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

Front Cover
University Press, 1902 - Calculus - 378 pages
 

Contents

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 Riemanns theorem on semiconvergent series
22
Infinite products
23
Cauchys theorem on the multiplication of absolutely convergent series
24
Mertens theorem on the multiplication of a semiconvergent series by an absolutely convergent series 21 222 22
25
Infinite determinants
26
Convergence of an infinite determinant
27
Abels result on the multiplication of series
27
PAGE
27
Powerseries
28
26
35
MISCELLANEOUS EXAMPLES 37
37
CHAPTER III
40
Continuity
41
Definite integrals
42
Limit to the value of a definite integral 33 Property of the elementary functions
44
Occasional failure of the property singularities 35 The analytic function
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
55
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 48 Functions with no essential singularities MISCELLANEOUS EXAMPLES
69
THE UNIFORM CONVERGENCE OF INFINITE SERIES
73
Uniform convergence
75
Connexion of discontinuity with nonuniform convergence
76
Distinction between absolute and uniform convergence 76
77
Condition for uniform convergence 53 Integration of infinite series
78
Differentiation of infinite series 55 Uniform convergence of powerseries
81
MISCELLANEOUS EXAMPLES
82
CHAPTER V
83
PAGE 83
84
Evaluation of the definite integral of a rational function
91
Cauchys integral
92
Connexion between the zeros of a function and the zeros of its derivate
93
MISCELLANEOUS EXAMPLES 93
94
CHAPTER VI
96
Darbouxs formula 64 The MaclaurinBernoullian expansion 96 The Bernoullian numbers and the Bernoullian polynomials
97
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 70 Rouchés extension of Lagranges theorem 71 Teixeiras generalisation of ...
105
Laplaces extension of Lagranges theorem
109
A further generalisation of Taylors theorem 109
110
Expansion of a periodic function as a series of cotangents 77 Expansion in inverse factorials
111
78
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 80
131
81
137
to 2π 82 The sine and cosine series
138
Alternative proof of Fouriers theorem
140
84
146
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
SECTION
163
93
173
97
179
Gauss expression of the logarithmic derivate of the Gammafunction
185
Evaluation of trigonometric integrals in terms of the Gammafunction
191
CHAPTER X
204
Schläflis integral for P 2
205
Rodrigues formula for the Legendre polynomials
206
The integralproperties of the Legendre polynomials
207
The recurrenceformulae for the Legendre function of the second kind
224
Laplaces integral for the Legendre function of the second kind
225
Relation between P 2 and Qn 2 when n is an integer
226
127
228
Neumanns expansion of an arbitrary function as a series of Legendre polynomials
230
The associated functions Pm z and Qm 2
231
130
232
131
233
Alternative expression of Pm 2 as a definite integral of Laplaces type
234
The function C₂ z
235
MISCELLANEOUS EXAMPLES
236
CHAPTER XI
240
Value of the series F a b c 1
241
137
242
138
245
Transformations of the general hypergeometric function
246
140
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
The Bessel coefficients
266
147
268
The general solution of Bessels equation by Bessel functions whose order
272
151
275
152
277
157
284
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
MISCELLANEOUS EXAMPLES
321
CHAPTER XIV
322
THE ELLIPTIC FUNCTION
323
Periodicity and other properties of P z
324
Expression of the function 2 by means of an integral
325
The homogeneity of the function z
329
Another form of the addition theorem
332
The roots e₁ eg ez
333
Addition of a halfperiod to the argument of z
334
Integration of ax+4bx³+6cx²+4dx+e
335
Another solution of the integrationproblem
336
Uniformisation of curves of genus unity
338
MISCELLANEOUS EXAMPLES
340
CHAPTER XV
342
Expression of the function f2 by means of an integral
343
The function snz
346
Expression of enz and dnz by means of integrals
347
The additiontheorem for the function dnz
348
The additiontheorems for the functions sn z and cn 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 sn z cn z dn z at the point ziK
354
General description of the functions sn z cn z dn z
355
Connexion of the function snz with the function 2
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 2
366
The quasiperiodicity of the function z
367
The function σ 2
368
The quasiperiodicity of the function σ 2
369
The integration of an elliptic function
372
MISCELLANEOUS EXAMPLES
374
INDEX
377

Other editions - View all

Common terms and phrases

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