Mathieu Functions and Spheroidal Functions and Their Mathematical Foundations
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837 Josef Meixner Friedrich W. Schafke Gerhard Wolf
Mathieu Functions and Spheroidal Functions and Their Mathematical Foundations Further Studies
Springer-Verlag Berlin Heidelberg New York 1980
Authors
Josef Meixner Am Blockhaus 31 5100 Aachen Federal Republic of Germany Friedrich W. Schafke Fakultat fur Mathematik Universitat Konstanz Postfach 5560 7750 Konstanz Federal Republic of Germany Gerhard Wolf FB 6 Mathematik Universitat-Gesarnthochschule Universitatsstrabe 3 Postfach 6843 4300 Essen 1 Federal Republic of Germany
AMS Subject Classifications (1980): 33A40, 33A45, 33A55, 34A20, 34B25, 34 B30, 34D05, 34E05, 35J05, 47 A 70 ISBN 3-540-10282-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-10282-5 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by SpringerNeriag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
TABLE OF CONTENTS
,
Introduction and Preface. 1. Foundations. 1.1. Eigenvalue problems with two parameters. 1.1.0.
1
Introduction.
1
3 7
1.1.1.
First presuppositions. Preliminary remarks.
1.1.2.
Estimates for the resolvent.
1.1.3.
The eigenvalues to
O.
11
1.1.4.
Further presuppositions and conclusions
15
1.1.5.
The residues of the resolvent. Principal solutions.
16
1.1.6.
Equiconvergence.
22
1.1.7.
Holomorphy properties. Estimates.
23
1.1.8.
Additional estimates. On the appl ication to boundary value problems for ordinary
26 27
1.' .9.
differential equations and differential systems. 1.1.10. Appl ication to Hill's differential equation In the real
33
domain. 1.1.11. Application to Hill's differential equation in the complex
36
domain. 1.1.12. Application to the spheroidal differential equation in the real domain.
39
1.1.13. Appl ication to the spheroidal differential equation in the
42
complex domain. 1.2. Simply separated operators.
44
1.2. O.
Introduction.
1.2.1.
The algebraic problem.
44 46
1.2.2.
Adjoint mappings.
50
1.2.3.
The analytical problem. Expansion theorem.
1.2.4.
The symmetric case.
53
1.2.5.
Appi i ca t l ons .
2. Mathieu Functions. 2.1. Integral relations. 2.1.1.
Integral relations of the fi rst kind.
2.1_2.
Integral relations of the second kind (with variable boundaries) .
58
60 63 63 61 71
IV
2.2. Addition theorems. 2.2.1. Lemmas concerning the transformation equation.
73 74
2.2.2.
Integral relations.
75
2.2.3.
The addition theorems.
2.2.4.
Consequences and special cases.
77 80
2.3. On the computation of the characteristic exponent. 2.4. On the eigenvalues for com
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