Book:E.T. Whittaker/A Course of Modern Analysis

Subject Matter

 * Analysis

Contents

 * Preface


 * Part I The Processes of Analysis


 * Chapter I: Complex Numbers
 * 1. Real numbers
 * 2. Complex numbers
 * 3. The modulus of a complex quantity
 * 4. The geometrical interpretation of complex numbers
 * Miscellaneous Examples


 * Chapter II: The Theory of Convergence
 * 5. The limit of a sequence of quantities
 * 6. The necessary and sufficient conditions for the existence of a limit
 * 7. Convergence of an infinite series
 * 8. Absolute convergence and semi-convergence
 * 9. The geometric series, and the series $\sum n^{-s}$
 * 10. The comparison-theorem
 * 11. Discussion of a special series of importance
 * 12. A convergency-test which depends on the ratio of successive terms of a series
 * 13. A general theorem on those series for which $\lim_{n \to \infty} \left({ \frac{ u_{n+1} } {u_n} }\right)$ is 1
 * 14. Convergence of the hypergeometric series
 * 15. Effect on changing the order of terms in a series
 * 16. The fundamental property of absolutely convergent series
 * 17. Riemann's theorem of semi-convergent series
 * 18. Cauchy's theorem on the multiplication of absolutely convergent series
 * 19. Merten's theorem on the multiplication of a semi-convergent series by an absolutely convergent series
 * 20. Abel's result on the multiplication of series
 * 21. Power-series
 * 22. Convergence of series derived from a power-series
 * 23. Infinite products
 * 24. Some examples of infinite products
 * 25. Cauchy's theorem on products which are not absolutely convergent
 * 26. Infinite determinants
 * 27. Convergence of an infinite determinant
 * 28. Persistence of convergence when the elements are changed
 * Miscellaneous Examples


 * Chapter III: The fundamental properties of Analytic Functions; Taylor's, Laurent's, and Liouville's Theorems
 * 29. The dependence of one complex number on another
 * 30. Continuity
 * 31. Definite integrals
 * 32. Limit to the value of a definite integral
 * 33. Property of the elementary functions
 * 34. Occasional failure of the property; singularities
 * 35. The analytic function
 * 36. Cauchy's theorem on the integral of a function round a contour
 * 37. The value of a function at a point, expressed as an integral taken round a contour enclosing the point
 * 38. The higher derivates
 * 39. Taylor's theorem
 * 40. Forms of the remainder in Taylor's series
 * 41. The process of continuation
 * 42. The identity of a function
 * 43. Laurent's theorem
 * 44. The nature of the singularities of a one-valued function
 * 45. The point at infinity
 * 46. Many-valued functions
 * 47. Liouville's theorem
 * 48. Functions with no essential singularities
 * Miscellaneous Examples


 * Chapter IV: The Uniform Convergence of Infinite Series
 * 49. Uniform convergence
 * 50. Connexion of discontinuity with non-uniform convergence
 * 51. Distinction between absolute and uniform convergence
 * 52. Condition for uniform convergence
 * 53. Integration of infinite series
 * 54. Differentiation of infinite series
 * 55. Uniform convergence of power-series
 * Miscellaneous Examples


 * Chapter V: The Theory of Residues; application to the evaluation of Real Definite Integrals
 * 56. Residues
 * 57. Evaluation of real definite integrals
 * 58. Evaluation of the definite integral of a rational function
 * 59. Cauchy's integral
 * 60. The numbers of roots of an equation contained within a contour
 * 61. Connexion between the zeros of a function and the zeros of its derivate
 * Miscellaneous Examples


 * Chapter VI: The expansion of functions in Infinite Series
 * 62. Darboux's Formula
 * 63. The Bernoullian numbers and the Bernoullian polynomials
 * 64. The Maclaurin-Bernoullian expansion
 * 65. Burmann's theorem
 * 66. Teixeira's extended form of Burmann's theorem
 * 67. Evaluation of the coefficients
 * 68. Expansion of a function of a root of an equation in terms of a parameter occurring in the equation
 * 69. Lagrange's theorem
 * 70. Rouché's expansion of Lagrange's theorem
 * 71. Teixeira's generalisation of Lagrange's theorem
 * 72. Laplace's extension of Lagrange's theorem
 * 73. A further generalisation of Taylor's theorem
 * 74. The expansion of a function as a series of rational functions
 * 75. Expansion of a function as an infinite product
 * 76. Expansion of a periodic function as a series of cotangents
 * 77. Expansion in inverse factorials
 * Miscellaneous Examples


 * Chapter VII: Fourier Series
 * 78. Definition of Fourier Series; nature of the region within which a Fourier series converges
 * 79. Values of the coefficients in terms of the sum of a Fourier series, when the series converges at all points in a belt of finite breadth in the $z$-plane
 * 80. Fourier's theorem
 * 81. The representation of a function by Fourier series for ranges other than $0$ to $2\pi$
 * 82. The sine and cosine series
 * 83. Alternative proof of Fourier's theorem
 * 84. Nature of the convergence of a Fourier series
 * 85. Determination of points of discontinuity
 * 86. The uniqueness of the Fourier expansion
 * Miscellaneous Examples


 * Chapter VIII: Asymptotic Expansions
 * 87. Simple example of an asymptotic expansion
 * 88. Definition of an asymptotic expansion
 * 89. Another example of an asymptotic expansion
 * 90. Multiplication of asymptotic expansions
 * 91. Integration of asymptotic expansions
 * 92. Uniqueness of an asymptotic expansion
 * Miscellaneous Examples