Sources in the Development of Mathematics: Series and Products from the Fifteenth to the Twenty-first Century

Cover......Page 1
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 19
1.1 Preliminary Remarks......Page 23
1.2 Transformation of Series......Page 26
1.3 Jyesthadeva on Sums of Powers......Page 27
1.4 Arctangent Series in the Yuktibhasa......Page 29
1.5 Derivation of the Sine Series in the Yuktibhasa......Page 30
1.6 Continued Fractions......Page 32
1.7 Exercises......Page 34
1.8 Notes on the Literature......Page 36
2.1 Preliminary Remarks......Page 38
2.2 Johann Faulhaber and Sums of Powers......Page 41
2.3 Jakob Bernoulli\'s Polynomials......Page 42
2.4 Proof of Bernoulli\'s Formula......Page 46
2.5 Exercises......Page 47
2.6 Notes on the Literature......Page 48
3.1 Preliminary Remarks......Page 50
3.2 Wallis\'s Infinite Product for bold0mu mumu Raw......Page 54
3.3 Brouncker and Infinite Continued Fractions......Page 55
3.4 Stieltjes: Probability Integral......Page 58
3.5 Euler: Series and Continued Fractions......Page 60
3.6 Euler: Products and Continued Fractions......Page 62
3.7 Euler: Continued Fractions and Integrals......Page 65
3.8 Sylvester: A Difference Equation and Euler\'s Continued Fraction......Page 67
3.9 Euler: Riccati\'s Equation and Continued Fractions......Page 68
3.10 Exercises......Page 70
3.11 Notes on the Literature......Page 72
4.1 Preliminary Remarks......Page 73
4.2 Landen\'s Derivation of the Binomial Theorem......Page 79
4.3 Euler\'s Proof for Rational Indices......Page 80
4.4 Cauchy: Proof of the Binomial Theorem for Real Exponents......Page 82
4.5 Abel\'s Theorem on Continuity......Page 84
4.6 Harkness and Morley\'s Proof of the Binomial Theorem......Page 88
4.7 Exercises......Page 89
4.8 Notes on the Literature......Page 91
5.1 Preliminary Remarks......Page 93
5.2 Descartes\'s Method of Finding the Normal......Page 95
5.3 Hudde\'s Rule for a Double Root......Page 96
5.4 Van Heuraet\'s Letter on Rectification......Page 97
5.5 Newton\'s Rectification of a Curve......Page 98
5.6 Leibniz\'s Derivation of the Arc Length......Page 99
5.7 Exercises......Page 100
5.8 Notes on the Literature......Page 101
6.1 Preliminary Remarks......Page 103
6.2 Harriot\'s Proof of the Arithmetic and Geometric Means Inequality......Page 109
6.3 Maclaurin\'s Inequalities......Page 110
6.4 Jensen\'s Inequality......Page 111
6.5 Reisz\'s Proof of Minkowski\'s Inequality......Page 112
6.6 Exercises......Page 113
6.7 Notes on the Literature......Page 118
7.1 Preliminary Remarks......Page 119
7.2 Pascal\'s Evaluation of ∫ sin x dx......Page 122
7.3 Gregory\'s Evaluation of a Beta Integral......Page 123
7.4 Gregory\'s Evaluation of ∫secθ dθ......Page 126
7.5 Barrow\'s Evaluation of ∫secθ dθ......Page 128
7.6 Barrow and the Integral ∫√x2+a2dx......Page 130
7.7 Barrow\'s Proof of d/dθ tanθ = sec2 θ......Page 132
7.8 Barrow\'s Product Rule for Derivatives......Page 133
7.10 Exercises......Page 136
7.11 Notes on the Literature......Page 140
8.1 Preliminary Remarks......Page 142
8.2 Newton\'s 1671 Calculus Text......Page 145
8.3 Leibniz: Differential Calculus......Page 148
8.4 Leibniz on the Catenary......Page 151
8.5 Johann Bernoulli on the Catenary......Page 153
8.6 Johann Bernoulli: The Brachistochrone......Page 154
8.7 Newton\'s Solution to the Brachistochrone......Page 155
8.8 Newton on the Radius of Curvature......Page 157
8.9 Johann Bernoulli on the Radius of Curvature......Page 158
8.10 Exercises......Page 159
8.11 Notes on the Literature......Page 160
9.1 Preliminary Remarks......Page 162
9.2 Algebra of Infinite Series......Page 164
9.3 Newton\'s Polygon......Page 167
9.4 Newton on Differential Equations......Page 168
9.5 Newton\'s Earliest Work on Series......Page 169
9.6 De Moivre on Newton\'s Formula for bold0mu mumu sinnsinnRawsinnsinnsinnsinn......Page 171
9.7 Stirling\'s Proof of Newton\'s Formula......Page 172
9.8 Zolotarev: Lagrange Inversion with Remainder......Page 174
9.9 Exercises......Page 175
9.10 Notes on the Literature......Page 178
10.1 Preliminary Remarks......Page 180
10.2 Newton: Divided Difference Interpolation......Page 185
10.4 Waring, Lagrange: Interpolation Formula......Page 187
10.5 Cauchy, Jacobi: Lagrange Interpolation Formula......Page 188
10.6 Newton on Approximate Quadrature......Page 190
10.7 Hermite: Approximate Integration......Page 192
10.8 Chebyshev on Numerical Integration......Page 194
10.9 Exercises......Page 195
10.10 Notes on the Literature......Page 197
11.1 Preliminary Remarks......Page 198
11.2 Newton\'s Transformation......Page 203
11.3 Montmort\'s Transformation......Page 204
11.4 Euler\'s Transformation Formula......Page 206
11.5 Stirling\'s Transformation Formulas......Page 209
11.6 Nicole\'s Examples of Sums......Page 212
11.7 Stirling Numbers......Page 213
11.8 Lagrange\'s Proof of Wilson\'s Theorem......Page 216
11.9 Taylor\'s Summation by Parts......Page 217
11.10 Exercises......Page 218
11.11 Notes on the Literature......Page 221
12.1 Preliminary Remarks......Page 222
12.2 Gregory\'s Discovery of the Taylor Series......Page 228
12.3 Newton: An Iterated Integral as a Single Integral......Page 231
12.4 Bernoulli and Leibniz: A Form of the Taylor Series......Page 232
12.5 Taylor and Euler on the Taylor Series......Page 233
12.6 Lacroix on d\'Alembert\'s Derivation of the Remainder......Page 234
12.7 Lagrange\'s Derivation of the Remainder Term......Page 235
12.8 Laplace\'s Derivation of the Remainder Term......Page 237
12.9 Cauchy on Taylor\'s Formula and l\'Hôpital\'s Rule......Page 238
12.10 Cauchy: The Intermediate Value Theorem......Page 240
12.11 Exercises......Page 241
12.12 Notes on the Literature......Page 242
13.1 Preliminary Remarks......Page 244
13.2 Newton\'s 1666 Basic Integrals......Page 250
13.3 Newton’s Factorization of xn ±1......Page 252
13.4 Cotes and de Moivre\'s Factorizations......Page 253
13.5 Euler: Integration of Rational Functions......Page 255
13.6 Euler\'s Generalization of His Earlier Work......Page 256
13.7 Hermite\'s Rational Part Algorithm......Page 259
13.8 Johann Bernoulli: Integration of √ax2 + bx + c......Page 260
13.9 Exercises......Page 261
13.10 Notes on the Literature......Page 265
14.1 Preliminary Remarks......Page 267
14.2 De Moivre on Recurrent Series......Page 269
14.3 Stirling\'s Method of Ultimate Relations......Page 272
14.4 Daniel Bernoulli on Difference Equations......Page 274
14.5 Lagrange: Nonhomogeneous Equations......Page 276
14.6 Laplace: Nonhomogeneous Equations......Page 279
14.7 Exercises......Page 280
14.8 Notes on the Literature......Page 281
15.1 Preliminary Remarks......Page 282
15.2 Leibniz: Equations and Series......Page 290
15.3 Newton on Separation of Variables......Page 292
15.4 Johann Bernoulli\'s Solution of a First-Order Equation......Page 293
15.5 Euler on General Linear Equations with Constant Coefficients......Page 294
15.6 Euler: Nonhomogeneous Equations......Page 296
15.7 Lagrange\'s Use of the Adjoint......Page 298
15.9 Riccati\'s Equation......Page 300
15.10 Singular Solutions......Page 301
15.11 Mukhopadhyay on Monge\'s Equation......Page 305
15.12 Exercises......Page 307
15.13 Notes on the Literature......Page 309
16.1 Preliminary Remarks......Page 311
16.2 Euler: Series for Elementary Functions......Page 314
16.3 Euler: Products for Trigonometric Functions......Page 315
16.4 Euler\'s Finite Product for bold0mu mumu sinnxsinnxRawsinnxsinnxsinnxsinnx......Page 316
16.5 Cauchy\'s Derivation of the Product Formulas......Page 317
16.6 Euler and Niklaus I Bernoulli: Partial Fractions Expansions of Trigonometric Functions......Page 320
16.7 Euler: Dilogarithm......Page 323
16.8 Landen\'s Evaluation of ζ(2)......Page 324
16.9 Spence: Two-Variable Dilogarithm Formula......Page 326
16.10 Exercises......Page 328
16.11 Notes on the Literature......Page 332
17.1 Preliminary Remarks......Page 333
17.2 Viète\'s Trigonometric Solution of the Cubic......Page 338
17.3 Descartes\'s Solution of the Quartic......Page 340
17.4 Euler\'s Solution of a Quartic......Page 341
17.5 Gauss: Cyclotomy, Lagrange Resolvents, and Gauss Sums......Page 342
17.6 Kronecker: Irreducibility of the Cyclotomic Polynomial......Page 346
17.8 Notes on the Literature......Page 347
18.1 Preliminary Remarks......Page 348
18.2 Euler\'s Proofs of Newton\'s Rule......Page 353
18.3 Maclaurin\'s Proof of Newton\'s Rule......Page 354
18.5 Gauss\'s Fundamental Theorem of Symmetric Functions......Page 356
18.6 Cauchy: Fundamental Theorem of Symmetric Functions......Page 357
18.7 Cauchy: Elementary Symmetric Functions as Rational Functions of Odd Power Sums......Page 358
18.8 Laguerre and Pólya on Symmetric Functions......Page 359
18.9 MacMahon\'s Generalization of Waring\'s Formula......Page 362
18.10 Exercises......Page 365
18.11 Notes on the Literature......Page 366
19.1 Preliminary Remarks......Page 368
19.2 Homogeneous Functions......Page 373
19.3 Cauchy: Taylor Series in Several Variables......Page 374
19.4 Clairaut: Exact Differentials and Line Integrals......Page 376
19.5 Euler: Double Integrals......Page 378
19.6 Lagrange\'s Change of Variables Formula......Page 380
19.7 Green\'s Integral Identities......Page 381
19.8 Riemann\'s Proof of Green\'s Formula......Page 383
19.9 Stokes\'s Theorem......Page 384
19.11 Notes on the Literature......Page 387
20.1 Preliminary Remarks......Page 389
20.2 Lagrange\'s Extension of the Euler--Maclaurin Formula......Page 397
20.3 Français\'s Method of Solving Differential Equations......Page 401
20.4 Herschel: Calculus of Finite Differences......Page 402
20.5 Murphy\'s Theory of Analytical Operations......Page 404
20.6 Duncan Gregory\'s Operational Calculus......Page 406
20.7 Boole\'s Operational Calculus......Page 409
20.8 Jacobi and the Symbolic Method......Page 412
20.9 Cartier: Gregory\'s Proof of Leibniz\'s Rule......Page 414
20.10 Hamilton\'s Algebra of Complex Numbers and Quaternions......Page 415
20.11 Exercises......Page 419
20.12 Notes on the Literature......Page 420
21.1 Preliminary Remarks......Page 422
21.2 Euler: Trigonometric Expansion of a Function......Page 428
21.3 Lagrange on the Longitudinal Motion of the Loaded Elastic String......Page 429
21.4 Euler on Fourier Series......Page 432
21.5 Fourier: Linear Equations in Infinitely Many Unknowns......Page 434
21.6 Dirichlet\'s Proof of Fourier\'s Theorem......Page 439
21.7 Dirichlet: On the Evaluation of Gauss Sums......Page 443
21.8 Exercises......Page 446
21.9 Notes on the Literature......Page 447
22.1 Preliminary Remarks......Page 449
22.2 The Riemann Integral......Page 451
22.3 Smith: Revision of Riemann and Discovery of the Cantor Set......Page 453
22.4 Riemann\'s Theorems on Trigonometric Series......Page 454
22.6 Schwarz\'s Lemma on Generalized Derivatives......Page 458
22.7 Cantor\'s Uniqueness Theorem......Page 459
22.8 Exercises......Page 461
22.9 Notes on the Literature......Page 465
23.1 Preliminary Remarks......Page 466
23.2 Stirling: Γ(1/2) by Newton–Bessel Interpolation......Page 472
23.3 Euler\'s Evaluation of the Beta Integral......Page 475
23.4 Gauss\'s Theory of the Gamma Function......Page 479
23.5 Poisson, Jacobi, and Dirichlet: Beta Integrals......Page 482
23.6 Bohr, Mollerup, and Artin on the Gamma Function......Page 484
23.7 Kummer\'s Fourier Series for bold0mu mumu ln (x)ln (x)Rawln (x)ln (x)ln (x)ln (x)......Page 487
23.8 Exercises......Page 489
23.9 Notes on the Literature......Page 496
24.1 Preliminary Remarks......Page 498
24.2 De Moivre\'s Asymptotic Series......Page 503
24.3 Stirling\'s Asymptotic Series......Page 505
24.4 Binet’s Integrals for ln Γ (x)......Page 508
24.5 Cauchy\'s Proof of the Asymptotic Character of de Moivre\'s Series......Page 510
24.6 Exercises......Page 511
24.7 Notes on the Literature......Page 515
25.1 Preliminary Remarks......Page 516
25.2 Euler on the Euler--Maclaurin Formula......Page 521
25.3 Maclaurin\'s Derivation of the Euler--Maclaurin Formula......Page 523
25.4 Poisson\'s Remainder Term......Page 525
25.5 Jacobi\'s Remainder Term......Page 527
25.6 Euler on the Fourier Expansions of Bernoulli Polynomials......Page 529
25.7 Abel\'s Derivation of the Plana--Abel Formula......Page 530
25.8 Exercises......Page 531
25.9 Notes on the Literature......Page 535
26.1 Preliminary Remarks......Page 537
26.2 Euler’s First Evaluation of Σ1/n2k......Page 543
26.3 Euler: Bernoulli Numbers and Σ1/n2k......Page 544
26.4 Euler\'s Evaluation of Some L-Series Values by Partial Fractions......Page 546
26.5 Euler\'s Evaluation of Σ1/n2 by Integration......Page 547
26.6 N. Bernoulli\'s Evaluation of Σ1/(2n+1)2......Page 549
26.7 Euler and Goldbach: Double Zeta Values......Page 550
26.8 Dirichlet\'s Summation of L(1,x)......Page 554
26.9 Eisenstein\'s Proof of the Functional Equation......Page 557
26.10 Riemann\'s Derivations of the Functional Equation......Page 558
26.11 Euler\'s Product for Σ1/ns......Page 561
26.12 Dirichlet Characters......Page 562
26.13 Exercises......Page 564
26.14 Notes on the Literature......Page 567
27.1 Preliminary Remarks......Page 569
27.2 Euler\'s Derivation of the Hypergeometric Equation......Page 577
27.3 Pfaff\'s Derivation of the 3F2 Identity......Page 578
27.4 Gauss\'s Contiguous Relations and Summation Formula......Page 579
27.5 Gauss\'s Proof of the Convergence of F(a,b,c,x) for c - a - b > 0......Page 581
27.6 Gauss\'s Continued Fraction......Page 582
27.7 Gauss: Transformations of Hypergeometric Functions......Page 583
27.8 Kummer\'s 1836 Paper on Hypergeometric Series......Page 586
27.9 Jacobi\'s Solution by Definite Integrals......Page 587
27.10 Riemann\'s Theory of Hypergeometric Functions......Page 589
27.11 Exercises......Page 591
27.12 Notes on the Literature......Page 594
28.1 Preliminary Remarks......Page 596
28.2 Legendre\'s Proof of the Orthogonality of His Polynomials......Page 600
28.3 Gauss on Numerical Integration......Page 601
28.4 Jacobi\'s Commentary on Gauss......Page 604
28.5 Murphy and Ivory: The Rodrigues Formula......Page 605
28.6 Liouville\'s Proof of the Rodrigues Formula......Page 607
28.7 The Jacobi Polynomials......Page 609
28.8 Chebyshev: Discrete Orthogonal Polynomials......Page 612
28.10 Chebyshev\'s Discrete Legendre and Jacobi Polynomials......Page 616
28.11 Exercises......Page 618
28.12 Notes on the Literature......Page 619
29.1 Preliminary Remarks......Page 621
29.3 Euler\'s bold0mu mumu qqRawqqqq-series Identities......Page 627
29.4 Euler\'s Pentagonal Number Theorem......Page 628
29.5 Gauss: Triangular and Square Numbers Theorem......Page 630
29.6 Gauss Polynomials and Gauss Sums......Page 633
29.7 Gauss\'s q-Binomial Theorem and the Triple Product Identity......Page 637
29.8 Jacobi: Triple Product Identity......Page 639
29.9 Eisenstein: q-Binomial Theorem......Page 640
29.10 Jacobi\'s q-Series Identity......Page 641
29.11 Cauchy and Ramanujan: The Extension of the Triple Product......Page 643
29.12 Rodrigues and MacMahon: Combinatorics......Page 644
29.13 Exercises......Page 645
29.14 Notes on the Literature......Page 647
30.1 Preliminary Remarks......Page 649
30.2 Sylvester on Partitions......Page 660
30.3 Cayley: Sylvester\'s Formula......Page 664
30.4 Ramanujan: Rogers--Ramanujan Identities......Page 666
30.5 Ramanujan\'s Congruence Properties of Partitions......Page 668
30.6 Exercises......Page 671
30.7 Notes on the Literature......Page 673
31.1 Preliminary Remarks......Page 675
31.2 Heine\'s Transformation......Page 683
31.3 Rogers: Threefold Symmetry......Page 684
31.4 Rogers: Rogers--Ramanujan Identities......Page 687
31.5 Rogers: Third Memoir......Page 692
31.6 Rogers--Szego Polynomials......Page 693
31.7 Feldheim and Lanzewizky: Orthogonality ofbold0mu mumu qqRawqqqq-Ultraspherical Polynomials......Page 695
31.8 Exercises......Page 699
31.9 Notes on the Literature......Page 701
32.1 Preliminary Remarks......Page 702
32.2 Euler: Sum of Prime Reciprocals......Page 704
32.3 Dirichlet: Infinitude of Primes in an Arithmetic Progression......Page 705
32.4 Class Number and Lχ (1)......Page 708
32.5 De la Vallée Poussin\'s Complex Analytic Proof of L(1) # 0......Page 710
32.6 Gelfond and Linnik: Proof of Lχ (1) # 0......Page 711
32.7 Monsky\'s Proof That L(1) # 0......Page 713
32.8 Exercises......Page 714
32.9 Notes on the Literature......Page 716
33.1 Preliminary Remarks......Page 717
33.2 Chebyshev on Legendre\'s Formula......Page 723
33.3 Chebyshev\'s Proof of Bertrand\'s Conjecture......Page 727
33.5 Mertens’s Evaluation of Πp≤x(1− 1/p)-1......Page 732
33.6 Riemann\'s Formula for π(x)......Page 736
33.7 Exercises......Page 739
33.8 Notes on the Literature......Page 741
34.1 Preliminary Remarks......Page 742
34.2 Boole\'s Derivation of an Invariant......Page 751
34.3 Differential Operators of Cayley and Sylvester......Page 755
34.4 Cayley\'s Generating Function for the Number of Invariants......Page 758
34.5 Sylvester\'s Fundamental Theorem of Invariant Theory......Page 762
34.6 Hilbert\'s Finite Basis Theorem......Page 765
34.8 Exercises......Page 768
34.9 Notes on the Literature......Page 769
35.1 Preliminary Remarks......Page 771
35.2  Fejér: Summability of Fourier Series......Page 782
35.3 Karamata\'s Proof of the Hardy--Littlewood Theorem......Page 785
35.4 Wiener\'s Proof of Littlewood\'s Theorem......Page 786
35.5 Hardy and Littlewood: The Prime Number Theorem......Page 788
35.6 Wiener\'s Proof of the PNT......Page 790
35.7 Kac\'s Proof of Wiener\'s Theorem......Page 793
35.8 Gelfand: Normed Rings......Page 794
35.9 Exercises......Page 797
35.10 Notes on the Literature......Page 799
36.1 Preliminary Remarks......Page 800
36.2 Fagnano Divides the Lemniscate......Page 808
36.3 Euler: Addition Formula......Page 812
36.4 Cayley on Landen\'s Transformation......Page 813
36.5 Lagrange, Gauss, Ivory on the agM......Page 816
36.6 Remarks on Gauss and Elliptic Functions......Page 822
36.7 Exercises......Page 833
36.8 Notes on the Literature......Page 835
37.1 Preliminary Remarks......Page 838
37.2 Abel: Elliptic Functions......Page 843
37.3 Abel: Infinite Products......Page 845
37.4 Abel: Division of Elliptic Functions and Algebraic Equations......Page 848
37.5 Abel: Division of the Lemniscate......Page 852
37.6 Jacobi\'s Elliptic Functions......Page 854
37.7 Jacobi: Cubic and Quintic Transformations......Page 856
37.8 Jacobi\'s Transcendental Theory of Transformations......Page 861
37.9 Jacobi: Infinite Products for Elliptic Functions......Page 866
37.10 Jacobi: Sums of Squares......Page 869
37.11 Cauchy: Theta Transformations and Gauss Sums......Page 871
37.12 Eisenstein: Reciprocity Laws......Page 874
37.13 Liouville\'s Theory of Elliptic Functions......Page 880
37.14 Exercises......Page 885
37.15 Notes on the Literature......Page 887
38.1 Preliminary Remarks......Page 889
38.2 Liouville Numbers......Page 900
38.3 Hermite\'s Proof of the Transcendence of bold0mu mumu eeRaweeee......Page 902
38.4 Hilbert\'s Proof of the Transcendence of e......Page 906
38.5 Exercises......Page 907
38.6 Notes on the Literature......Page 908
39.1 Preliminary Remarks......Page 909
39.2 Jacobi on Jensen\'s Formula......Page 914
39.3 Jensen\'s Proof......Page 916
39.4 Bäcklund Proof of Jensen\'s Formula......Page 917
39.5 R. Nevanlinna\'s Proof of the Poisson--Jensen Formula......Page 918
39.6 Nevanlinna\'s First Fundamental Theorem......Page 920
39.7 Nevanlinna\'s Factorization of a Meromorphic Function......Page 923
39.9 Borel\'s Theorem......Page 924
39.10 Nevanlinna\'s Second Fundamental Theorem......Page 925
39.11 Exercises......Page 927
39.12 Notes on the Literature......Page 928
40.1 Preliminary Remarks......Page 929
40.2 Gronwall: Area Inequalities......Page 936
40.3 Bieberbach\'s Conjecture......Page 938
40.4 Littlewood: |an| ≤ en......Page 939
40.5 Littlewood and Paley on Odd Univalent Functions......Page 940
40.6 Karl Löwner and the Parametric Method......Page 942
40.7 De Branges: Proof of Bieberbach\'s Conjecture......Page 945
40.8 Exercises......Page 949
40.9 Notes on the Literature......Page 950
41.1 Preliminary Remarks......Page 951
41.3 Gauss\'s Proof that Z×p Is Cyclic......Page 954
41.4 Gauss on Irreducible Polynomials Modulo a Prime......Page 955
41.5 Galois on Finite Fields......Page 958
41.6 Dedekind\'s Formula......Page 961
41.7 Exercises......Page 962
41.8 Notes on the Literature......Page 963
Referenes......Page 965
Index......Page 981