transfer, 586

functional, 728

functional derivative, 730

fundamental theorem of algebra, 478

fundamental theorem of calculus, 87

G-orthogonal, 187, 219

matrix, 191, 222

space, 200

vector

in space, 199

gain function, 586

Galileo, 26, 90, 97, 325

gamma function, 318–319

Stirling approximation, 319

gauge transformation, 418

Gauss, 279, 321, 326, 503, 617, 641

Gauss elimination, 231

Gauss’s law, 369

diﬀerential form, 378

integral form, 377

Gauss, Johann Carl Friedrich

biography, 330

Gaussian

Fourier transform of, 699

Gay–Lussac, 594

generalized coordinates, 741

generalized momentum, 748

generating function

Hermite polynomials, 673

geodesic, 465

relativity, 466

sphere, 466

geometric series, 271

geometry

and metric tensor, 456

distance formula, 241

Gibb’s phenomenon, 302

Gibbs, 370

Gibbs, Josiah Willard

biography, 381

Goldbach, 320

gradient, 355–361, 445

components, 440

curvilinear coordinates, 425–427

normal to surface, 358

three dimensions, 357

two dimensions, 357

Gram–Schmidt process, 221

for space, 199

Grassmann, 382

Green, 210

Green’s function

advanced, 712

diﬀerebtial eq. for, 707

heat equation, 709–710

Laplacian, 708–709

Poisson equation, 709

retarded, 712

wave equation, 711–712

Green’s Functions, 705–712

Gregory, 272, 294

guided wave, 682–686

TE, 684

TEM, 685

TM, 684

Halley, 641

Hamilton, 369, 382

Hamilton, William R.

biography, 10

Hamiltonian, 747–749

harmonic oscillator

quantum, 667

Hermite DE, 668

heat-conducting plate

circular, 664–665

rectangular, 663–664

heat-conducting rod, 662–663

heat conductor, 598

heat equation, 543, 661–665

Green’s function, 709–710

one-dimensional, 704

heat transfer

time-dependent, 663