Applied pseudoanalytic function theory:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Basel [u.a.]
Birkhäuser
2009
|
| Schriftenreihe: | Frontiers in Mathematics
|
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017321361&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XII, 184 S. 240 mm x 170 mm |
| ISBN: | 9783034600033 9783034600040 |
Internformat
MARC
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| 100 | 1 | |a Kravchenko, Vladislav V. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Applied pseudoanalytic function theory |c Vladislav V. Kravchenko |
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 2009 | |
| 300 | |a XII, 184 S. |c 240 mm x 170 mm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Frontiers in Mathematics | |
| 650 | 4 | |a Differential equations, Partial | |
| 650 | 4 | |a Functions of complex variables | |
| 650 | 0 | 7 | |a Pseudoanalytische Funktion |0 (DE-588)4176118-2 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Pseudoanalytische Funktion |0 (DE-588)4176118-2 |D s |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-017321361 | |
Datensatz im Suchindex
| DE-BY-TUM_call_number | 0102 MAT 307f 2009 A 4026 |
|---|---|
| DE-BY-TUM_katkey | 1682573 |
| DE-BY-TUM_location | 01 |
| DE-BY-TUM_media_number | 040010151658 |
| _version_ | 1821933637258444801 |
| adam_text | Titel: Applied pseudoanalytic function theory
Autor: Kravchenko, Vladislav V.
Jahr: 2009
Contents
Foreword..................................... xi
1 Introduction ................................. 1
Part I Pseudoanalytic Function Theory and
Second-order Elliptic Equations
2 Definitions and Results from Bers Theory
2.1 Generating pairs and differentiation................. 9
2.2 Pseudoanalytic functions....................... 11
2.3 Derivatives and integrals of pseudoanalytic functions....... 13
2.3.1 Equivalent generating pairs................. 13
2.3.2 Vekua s equation for (F, G)-derivatives........... 14
2.3.3 Integration .......................... 16
3 Solutions of Second-order Elliptic Equations as Real Components
of Complex Pseudoanalytic Functions
3.1 Factorization of the stationary Schrodinger operator....... 21
3.2 Factorization of the operator divpgrad+g............. 23
3.3 Conjugate metaharmonic functions................. 27
3.4 The main Vekua equation...................... 29
3.5 Cauchy s integral theorem for the Schrodinger equation...... 31
3.6 p-analytic functions.......................... 32
4 Formal Powers
4.1 Definition............................... 35
4.2 An important special case...................... 37
4.3 Similarity principle.......................... 38
4.4 Taylor series in formal powers.................... 41
4.5 The Runge theorem ......................... 43
4.6 Complete systems of solutions for second-order equations..... 43
4.7 A remark on orthogonal coordinate systems in a plane...... 45
i Contents
4.8 Explicit construction of a generating sequence........... 46
4.9 Explicit construction of complete systems of solutions
of second-order elliptic equations.................. 50
4.9.1 Explicit construction of complete systems of solutions
for a stationary Schrodinger equation............ 51
4.9.2 Complete systems of solutions for the
conductivity equation .................... 51
4.10 Numerical solution of boundary value problems.......... 52
Cauchy s Integral Formula
5.1 Preliminary information on the Cauchy integral formula
for pseudoanalytic functions..................... 55
5.2 Relation between the main Vekua equation and the
system describing p-analytic functions............... 57
5.3 The transplant operator....................... 58
5.4 Cauchy integral formulas for a;fc-analytic functions ........ 60
Complex Riccati Equation
6.1 Preliminary notes........................... 65
6.2 The two-dimensional stationary Schrodinger equation and
the complex Riccati equation.................... 67
6.3 Generalizations of classical theorems................ 69
6.4 Cauchy s integral theorem...................... 71
Part II Applications to Sturm-Liouville Theory
7 A Representation for Solutions of the Sturm-Liouville Equation
7.1 Solving the one-dimensional Schrodinger equation......... 75
7.2 The + -case............................. 76
7.3 Two sets of Taylor coefficients.................... 80
7.4 Solution of the one-dimensional Schrodinger equation....... 81
7.5 Validating the result......................... 83
7.6 The - case............................. 84
7.7 Complex potential.......................... 85
7.8 Solution of the Sturm-Liouville equation.............. 86
7.9 Numerical method for solving Sturm-Liouville equations..... 90
8 Spectral Problems and Darboux Transformation
8.1 Sturm-Liouville problem as a problem of finding zeros
of an analytic function........................ 93
8.1.1 Sturm-Liouville problems with spectral parameter
dependent boundary conditions............... 95
Contents
8.2 Numerical method for solving Sturm-Liouville problems..... 96
8.3 A remark on the Darboux transformation............. 98
Part III Applications to Real First-order Systems
9 Beltrami Fields
9.1 Description of the result....................... 103
9.2 Reduction of (9.1) to a Vekua equation............... 104
9.3 Solution in the case when a is a function of one
Cartesian variable........................... 105
10 Static Maxwell System in Axially Symmetric Inhomogeneous Media
10.1 Meridional and transverse fields................... Ill
10.2 Reduction of the static Maxwell system to
p-analytic functions.......................... 112
10.2.1 The meridional case..................... 112
10.2.2 The transverse case...................... 112
10.3 Construction of formal powers.................... 113
10.3.1 Formal powers in the meridional case............ 113
10.3.2 Formal powers in the transverse case............ 114
Part IV Hyperbolic Pseudoanalytic Functions
11 Hyperbolic Numbers and Analytic Functions
12 Hyperbolic Pseudoanalytic Functions
12.1 Differential operators......................... 125
12.2 Hyperbolic pseudoanalytic function theory............. 126
12.3 Generating sequences......................... 130
13 Relationship between Hyperbolic Pseudoanalytic Functions
and Solutions of the Klein-Gordon Equation
13.1 Factorization of the Klein-Gordon equation............133
13.2 The main hyperbolic Vekua equation................135
13.3 Generating sequence for the main hyperbolic
Vekua equation............................138
x Contents
Part V Bicomplex and Biquaternionic Pseudoanalytic
Functions and Applications
14 The Dirac Equation
14.1 Notation................................ 150
14.2 Quaternionic form of the Dirac equation.............. 151
14.3 The Dirac equation in a two-dimensional case as
a bicomplex Vekua equation..................... 153
14.4 Some definitions and results from Bers theory for
bicomplex pseudoanalytic functions................. 155
14.4.1 Generating pair, derivative and antiderivative....... 155
14.4.2 Generating sequences and Taylor series
in formal powers....................... 156
14.5 The main bicomplex Vekua equation................ 158
14.6 Dirac equation with a scalar potential............... 159
15 Complex Second-order Elliptic Equations and
Bicomplex Pseudoanalytic Functions....................163
16 Multidimensional Second-order Equations
16.1 Factorization.............................167
16.2 The main quaternionic Vekua equation...............169
Open Problems..................................171
Bibliography...................................173
Index.......................................183
|
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| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV035400714 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T13:31:39Z |
| institution | BVB |
| isbn | 9783034600033 9783034600040 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017321361 |
| oclc_num | 403388292 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-20 DE-11 DE-83 |
| owner_facet | DE-91G DE-BY-TUM DE-20 DE-11 DE-83 |
| physical | XII, 184 S. 240 mm x 170 mm |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Birkhäuser |
| record_format | marc |
| series2 | Frontiers in Mathematics |
| spellingShingle | Kravchenko, Vladislav V. Applied pseudoanalytic function theory Differential equations, Partial Functions of complex variables Pseudoanalytische Funktion (DE-588)4176118-2 gnd |
| subject_GND | (DE-588)4176118-2 |
| title | Applied pseudoanalytic function theory |
| title_auth | Applied pseudoanalytic function theory |
| title_exact_search | Applied pseudoanalytic function theory |
| title_full | Applied pseudoanalytic function theory Vladislav V. Kravchenko |
| title_fullStr | Applied pseudoanalytic function theory Vladislav V. Kravchenko |
| title_full_unstemmed | Applied pseudoanalytic function theory Vladislav V. Kravchenko |
| title_short | Applied pseudoanalytic function theory |
| title_sort | applied pseudoanalytic function theory |
| topic | Differential equations, Partial Functions of complex variables Pseudoanalytische Funktion (DE-588)4176118-2 gnd |
| topic_facet | Differential equations, Partial Functions of complex variables Pseudoanalytische Funktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017321361&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT kravchenkovladislavv appliedpseudoanalyticfunctiontheory |
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Teilbibliothek Mathematik & Informatik
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0102 MAT 307f 2009 A 4026
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