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 (Google eBook)

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University Press, 1902 - Functions - 378 pages
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Contents

The geometric series and the series 2n
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
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
Merteus 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 one valued function
63
The point at infinity
64
Manyvalued functions
66
Liouvillcs 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
83
Evaluation of real definite integrals
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
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
Rouches 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 HI 75 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 2ir
137
The sine and cosine series 1 38
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 rz as a contourintegral
181
Expression of r z 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 rz 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
Evaluation of the integralexpression for PKz M a powerseries
213
Laplaces integralexpression for Pz
215
The MehlerDirichlet definite integral for Pnt
218
Expansion of Pnz as a series of powers of z
220
The Legendre functions of the second kind
221
Expansion of Qn z 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 PK 2 and Qn z when n is an integer
226
Expansion of txl as a series of Legendre polynomials
228
Neumanns expansion of an arbitrary function as a series of Legendre polynomials
230
The associated functions Pnm z and QHmz
231
The definite integrals of the associated Legendre functions
232
Expansion of Pnmz as a definite integral of Laplaces type
233
Alternative expression of Pnm z as a definite integral of Laplaces type
234
The function Cnz
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 PI
257
Evaluation of a doublecontour integral
259
Relations between contiguous hypergeometric functions
260
Miscellaneous Examples
263
CHAPTER XII
266
Bessels differential equation
268
Bessels equation as a case of the hypergeomctric equation
269
The general solution of Bessels equation by Bessel functions whose order is not necessarily an integer
272
The recurrenceformulae for the Bessel functions
274
Relation between two Bessel functions whose orders differ by an integer
275
The roots of Bessel functions
277
Extension of the integralformula to the case in which n is not an integer
279
A second expression of Jn z as a definite integral whose path of integration is real
282
Hankels definiteintegral solution of Bessels differential equation
283
Expression of Jn z for all values of n and z by an integral of Hankels type
284
Bessel functions as a limiting case of Legendre functions
287
Bessel functions whose order is half an odd integer
288
Expression of Jn z in a form which furnishes an approximate value to J z for large real positive values of z
289
The asymptotic expansion of the Bessel functions
292
The second solution of Bessels equation when the order is an integer
294
Neumanns expansion determination of the coefficients
299
Proof of Neumanns expansion
300
Schlomilchs expansion of an arbitrary function in terms of Bessel functions of order zero
302
Tabulation of the Bessel functions
304
CHAPTER XIII
309
Laplaces equation the general solution certain particular solutions
311
The seriessolution of Laplaces equation
314
Determination of a solution of Laplaces equation which satisfies given boundaryconditions
315
Particular solutions of Laplaces equation which depend on Bessel functions
317
Solution of the equation + + V0
318
Solution of the equation +2+2+I0
319
Miscellaneous Examples
321
CHAPTER XIV
322
Definition of f
323
Periodicity and other properties of I? z
324
Expression of the function fl z by means of an integral
325
The homogeneity of the function fr z
329
Another form of the addition theorem
332
The roots e2 e3
333
Addition of a halfperiod to the argument of jpz
334
Integration of ttjut + 4bx3 + 6cx2 + 4dj+ei
335
Another solution of the integrationproblem
336
Uniformisation of curves of genus unity
338
Miscellaneous Examples
340
CHAPTER XV
342
Expression of the function z by means of an integral
343
The function snz
346
Expression of en z and dnz by means of integrals
347
The additiontheorem for the function dnz
348
The additiontheorems for the functions sn z 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 sn z en z dn z at the point z i A
354
General description of the functions snz enz dnz
355
Connexion of the function sn z with the function jf z
356
Expansion of sin z as a trigonometric series
357
Miscellaneous Examples
359
CHAPTER XVI
362
Expression of any elliptic function in terms of z and P z
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 f
366
The quasiperiodicity of the function z
367
The function r
368
The quasiperiodicity of the function tr z
369
The integration of an elliptic function
372
Miscellaneous Examples
374
Index
377

Common terms and phrases

Popular passages

Page 90 - 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 313 - 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 40 - 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 267 - 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 315 - 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 110 - Consider now a function /(*), whose only singularities in the finite part of the plane are...
Page 123 - Although the persymmetric determinant where sr is the sum of the rth powers of the roots of the equation x...
Page 182 - ... by the stipulation that the variable is not to cross the real axis at any point on the positive side of the origin.

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A Course of Modern Analysis - Cambridge University Press
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Zentralblatt MATH Database 1931 – 2008 0951.30002
A course of modern analysis. An introduction to the general theory of. infinite processes and of analytic functions; with an account of the principal ...
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