Special functions:
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|---|---|
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| Language: | English Chinese |
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Singapore [u.a.]
World Scientific
1989
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| Online Access: | Inhaltsverzeichnis |
| Item Description: | Aus dem Chines. übers. |
| Physical Description: | XVI, 695 S. |
| ISBN: | 9971506599 997150667X |
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| 100 | 1 | |a Wang, Z. X. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Special functions |c Z. X. Wang ; D. R. Guo |
| 264 | 1 | |a Singapore [u.a.] |b World Scientific |c 1989 | |
| 300 | |a XVI, 695 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Aus dem Chines. übers. | ||
| 650 | 7 | |a Fonctions spéciales |2 ram | |
| 650 | 7 | |a Funcoes (matematica) |2 larpcal | |
| 650 | 4 | |a Functions, Special | |
| 650 | 0 | 7 | |a Spezielle Funktion |0 (DE-588)4182213-4 |2 gnd |9 rswk-swf |
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| 700 | 1 | |a Guo, D. R. |e Verfasser |4 aut | |
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CONTENTS
Preface
.
v
Chapter
1.
THE EXPANSION OF FUNCTIONS IN
INFINITE SERIES AND INFINITE PRODUCTS
1.1.
Bernoulli Polynomials and Bernoulli Numbers
. 1
1.2.
Euler
Polynomials and
Euler
Numbers
. 5
1.3.
Euler-Maclaurin Formula
. 8
1.4.
Lagrange's Expansion Formula
. 14
1.5.
Expansion of Meromorphic Functions in Rational Fractions
. 17
1.6.
Infinite Product
. 21
1.7.
The Expansion of a Function in Infinite Product.
Weierstrass
Theorem
. 25
1.8.
Asymptotic Expansion
. 29
1.9.
The Asymptotic Expansion of the Laplace Integral,
Watson's Lemma
. 34
1.10.
Expansion in Terms of Functions of an
Orthonormal
Set
. 36
Exercise
1.41
Chapter
2.
LINEAR ORDINARY DIFFERENTIAL
EQUATIONS OF THE SECOND ORDER
2.1.
Singular Points of Linear Ordinary Differential Equations of the
x
Special
Functions
Second Order
. 47
2.2.
Solution of the Equation in the Vicinity of an Ordinary Point
. . 48
2.3.
Solutions of the Equation in the Vicinity of a Singular Point
. . 51
2.4.
Regular Solution. Regular Singularities
. 55
2.5.
Frobenius Method
. 61
2.6.
Point at Infinity
.· · 63
2.7.
Equations of Fuchsian Type
. 64
2.8.
Equations of Fuchsian Type Having Five Regular Singular Points
66
2.9.
Equations of Fuchsian Type Having Three Regular Singularities
. 68
2.10.
Irregular Singularities. Formal Regular Solution
. 71
2.11.
Irregular Singularities. Normal Solutions and Subnormal
Solutions
. 73
2.12.
Method of Solution by Integrals. Basic Principle
. 78
2.13.
Equations of Laplacian Type and Laplace Transform
. 81
2.14.
Euler
Transform
. 85
Exercise
2. 88
Chapter
3.
THE GAMMA FUNCTION
3.1.
Definition of the Gamma Function
. 93
3.2.
Recurrence Relation
. 94
3.3.
The Infinite-Product Expression of
Euler
. 95
3.4.
Weierstrass'
Infinite Product
. 97
3.5.
Relation between the
Г
-Function
and the Trigonometric
Function
. 98
3.6.
Multiplication Formula
. 99
3.7.
Contour Integral
. 101
3.8.
Euler
Integral of the First Kind. B-Function
. 103
3.9.
Double-Contour Integral
. 105
3.10.
Dirichlet Integral
. 106
3.11.
Logarithmic Derivative of the
Г
-Function
. 107
3.12.
Asymptotic Expansions
.
Ill
3.13.
Another Derivation of the Asymptotic Expansion
. 112
3.14.
Riemann ^-Function
. 114
3.15.
The Functional Equation of the
ς-
Function
. 115
3.16.
The Value of f(s,a) when
s
is an Integer
.
Ц7
3.17.
Hermite Formula
.
ng
3.18.
Relation to the
Γ-
Function
. 120
3.19.
Euler
Product of the f-Function
. 123
3.20.
Riemann Integral of the ^-Function
. 124
Contents xi
3.21.
Another Derivation of the Asymptotic Expansion of the
T-Function
.125
3.22.
Evaluation of the ^-Function
.128
Exercise
3. 128
Chapter
4.
HYPERGEOMETRIC FUNCTION
4.1.
Hypergeometric Series and Hypergeometric Function
. 135
4.2.
Recurrence Relations
. 137
4.3.
Other Solutions of the Hypergeometric Equation Expressed in
Terms of Hypergeometric Functions
. 139
4.4.
The Second Solution of the Hypergeometric Equation when the
Difference of the Exponents is an Integer
. 144
4.5.
Integral Representations of the Hypergeometric Function
. 150
4.6.
Barnes' Integral Representation of the Hypergeometric Function
. 153
4.7.
The Value of
F(a,ß, 7,1) . 156
4.8.
Connections between the Fundamental Solutions at the Singular
Points
0,1, 00.
Analytic Continuation
. 159
4.9.
When
7-а-
β, α- β
are Integers
. 162
4.10.
Jacobi Polynomials
. 169
4.11.
Chebyshev Polynomials
. 173
4.12.
Quadratic Transformations
. 177
4.13.
Kummer's Formula and Summation Formula Derived from It
. . 184
4.14.
Asymptotic Expansions for Large Parameters
. 186
4.15.
Generalized Hypergeometric Series
. 189
4.16.
Hypergeometric Series with Two Variables
. 191
4.17.
The Transformation Formulae of
Fi
and F2
. 195
4.18.
Reducible Cases
. 197
Exercise
4 . 202
Chapter
б.
LEGENDRE FUNCTIONS
5.1.
Legendre Equation
. 210
5.2.
Legendre Polynomials
.,.,.,., 212
5.3.
The Generating Function of Pn(x).
Rodrigues
Formula
, , , , , 215
5.4.
Integral Representations of Pn
(ж)
. 216
5.5.
Recurrence Relations of Pn
(%) . 218
5.6.
Legendre Polynomials as a Complete Set of
Orthonormal
Functions
. 219
5.7.
Zeros of Pn(x)
.,.,,.,. 223
5.8.
Legendre Functions of the Second Kind, Qn (x)
. 224
xü Special
Functions
5.9.
Recurrence Relations of Qn{x)
.
5.10.
Expansion of the Function (i
-
t)"1 in Terms of Legendre
Functions. Neumann Expansion
.231
5.11.
Associate Legendre Functions
P¡m(x)
.233
5.12.
Orthogonality Relations of
Ρ,"1 (ζ)
.235
5.13.
Recurrence Relations for P,m(z) and
QŢ{x)
. 239
5.14.
Addition Formula
. 241
5.15.
Spherical Surface Harmonics
¥іт(в,<р)
.244
5.16.
The General Associate Legendre Functions P£{z)
. 247
5.17.
Q4(z)
. 251
5.18.
Definition of PP{x) on the Cut: -oo
<
χ
< 1.255
5.19.
Definition of Qa(x) on the Cut: -oo
<
χ
< 1. . . . 258
5.20.
Other Integral Expressions for Pu (z) and P? {z)
. 262
5.21.
Addition Formulae
. 267
5.22.
Asymptotic Expansions of P£(cos0) and Q£(cos0) when v
-»
oo
. 270
5.23.
Ultra-Spherical Polynomials C* (x)
.274
Exercise
5.277
Chapter
6.
CONFLUENT HYPERGEOMETRIC FUNCTIONS
6.1.
Confluent Hypergeometric Functions
. . 296
6.2.
Relations among the Consecutive Functions
.299
6.3.
Whittaker Equation and Whittaker Functions Mk,m(z)
. 300
6.4.
Integral Representations
.302
6.5.
Whittaker Functions Wk<m(z)
.305
6.6.
Asymptotic Expansion of W/cim
(2)
when
г
—»
oo
.307
6.7.
Barnes' Integral Representation of Wk,m(z)
. 310
6.8.
Relations between W±k<m{±z) and M±k,±m{±z). Asymptotic
Expansion of F(a,
η,ζ).
Stokes Phenomenon
. 313
6.9.
The Case when
7
(or 2m) is an Integer
. 316
6.10.
The Asymptotic Expansions of F(a,
η,ζ)
for Large
|a|, j-yj
. . .318
6.11.
Differential Equations Reducible to the Confluent Hypergeometric
Equation
. 318
6.12.
Weber Equation. Parabolic Cylinder Functions Dn(z)
. 320
6.13.
Hermite Functions and Hermite Polynomials
. 325
6.14.
Laguerre Polynomials
. 327
6.15.
Other Special Functions Expressible by Whittaker Functions
. . 332
Exercise
6. 335
Contents xiii
Chapter
7. BESSEL
FUNCTIONS
7.1. Bessel
Equation. Its Relation to the Confluent Hypergeometric
Equation
.345
7.2.
Bessel Functions of the First Kind: J±u(z), 2v
φ
integer
. 347
7.3.
Bessel Functions of Order Half an Odd Integer: Jn+i
(г)
(n = 0,±l,±2,.)
. . . .' . . . . , .350
7.4.
Integral Representations of Ju(z)
. 351
7.5.
Bessel Functions of Integral Order Jn(z)(n
— 0,1,2,.) . , . .359
7.6.
Bessel Functions of the Second Kind Yu (z)
. 365
7.7.
Bessel Functions of the Third Kind (Hankel Functions)
НІ1)(г),Нр\г)
. 368
7.8.
Modified (or Imaginary Argument) Bessel Functions
Iu{z)
and
Ku(z). Thomson Functions
Ьегу(г)
and
bei„(z);
etc.
. 374
7.9.
Spherical Bessel Functions ji{z),ni[z),h\l\z), h^\z)
. . . . . 376
7.10.
Asymptotic Expansions for the Case ¡ar|
—»■
oo
.378
7.11.
The Method of Steepest Descent
. 381
7.12.
Asymptotic Expansions of Bessel Functions of Order
и
for Large
\v\ and
|г|
.384
7.13.
Addition Formulae
.395
7.14.
Integrals Containing Bessel Functions.
(1)
Finite Integrals
. . . 399
7.15.
Integrals Containing Bessel Functions.
(2)
Infinite Integrals
. . . 401
7.16.
Neumann Expansion
.412
7.17.
Kapteyn Expansion
. 415
7.18.
The Zeros of Bessel Functions
. 420
7.19.
Fourier-Bessel Expansion
. 424
Exercise
7. 425
Chapter
8.
WEIERSTRASS
ELLIPTIC FUNCTIONS
8.1.
Elliptic Integrals and Elliptic Functions
. 456
8.2.
The Periods of Elliptic Integrals
. 460
8.3.
The General Properties of Doubly-Periodic Functions and
Elliptic Functions
. 462
8.4.
The Function p(z)
. 466
8.5.
Algebraic Relation between p(z) and p'(z)
.,., , . , 468
8.6.
The Function
ζ(ζ)
.,,,. 471
8.7.
The Function
σ{ζ)
. 473
8.8.
Homogeneity of the
Weierstrass
Elliptic Function
. 476
8.9.
Representations of a General Elliptic Function
.,.,. 476
8.10.
Addition Formulae
. 481
xiv
Special Functions
8.11.
Expressing the Coordinates of a Cubic Curve by
Elliptic Functions
.".485
8.12.
The Problem of a Quartic Polynomial
.486
8.13.
Curves of Genus (Deficiency)
1 .489
Exercise
8. . 493
Chapter
9.
Theta Functions
9.1.
The Theta Function
θ(ν)
. 498
9.2.
The Functions
âh(v)
. 500
9.3.
Elliptic Functions Represented by Theta Functions
. 502
9.4.
Relations among the Squares of
#*(ν)
. 503
9.5.
Addition Formulae
. 504
9.6.
Differential Equations Satisfied by Theta Functions
. 506
9.7.
The Values of Some Constants
.508
9.8.
Legendre's Elliptic Integral of the First Kind
. 510
9.9.
Jacobi's Imaginary Transformation
. 514
9.10.
Transformation of
Landen-
Type
.516
9.11.
Representation of Theta Functions by Infinite Product
.517
9.12.
Fourier Expansion of the Logarithmic Derivatives of Theta
Functions
.521
9.13.
The Functions 0(u) and #(u)
.522
Exercise
9. 523
Chapter
10.
JACOBIAN ELLIPTIC FUNCTIONS
10.1.
Jacobian Elliptic Functions sn u, en
u, dn u
. 530
10.2.
Geometric Representations of Jacobian Elliptic Functions
. . . 532
10.3.
Complete Elliptic Integrals
. 535
10.4.
Addition Formulae
. . 537
10.5.
The Periodicity of Jacobian Elliptic Functions
. 539
10.6.
The Zeros and Poles of Jacobian Elliptic Functions
. 540
10.7.
Transformations of Elliptic Functions
. 542
10.8.
Reductions of Elliptic Integrals
. 545
10.9.
Elliptic Integral of the Second Kind
. 552
10.10.
Elliptic Integral of the Third Kind
. 553
10.11.
Properties of the Function E(u)
. 555
10.12.
Differential Equations Satisfied by
К
and
E
with Respect
to
к
and Expansions of
К
and
E
with Respect to
к
. 558
10.13.
Relations between Jacobian Elliptic Functions and Theta
Functions
.,. 561
Contents xv
10.14.
Expressing Jacobian Elliptic Functions in Infinite Products and
Fourier Series
.566
Exercise
10. 569
Chapter
11.
LAMÉ
FUNCTIONS
11.1.
Ellipsoidal Coordinates
.,,, 575
11.2.
Representing the Coordinates with Elliptic Functions
, . . . , 578
11.3.
Lamé
Equation
. 580
11.4.
Four Types of
Lamé
Functions
.,,,,,,.,., 583
11.5.
Ellipsoidal Harmonics
. 589
11.6.
Niven's Representation
. 591
11.7.
On the Zeros of
Lamé
Polynomials
. 595
11.8.
Lamé
Functions of the Second Kind
. 597
11.9.
Generalized
Lamé
Functions
. 599
11.10.
Integral Equations of
Lamé
Functions
. 602
11.11.
The Integral Representation of Ellipsoidal Harmonics
. 604
Exercise
11.607
Chapter
12.
MATHIEU
FUNCTIONS
12.1.
Mathieu
Equation
.610
12.2.
General Properties of the Solution. Fundamental Solutions
. . . 612
12.3.
Floquet Solution
. 614
12.4.
Periodic Solutions of
Mathieu
Equation
.615
12.5.
Fourier Expansion of the Floquet Solution
. 617
12.6.
Formulae for Computing Eigenvalues X(q)
. 620
12.7.
Mathieu
Functions cem
(z), m
— 0,1,2,.
and
ѕе^г),
m=
1,2,. . 624
12.8.
Expansion of \u(q) in Powers of
q
. 627
12.9.
Fourier Expansions of cem (z) and sem (z) for Small
q
. 630
12.10.
Infinite Determinant
. 632
12.11.
НШ
Equation
.,.,.,.,,.,,. 633
12.12.
Stable and Unstable Solutions of
Mathieu
Equation.,
Stable Region and Unstable Region
.,,., 637
12.13.
Approximate Solutions
oî Mathieu
Equatio». for A
>
q
:
ι) , ,
640
12.14.
Integral Equations for
Mathieu
Functions
.,,. = ,,,,, 643
Exercise
12 . .,.,,.,-,,.,., 646
Appendices
Appendix I. Roots of a Cubic Equation
. 654
xvi
Special
Functions
Appendix
II. Roots of the Quartic Equation
.656
Appendix III. Orthogonal Curvilinear Coordinate Systems
.658
Bibliography
.677
Glossary
.679
Index
. . . 683 |
| any_adam_object | 1 |
| author | Wang, Z. X. Guo, D. R. |
| author_facet | Wang, Z. X. Guo, D. R. |
| author_role | aut aut |
| author_sort | Wang, Z. X. |
| author_variant | z x w zx zxw d r g dr drg |
| building | Verbundindex |
| bvnumber | BV004058289 |
| callnumber-first | Q - Science |
| callnumber-label | QA331 |
| callnumber-raw | QA331 QC20.7.F87 |
| callnumber-search | QA331 QC20.7.F87 |
| callnumber-sort | QA 3331 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 680 |
| classification_tum | MAT 330f |
| ctrlnum | (OCoLC)243435756 (DE-599)BVBBV004058289 |
| dewey-full | 515/.5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.5 |
| dewey-search | 515/.5 |
| dewey-sort | 3515 15 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV004058289 |
| illustrated | Not Illustrated |
| indexdate | 2024-08-14T00:45:14Z |
| institution | BVB |
| isbn | 9971506599 997150667X |
| language | English Chinese |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-002537942 |
| oclc_num | 243435756 |
| open_access_boolean | |
| owner | DE-12 DE-91 DE-BY-TUM DE-824 DE-703 DE-706 DE-11 DE-739 DE-188 |
| owner_facet | DE-12 DE-91 DE-BY-TUM DE-824 DE-703 DE-706 DE-11 DE-739 DE-188 |
| physical | XVI, 695 S. |
| publishDate | 1989 |
| publishDateSearch | 1989 |
| publishDateSort | 1989 |
| publisher | World Scientific |
| record_format | marc |
| spelling | Wang, Z. X. Verfasser aut Special functions Z. X. Wang ; D. R. Guo Singapore [u.a.] World Scientific 1989 XVI, 695 S. txt rdacontent n rdamedia nc rdacarrier Aus dem Chines. übers. Fonctions spéciales ram Funcoes (matematica) larpcal Functions, Special Spezielle Funktion (DE-588)4182213-4 gnd rswk-swf Funktion Mathematik (DE-588)4071510-3 gnd rswk-swf Analysis (DE-588)4001865-9 gnd rswk-swf Funktion Mathematik (DE-588)4071510-3 s DE-604 Spezielle Funktion (DE-588)4182213-4 s Analysis (DE-588)4001865-9 s Guo, D. R. Verfasser aut Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002537942&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Wang, Z. X. Guo, D. R. Special functions Fonctions spéciales ram Funcoes (matematica) larpcal Functions, Special Spezielle Funktion (DE-588)4182213-4 gnd Funktion Mathematik (DE-588)4071510-3 gnd Analysis (DE-588)4001865-9 gnd |
| subject_GND | (DE-588)4182213-4 (DE-588)4071510-3 (DE-588)4001865-9 |
| title | Special functions |
| title_auth | Special functions |
| title_exact_search | Special functions |
| title_full | Special functions Z. X. Wang ; D. R. Guo |
| title_fullStr | Special functions Z. X. Wang ; D. R. Guo |
| title_full_unstemmed | Special functions Z. X. Wang ; D. R. Guo |
| title_short | Special functions |
| title_sort | special functions |
| topic | Fonctions spéciales ram Funcoes (matematica) larpcal Functions, Special Spezielle Funktion (DE-588)4182213-4 gnd Funktion Mathematik (DE-588)4071510-3 gnd Analysis (DE-588)4001865-9 gnd |
| topic_facet | Fonctions spéciales Funcoes (matematica) Functions, Special Spezielle Funktion Funktion Mathematik Analysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002537942&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT wangzx specialfunctions AT guodr specialfunctions |