Holomorphic functions in the plane and n-dimensional space:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel [u.a.]
Birkhäuser
2008
|
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 377 - 384 |
| Physical Description: | XIII, 394 S. Ill., graph. Darst. 24 cm CD-ROM (12 cm) |
| ISBN: | 9783764382711 9783764382728 3764382716 |
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MARC
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| 100 | 1 | |a Gürlebeck, Klaus |d 1954- |e Verfasser |0 (DE-588)1015214991 |4 aut | |
| 240 | 1 | 0 | |a Funktionentheorie in der Ebene und im Raum |
| 245 | 1 | 0 | |a Holomorphic functions in the plane and n-dimensional space |c Klaus Gürlebeck ; Klaus Habetha ; Wolfgang Sprößig |
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 2008 | |
| 300 | |a XIII, 394 S. |b Ill., graph. Darst. |c 24 cm |e CD-ROM (12 cm) | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Literaturverz. S. 377 - 384 | ||
| 650 | 4 | |a Holomorphic functions | |
| 650 | 4 | |a Holomorphic functions |v Problems, exercises, etc | |
| 650 | 0 | 7 | |a Holomorphe Funktion |0 (DE-588)4025645-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Funktionentheorie |0 (DE-588)4018935-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Clifford-Analysis |0 (DE-588)4484012-3 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Funktionentheorie |0 (DE-588)4018935-1 |D s |
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| 689 | 1 | 0 | |a Clifford-Analysis |0 (DE-588)4484012-3 |D s |
| 689 | 1 | |5 DE-604 | |
| 689 | 2 | 0 | |a Holomorphe Funktion |0 (DE-588)4025645-5 |D s |
| 689 | 2 | |5 DE-604 | |
| 700 | 1 | |a Habetha, Klaus |d 1932-2024 |e Verfasser |0 (DE-588)1078136440 |4 aut | |
| 700 | 1 | |a Sprößig, Wolfgang |d 1946- |e Verfasser |0 (DE-588)1055083103 |4 aut | |
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Record in the Search Index
| _version_ | 1805090448088236032 |
|---|---|
| adam_text |
Contents
Preface
to the German Edition
xi
Preface to the English Edition
xiv
I Numbers
1
1
Complex numbers
. 2
1.1
The History of Their Discovery
. 2
1.2
Definition and Properties
. 3
1.3
Representations and geometric aspects
. 10
1.4
Exercises
. 13
2
Quaternions
. 15
2.1
The history of their discovery
. 15
2.2
Definition and properties
. 16
2.3
Mappings and representations
. 24
2.3.1
Basic maps
. 24
2.3.2
Rotations in R3
. 26
2.3.3
Rotations of
Ж4
. 30
2.3.4
Representations
. 31
2.4
Vectors and geometrical aspects
. 33
2.4.1
Bilinear products
. 37
2.4.2
Multilinear products
. 42
2.5
Applications
. 46
2.5.1
Visualization of the sphere S3
. 46
2.5.2
Elements of spherical trigonometry
. 47
2.6
Exercises
. 49
3
Clifford numbers
. 50
3.1
History of the discovery
. 50
3.2
Definition and properties
. 52
3.2.1
Definition of the Clifford algebra
. 52
3.2.2
Structures and automorphisms
. 55
3.2.3
Modulus
. 58
3.3
Geometrie
applications
. 61
3.3.1
Spin groups
. 61
3.3.2
Construction of rotations of
Жп
. 63
3.3.3
Rotations of Rn+1
. 66
3.4
Representations
. 67
3.5
Exercises
. 71
II Functions
73
4
Topological aspects
. 74
4.1
Topology and continuity
. 74
4.2
Series
. 79
4.3
Riemann spheres
. 83
4.3.1
Complex case
. 83
4.3.2
Higher dimensions
. 87
4.4
Exercises
. 88
5
Holomorphic functions
. 90
5.1
Differentiation in
С
. 90
5.2
Differentiation in
И
. 95
5.2.1
Mejlikhzhon's result
. 96
5.2.2
H-holomorphic functions
. 97
5.2.3
Holomorphic functions and differential forms
. 101
5.3
Differentiation in
С£(п)
. 104
5.4
Exercises
. 107
6
Powers and
Möbius
transforms
. 108
6.1
Powers
. 108
6.1.1
Powers in
С
. 108
6.1.2
Powers in higher dimensions
. 109
6.2
Möbius
transformations
. 114
6.2.1
Möbius
transformations in
С
. 114
6.2.2
Möbius
transformations in higher dimensions
. . . 118
6.3
Exercises
. 124
III Integration and integral theorems
125
7
Integral theorems and integral formulae
. 126
7.1
Cauchy's integral theorem and its inversion
. 126
7.2
Formulae of Borel-Pompeiu and Cauchy
. 129
7.2.1
Formula of Borel-Pompeiu
. 129
7.2.2
Formula of Cauchy
. 131
7.2.3
Formulae of Plemelj-Sokhotski
. 133
7.2.4
History of Cauchy and Borel-Pompeiu formulae
. 138
7.3
Consequences of Cauchy's integral formula
. 141
7.3.1
Higher order derivatives of holomorphic functions
141
7.3.2
Mean value property and maximum principle
. . . 144
7.3.3
Liouville's theorem
. 146
7.3.4
Integral formulae of
Schwarz
and
Poisson
. 147
7.4
Exercises
. 149
8
Teodorescu transform
. 151
8.1
Properties of the Teodorescu transform
. 151
8.2
Hodge decomposition of the quaternionic Hubert space
. 156
8.2.1
Hodge decomposition
. 156
8.2.2
Representation theorem
. 159
8.3
Exercises
. 160
IV Series expansions and local behavior
161
9
Power series
. 162
9.1 Weierstraß'
convergence theorems, power series
. 162
9.1.1
Convergence theorems according to
Weierstraß . . 162
9.1.2
Power series in
С
. 164
9.1.3
Power series in Ci{n)
. 167
9.2
Taylor and Laurent series in
С
. 169
9.2.1
Taylor series
. 169
9.2.2
Laurent series
. 173
9.3
Taylor and Laurent series in
Ci(n)
. 175
9.3.1
Taylor series
. 175
9.3.2
Laurent series
. 181
9.4
Exercises
. 184
10
Orthogonal expansions in
H
. 186
10.1
Complete H-holomorphic function systems
. 186
10.1.1
Polynomial systems
. 188
10.1.2
Inner and outer spherical functions
. 191
10.1.3
Harmonic spherical functions
. 194
10.1.4
H-holomorphic spherical functions
. 196
10.1.5
Completeness in L2{B3)
Π
ker
д
. 202
10.2
Fourier expansion in
И
. 203
10.3
Applications
. 203
10.3.1
Derivatives of H-holomorphic polynomials
. 203
10.3.2
Primitives of H-holomorphic functions
. 207
10.3.3
Decomposition theorem and Taylor expansion
. . 213
10.4
Exercises
. 215
11
Elementary functions
. 218
11.1
Elementary functions in
С
. 218
11.1.1
Exponential function
. 218
11.1.2
Trigonometric functions
. 219
11.1.3
Hyperbolic functions
. 221
11.1.4
Logarithm
. 223
11.2
Elementary functions in C£(n)
. 225
11.2.1
Polar decomposition of the Cauchy-Riemann
operator
. 225
11.2.2
Elementary radial functions
. 229
11.2.3
Pueter—See construction of holomorphic functions
234
11.2.4
Cauehy-Kovalevsky extension
. 239
11.2.5
Separation of variables
. 244
11.3
Exercises
. 249
12
Local structure of holomorphic functions
. 252
12.1
Behavior at zeros
. 252
12.1.1
Zeros in
С
. 252
12.1.2
Zeros in C£(n)
. 255
12.2
Isolated singularities of holomorphic functions
. 259
12.2.1
Isolated singularities in
С
. 259
12.2.2
Isolated singularities in Ct(n)
. 265
12.3
Residue theorem and the argument principle
. 267
12.3.1
Residue theorem in
С
. 267
12.3.2
Argument principle in
С
. 270
12.3.3
Residue theorem in
Ci(n)
. 274
12.3.4
Argument principle in C£(n)
. 276
12.4
Calculation of real integrals
. 279
12.5
Exercises
. 285
13
Special functions
. 287
13.1
Euler's Gamma function
. 287
13.1.1
Definition and functional equation
. 287
13.1.2
Stirling's theorem
. 291
13.2
Riemann's
Zeta
function
. 296
13.2.1
Dirichlet series
. 296
13.2.2
Riemann's
Zeta
function
. 298
13.3
Automorphic forms and functions
. 302
13.3.1
Automorphic forms and functions in
С
. 302
13.3.2
Automorphic functions and forms in
Clin)
. 307
13.4
Exercises
. 321
Appendix
323
A.I Differential forms in Rn
. 324
A.
1.1
Alternating linear mappings
. 324
A.1.2 Differential forms
. 329
A.1.3 Exercises
. 336
A.
2
Integration and manifolds
. 338
A.2.1 Integration
. 338
A.2.1.1 Integration in
Шп+1
. 338
A.2.1.
2
Transformation of variables
. 339
A.2.1.3 Manifolds and integration
. 341
A.2.2 Theorems of Stokes,
Gauß,
and Green
. 351
A.2.2.1 Theorem of Stokes
. 351
A.2.2.2 Theorem of
Gauß. 352
A.2.2.3
Theorem of
Green. 354
A.2.3 Exercises
. 355
A.3 Some function spaces
. 357
A.3.1 Spaces of Holder continuous functions
. 357
A.
3.2
Spaces of differentiable functions
. 358
A.3.3 Spaces of
integrable
functions
. 359
A.3.4 Distributions
. 360
A.3.5 Hardy spaces
. 361
A.3.6 Sobolev spaces
. 361
A.4 Properties of holomorphic spherical functions
. 363
A.
4.1
Properties of Legendre polynomials
. 363
A.
4.2
Norm of holomorphic spherical functions
. 364
A.4.3 Scalar products of holomorphic spherical functions
. 368
A.
4.4
Complete
orthonormal
systems in
Тї^ш
. 370
A.4.
5
Derivatives of holomorphic spherical functions
. 374
A.4.6 Exercises
. 375
Bibliography
377
Index
385 |
| adam_txt |
Contents
Preface
to the German Edition
xi
Preface to the English Edition
xiv
I Numbers
1
1
Complex numbers
. 2
1.1
The History of Their Discovery
. 2
1.2
Definition and Properties
. 3
1.3
Representations and geometric aspects
. 10
1.4
Exercises
. 13
2
Quaternions
. 15
2.1
The history of their discovery
. 15
2.2
Definition and properties
. 16
2.3
Mappings and representations
. 24
2.3.1
Basic maps
. 24
2.3.2
Rotations in R3
. 26
2.3.3
Rotations of
Ж4
. 30
2.3.4
Representations
. 31
2.4
Vectors and geometrical aspects
. 33
2.4.1
Bilinear products
. 37
2.4.2
Multilinear products
. 42
2.5
Applications
. 46
2.5.1
Visualization of the sphere S3
. 46
2.5.2
Elements of spherical trigonometry
. 47
2.6
Exercises
. 49
3
Clifford numbers
. 50
3.1
History of the discovery
. 50
3.2
Definition and properties
. 52
3.2.1
Definition of the Clifford algebra
. 52
3.2.2
Structures and automorphisms
. 55
3.2.3
Modulus
. 58
3.3
Geometrie
applications
. 61
3.3.1
Spin groups
. 61
3.3.2
Construction of rotations of
Жп
. 63
3.3.3
Rotations of Rn+1
. 66
3.4
Representations
. 67
3.5
Exercises
. 71
II Functions
73
4
Topological aspects
. 74
4.1
Topology and continuity
. 74
4.2
Series
. 79
4.3
Riemann spheres
. 83
4.3.1
Complex case
. 83
4.3.2
Higher dimensions
. 87
4.4
Exercises
. 88
5
Holomorphic functions
. 90
5.1
Differentiation in
С
. 90
5.2
Differentiation in
И
. 95
5.2.1
Mejlikhzhon's result
. 96
5.2.2
H-holomorphic functions
. 97
5.2.3
Holomorphic functions and differential forms
. 101
5.3
Differentiation in
С£(п)
. 104
5.4
Exercises
. 107
6
Powers and
Möbius
transforms
. 108
6.1
Powers
. 108
6.1.1
Powers in
С
. 108
6.1.2
Powers in higher dimensions
. 109
6.2
Möbius
transformations
. 114
6.2.1
Möbius
transformations in
С
. 114
6.2.2
Möbius
transformations in higher dimensions
. . . 118
6.3
Exercises
. 124
III Integration and integral theorems
125
7
Integral theorems and integral formulae
. 126
7.1
Cauchy's integral theorem and its inversion
. 126
7.2
Formulae of Borel-Pompeiu and Cauchy
. 129
7.2.1
Formula of Borel-Pompeiu
. 129
7.2.2
Formula of Cauchy
. 131
7.2.3
Formulae of Plemelj-Sokhotski
. 133
7.2.4
History of Cauchy and Borel-Pompeiu formulae
. 138
7.3
Consequences of Cauchy's integral formula
. 141
7.3.1
Higher order derivatives of holomorphic functions
141
7.3.2
Mean value property and maximum principle
. . . 144
7.3.3
Liouville's theorem
. 146
7.3.4
Integral formulae of
Schwarz
and
Poisson
. 147
7.4
Exercises
. 149
8
Teodorescu transform
. 151
8.1
Properties of the Teodorescu transform
. 151
8.2
Hodge decomposition of the quaternionic Hubert space
. 156
8.2.1
Hodge decomposition
. 156
8.2.2
Representation theorem
. 159
8.3
Exercises
. 160
IV Series expansions and local behavior
161
9
Power series
. 162
9.1 Weierstraß'
convergence theorems, power series
. 162
9.1.1
Convergence theorems according to
Weierstraß . . 162
9.1.2
Power series in
С
. 164
9.1.3
Power series in Ci{n)
. 167
9.2
Taylor and Laurent series in
С
. 169
9.2.1
Taylor series
. 169
9.2.2
Laurent series
. 173
9.3
Taylor and Laurent series in
Ci(n)
. 175
9.3.1
Taylor series
. 175
9.3.2
Laurent series
. 181
9.4
Exercises
. 184
10
Orthogonal expansions in
H
. 186
10.1
Complete H-holomorphic function systems
. 186
10.1.1
Polynomial systems
. 188
10.1.2
Inner and outer spherical functions
. 191
10.1.3
Harmonic spherical functions
. 194
10.1.4
H-holomorphic spherical functions
. 196
10.1.5
Completeness in L2{B3)
Π
ker
д
. 202
10.2
Fourier expansion in
И
. 203
10.3
Applications
. 203
10.3.1
Derivatives of H-holomorphic polynomials
. 203
10.3.2
Primitives of H-holomorphic functions
. 207
10.3.3
Decomposition theorem and Taylor expansion
. . 213
10.4
Exercises
. 215
11
Elementary functions
. 218
11.1
Elementary functions in
С
. 218
11.1.1
Exponential function
. 218
11.1.2
Trigonometric functions
. 219
11.1.3
Hyperbolic functions
. 221
11.1.4
Logarithm
. 223
11.2
Elementary functions in C£(n)
. 225
11.2.1
Polar decomposition of the Cauchy-Riemann
operator
. 225
11.2.2
Elementary radial functions
. 229
11.2.3
Pueter—See construction of holomorphic functions
234
11.2.4
Cauehy-Kovalevsky extension
. 239
11.2.5
Separation of variables
. 244
11.3
Exercises
. 249
12
Local structure of holomorphic functions
. 252
12.1
Behavior at zeros
. 252
12.1.1
Zeros in
С
. 252
12.1.2
Zeros in C£(n)
. 255
12.2
Isolated singularities of holomorphic functions
. 259
12.2.1
Isolated singularities in
С
. 259
12.2.2
Isolated singularities in Ct(n)
. 265
12.3
Residue theorem and the argument principle
. 267
12.3.1
Residue theorem in
С
. 267
12.3.2
Argument principle in
С
. 270
12.3.3
Residue theorem in
Ci(n)
. 274
12.3.4
Argument principle in C£(n)
. 276
12.4
Calculation of real integrals
. 279
12.5
Exercises
. 285
13
Special functions
. 287
13.1
Euler's Gamma function
. 287
13.1.1
Definition and functional equation
. 287
13.1.2
Stirling's theorem
. 291
13.2
Riemann's
Zeta
function
. 296
13.2.1
Dirichlet series
. 296
13.2.2
Riemann's
Zeta
function
. 298
13.3
Automorphic forms and functions
. 302
13.3.1
Automorphic forms and functions in
С
. 302
13.3.2
Automorphic functions and forms in
Clin)
. 307
13.4
Exercises
. 321
Appendix
323
A.I Differential forms in Rn
. 324
A.
1.1
Alternating linear mappings
. 324
A.1.2 Differential forms
. 329
A.1.3 Exercises
. 336
A.
2
Integration and manifolds
. 338
A.2.1 Integration
. 338
A.2.1.1 Integration in
Шп+1
. 338
A.2.1.
2
Transformation of variables
. 339
A.2.1.3 Manifolds and integration
. 341
A.2.2 Theorems of Stokes,
Gauß,
and Green
. 351
A.2.2.1 Theorem of Stokes
. 351
A.2.2.2 Theorem of
Gauß. 352
A.2.2.3
Theorem of
Green. 354
A.2.3 Exercises
. 355
A.3 Some function spaces
. 357
A.3.1 Spaces of Holder continuous functions
. 357
A.
3.2
Spaces of differentiable functions
. 358
A.3.3 Spaces of
integrable
functions
. 359
A.3.4 Distributions
. 360
A.3.5 Hardy spaces
. 361
A.3.6 Sobolev spaces
. 361
A.4 Properties of holomorphic spherical functions
. 363
A.
4.1
Properties of Legendre polynomials
. 363
A.
4.2
Norm of holomorphic spherical functions
. 364
A.4.3 Scalar products of holomorphic spherical functions
. 368
A.
4.4
Complete
orthonormal
systems in
Тї^ш
. 370
A.4.
5
Derivatives of holomorphic spherical functions
. 374
A.4.6 Exercises
. 375
Bibliography
377
Index
385 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Gürlebeck, Klaus 1954- Habetha, Klaus 1932-2024 Sprößig, Wolfgang 1946- |
| author_GND | (DE-588)1015214991 (DE-588)1078136440 (DE-588)1055083103 |
| author_facet | Gürlebeck, Klaus 1954- Habetha, Klaus 1932-2024 Sprößig, Wolfgang 1946- |
| author_role | aut aut aut |
| author_sort | Gürlebeck, Klaus 1954- |
| author_variant | k g kg k h kh w s ws |
| building | Verbundindex |
| bvnumber | BV023305840 |
| callnumber-first | Q - Science |
| callnumber-label | QA331 |
| callnumber-raw | QA331 |
| callnumber-search | QA331 |
| callnumber-sort | QA 3331 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 700 |
| ctrlnum | (OCoLC)173239493 (DE-599)DNB981981453 |
| dewey-full | 515.98 515.9 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.98 515.9 |
| dewey-search | 515.98 515.9 |
| dewey-sort | 3515.98 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV023305840 |
| illustrated | Illustrated |
| index_date | 2024-07-02T20:48:27Z |
| indexdate | 2024-07-20T09:39:58Z |
| institution | BVB |
| isbn | 9783764382711 9783764382728 3764382716 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016490214 |
| oclc_num | 173239493 |
| open_access_boolean | |
| owner | DE-824 DE-19 DE-BY-UBM DE-355 DE-BY-UBR |
| owner_facet | DE-824 DE-19 DE-BY-UBM DE-355 DE-BY-UBR |
| physical | XIII, 394 S. Ill., graph. Darst. 24 cm CD-ROM (12 cm) |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Birkhäuser |
| record_format | marc |
| spelling | Gürlebeck, Klaus 1954- Verfasser (DE-588)1015214991 aut Funktionentheorie in der Ebene und im Raum Holomorphic functions in the plane and n-dimensional space Klaus Gürlebeck ; Klaus Habetha ; Wolfgang Sprößig Basel [u.a.] Birkhäuser 2008 XIII, 394 S. Ill., graph. Darst. 24 cm CD-ROM (12 cm) txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. 377 - 384 Holomorphic functions Holomorphic functions Problems, exercises, etc Holomorphe Funktion (DE-588)4025645-5 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Clifford-Analysis (DE-588)4484012-3 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 s DE-604 Clifford-Analysis (DE-588)4484012-3 s Holomorphe Funktion (DE-588)4025645-5 s Habetha, Klaus 1932-2024 Verfasser (DE-588)1078136440 aut Sprößig, Wolfgang 1946- Verfasser (DE-588)1055083103 aut text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2876002&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016490214&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gürlebeck, Klaus 1954- Habetha, Klaus 1932-2024 Sprößig, Wolfgang 1946- Holomorphic functions in the plane and n-dimensional space Holomorphic functions Holomorphic functions Problems, exercises, etc Holomorphe Funktion (DE-588)4025645-5 gnd Funktionentheorie (DE-588)4018935-1 gnd Clifford-Analysis (DE-588)4484012-3 gnd |
| subject_GND | (DE-588)4025645-5 (DE-588)4018935-1 (DE-588)4484012-3 |
| title | Holomorphic functions in the plane and n-dimensional space |
| title_alt | Funktionentheorie in der Ebene und im Raum |
| title_auth | Holomorphic functions in the plane and n-dimensional space |
| title_exact_search | Holomorphic functions in the plane and n-dimensional space |
| title_exact_search_txtP | Holomorphic functions in the plane and n-dimensional space |
| title_full | Holomorphic functions in the plane and n-dimensional space Klaus Gürlebeck ; Klaus Habetha ; Wolfgang Sprößig |
| title_fullStr | Holomorphic functions in the plane and n-dimensional space Klaus Gürlebeck ; Klaus Habetha ; Wolfgang Sprößig |
| title_full_unstemmed | Holomorphic functions in the plane and n-dimensional space Klaus Gürlebeck ; Klaus Habetha ; Wolfgang Sprößig |
| title_short | Holomorphic functions in the plane and n-dimensional space |
| title_sort | holomorphic functions in the plane and n dimensional space |
| topic | Holomorphic functions Holomorphic functions Problems, exercises, etc Holomorphe Funktion (DE-588)4025645-5 gnd Funktionentheorie (DE-588)4018935-1 gnd Clifford-Analysis (DE-588)4484012-3 gnd |
| topic_facet | Holomorphic functions Holomorphic functions Problems, exercises, etc Holomorphe Funktion Funktionentheorie Clifford-Analysis |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=2876002&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016490214&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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