A Course of Modern Analysis: An Introduction to the General Theory of Infinite Series and of Analytic Functions, with an Account of the Principal Transcendental Functions

Edmund Taylor Whittaker

University Press, 1902 - Calculus - 378 pages

Contents

• CHAPTER I: 3
• Complex numbers: 4
• The modulus of a complex quantity: 5
• The geometrical interpretation of complex numbers: 6
• MISCELLANEOUS EXAMPLES: 7
• CHAPTER II: 8
• Convergence of an infinite series: 10
• Absolute convergence and semiconvergence: 12
• The geometric series and the series En: 13
• The comparisontheorem: 14
• Discussion of a special series of importance: 16
• A convergencytest which depends on the ratio of the successive terms of a series: 17
• A general theorem on those series for which Limit Un+1 is 1: 18
• Convergence of the hypergeometric series: 20
• Effect of changing the order of the terms in a series: 21
• The fundamental property of absolutely convergent series: 22
• Cauchys theorem on the multiplication of absolutely convergent series: 24
• Mertens theorem on the multiplication of a semiconvergent series by an absolutely convergent series: 25
• Abels result on the multiplication of series: 26
• Powerseries: 28
• SECTION PAGE 22 Convergence of series derived from a powerseries: 30
• Infinite products: 31
• Some examples of infinite products: 32
• Cauchys theorem on products which are not absolutely convergent: 34
• Infinite determinants: 35
• Convergence of an infinite determinant: 36
• Persistence of convergence when the elements are changed: 37
• CHAPTER III: 40
• Continuity: 41
• Definite integrals: 42
• Limit to the value of a definite integral: 44
• Occasional failure of the property singularities: 45
• Cauchys theorem on the integral of a function round a contour: 47
• The value of a function at a point expressed as an integral taken round a contour enclosing the point: 50
• The higher derivates: 51
• Taylors theorem: 54
• Forms of the remainder in Taylors series: 56
• The process of continuation: 57
• The identity of a function: 59
• Laurents theorem: 60
• The nature of the singularities of a onevalued function: 63
• The point at infinity: 64
• Manyvalued functions: 66
• Liouvilles theorem: 69
• MISCELLANEOUS EXAMPLES: 70
• CHAPTER IV: 73
• Connexion of discontinuity with nonuniform convergence: 76
• Distinction between absolute and uniform convergence: 77
• Condition for uniform convergence: 78
• Differentiation of infinite series: 81
• MISCELLANEOUS EXAMPLES: 82
• THE THEORY OF RESIDUES APPLICATION TO THE EVALUATION OF REAL DEFINITE INTEGRALS PAGE SECTION 56 Residues: 83
• Evaluation of real definite integrals: 84
• Evaluation of the definite integral of a rational function: 91
• Cauchys integral 92: 92
• Connexion between the zeros of a function and the zeros of its derivate: 93
• MISCELLANEOUS EXAMPLES: 94
• CHAPTER VI: 96
• The Bernoullian numbers and the Bernoullian polynomials: 97
• The MaclaurinBernoullian expansion: 99
• Burmanns theorem: 100
• Teixeiras extended form of Burmanns theorem: 102
• Evaluation of the coefficients: 103
• Expansion of a function of a root of an equation in terms of a parameter occurring in the equation: 105
• Lagranges theorem: 106
• Rouchés extension of Lagranges theorem: 108
• Teixeiras generalisation of Lagranges theorem: 109
• A further generalisation of Taylors theorem: 110
• The expansion of a function as a series of rational functions: 111
• Expansion of a function as an infinite product: 114
• Expansion of a periodic function as a series of cotangents: 116
• Expansion in inverse factorials: 117
• MISCELLANEOUS EXAMPLES: 119
• CHAPTER VII: 127
• 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 zplane: 130
• Fouriers theorem: 131
• The representation of a function by Fourier series for ranges other than 0 to 2π: 137
• The sine and cosine series: 138
• Alternative proof of Fouriers theorem: 140
• Nature of the convergence of a Fourier series: 147
• Determination of points of discontinuity: 151
• The uniqueness of the Fourier expansion: 152
• MISCELLANEOUS EXAMPLES: 157
• CHAPTER VIII: 163
• Definition of an asymptotic expansion: 164
• Another example of an asymptotic expansion: 165
• Multiplication of asymptotic expansions: 167
• Integration of asymptotic expansions: 168
• MISCELLANEOUS EXAMPLES: 169
• TRANSCENDENTAL FUNCTIONS: 171
• CHAPTER IX: 173
• The Weierstrassian form for the Gammafunction: 174
• The differenceequation satisfied by the Gammafunction: 176
• Evaluation of a general class of infinite products: 177
• Connexion between the Gammafunction and the circular functions: 179
• Expansions for the logarithmic derivates of the Gammafunction: 180
• Heines expression of F z as a contourintegral: 181
• Expression of r 2 as a definite integral whose path of integration is real: 183
• Extension of the definiteintegral expression to the case in which the argument of the Gammafunction is negative: 184
• Gauss expression of the logarithmic derivate of the Gammafunction as a definite integral: 185
• Binets expression of log F z in terms of a definite integral: 186
• The Eulerian integral of the first kind: 189
• Expression of the Eulerian integral of the first kind in terms of Gamma functions: 190
• Evaluation of trigonometric integrals in terms of the Gammafunction: 191
• The asymptotic expansion of the logarithm of the Gammafunction Stirlings series: 193
• Asymptotic expansion of the Gammafunction: 194
• MISCELLANEOUS EXAMPLES: 195
• CHAPTER X: 204
• Schläffis integral for Pn 2: 205
• Rodrigues formula for the Legendre polynomials: 206
• The integralproperties of the Legendre polynomials: 207
• Legendre functions: 208
• The recurrenceformulae: 210
• Evaluation of the integralexpression for Pn 2 as a powerseries: 213
• Laplaces integralexpression for Pn 2: 215
• The MehlerDirichlet definite integral for Pn 2: 218
• Expansion of Pn 2 as a series of powers of 1z: 220
• The Legendre functions of the second kind: 221
• Expansion of Qn 2 as a powerseries: 222
• The recurrenceformulae for the Legendre function of the second kind: 224
• Laplaces integral for the Legendre function of the second kind: 225
• Relation between Pn 2 and Qn 2 when n is an integer: 226
• Expansion of tx1 as a series of Legendre polynomials: 228
• Neumanns expansion of an arbitrary function as a series of Legendre polynomials: 230
• The associated functions Pm z and Qm 2: 231
• The definite integrals of the associated Legendre functions: 232
• Expansion of Pm 2 as a definite integral of Laplaces type: 233
• Alternative expression of Pm z as a definite integral of Laplaces type: 234
• The function C 2: 235
• MISCELLANEOUS EXAMPLES: 236
• CHAPTER XI: 240
• Value of the series F a b c 1: 241
• The differential equation satisfied by the hypergeometric series: 242
• The Legendre functions as a particular case of the hypergeometric function: 245
• Transformations of the general hypergeometric function: 246
• The twentyfour particular solutions of the hypergeometric differential equation: 249
• Relations between the particular solutions of the hypergeometric differential equation: 251
• Solution of the general hypergeometric differential equation by a definite integral: 253
• Determination of the integral which represents Pa: 257
• Evaluation of a doublecontour integral: 259
• Relations between contiguous hypergeometric functions: 260
• MISCELLANEOUS EXAMPLES: 263
• BESSEL FUNCTIONS: 266
• The general solution of Bessels equation by Bessel functions whose order: 272
• Extension of the integralformula to the case in which n is not an integer: 279
• Bessel functions as a limiting case of Legendre functions: 287
• The second solution of Bessels equation when the order is an integer: 294
• Proof of Neumanns expansion: 300
• CHAPTER XIII: 309
• Determination of a solution of Laplaces equation which satisfies given: 315
• CHAPTER XIV: 322
• Definition ofz: 323
• Periodicity and other properties ofz: 324
• Expression of the function 2 by means of an integral: 325
• The homogeneity of the function 2: 329
• Another form of the addition theorem: 332
• The roots e₁ ед вз: 333
• Addition of a halfperiod to the argument of 2: 334
• Integration of ax +46x3+6cx²+4dx+e: 335
• Another solution of the integrationproblem: 336
• Uniforınisation of curves of genus unity: 338
• MISCELLANEOUS EXAMPLES: 340
• CHAPTER XV: 342
• Expression of the function fz by means of an integral: 343
• The function sn z: 345
• The functions en z and dnz: 346
• Expression of enz and dnz by means of integrals: 347
• The additiontheorem for the function dnz: 348
• The additiontheorems for the functions snz and en z: 350
• The constant K: 351
• The constant K: 352
• The periodicity of the elliptic functions with respect to K+iK: 353
• The behaviour of the functions snz en z dnz at the point ziK: 354
• General description of the functions snz enz dnz: 355
• Connexion of the function snz with the function2: 356
• Expansion of snz as a trigonometric series: 357
• MISCELLANEOUS EXAMPLES: 359
• CHAPTER XVI: 362
• Expression of any elliptic function in terms of 2 and 2: 363
• Relation between any two elliptic functions which admit the same periods: 364
• Relation between the zeros and poles of an elliptic function: 365
• The function 3: 366
• The quasiperiodicity of the function 2: 367
• The function σ2: 368
• The quasiperiodicity of the function σ 2: 369
• The integration of an elliptic function: 372
• MISCELLANEOUS EXAMPLES: 374
• INDEX: 377

Popular passages

Page 82 - The former denotes a family of straight lines whose distance from the origin is equal to a, the latter a circle whose centre is at the origin, and whose radius is equal to a. And here, as was noted generally by Lagrange, the singular solution seems to be, in relation to geometry, the more important of the two. 3. A more general class of problems is that in which it is required to determine the curves in which some one of the foregoing elements, Art. 1, is equal...

Page 305 - This is a linear differential equation of the second order with constant coefficients ; its solution, found in the usual way, is where A and B are arbitrary constants.

Page 32 - Those quantities which retain the same value are called constant; those whose values are varying are called variable. When variable quantities are so connected that the value of one of them is determined by the value ascribed to the others, that variable quantity is said to be a function of the others.

Page 259 - Untersuchung des Theils der planetarischen Storungen, welcher aus der Bewegung der Sonne entsteht.

Page 4 - A complex number is an expression of the form a + bi, where a and b are real numbers and i =V~ 1 -In the complex number, a is called the real part and bi is the imaginary part.

Page 307 - Laplace's equation, and equating to zero the coefficients of the various powers of x, y, and...

Page 6 - I by 6, then r and 6 are clearly the radius vector and vectorial angle of the point P, referred to the origin 0 and axis Ox. The representation of complex quantities thus afforded is often called the Argand diagram*.

Page 102 - Consider now a function /(*), whose only singularities in the finite part of the plane are...

Page 115 - Although the persymmetric determinant where sr is the sum of the rth powers of the roots of the equation x...

Page 174 - ... by the stipulation that the variable is not to cross the real axis at any point on the positive side of the origin.

Bibliographic information

• Title: A Course of Modern Analysis: An Introduction to the General Theory of Infinite Series and of Analytic Functions, with an Account of the Principal Transcendental Functions
• Author: Edmund Taylor Whittaker
• Publisher: University Press, 1902
• Length: 378 pages