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References
- Wolfgang Lay, Universität Stuttgart
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- Higher Special Functions
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- 16 May 2024
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- 23 May 2024, pp 296-297
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Appendix B - Curriculum Vitae of George Cecil Jaffé
- Wolfgang Lay, Universität Stuttgart
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- Higher Special Functions
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- 16 May 2024
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- 23 May 2024, pp 293-295
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Frontmatter
- Wolfgang Lay, Universität Stuttgart
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- Higher Special Functions
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- 16 May 2024
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- 23 May 2024, pp i-iv
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Contents
- Wolfgang Lay, Universität Stuttgart
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- Higher Special Functions
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- 23 May 2024, pp vii-viii
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Index
- Wolfgang Lay, Universität Stuttgart
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- Higher Special Functions
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- 23 May 2024, pp 298-300
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Dedication
- Wolfgang Lay, Universität Stuttgart
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- Higher Special Functions
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- 16 May 2024
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- 23 May 2024, pp v-vi
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4 - Central Two-Point Connection Problems and Higher Special Functions
- Wolfgang Lay, Universität Stuttgart
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- Higher Special Functions
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- 16 May 2024
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- 23 May 2024, pp 157-189
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Summary
We have a division into classical and higher special functions. The singular boundary eigenvalue problems producing classical special functions are not specifically denoted, while for higher special functions they are called central two-point connection problems (CTCPs). Solving a CTCP means determining the parameters of the underlying differential equation for which a particular solution obeys the singular boundary conditions. The method is developed in detail, and demonstrated for the Heun class. This is the main mathematical part of the book. We see the power of the Jaffé method in generalising such CTCPs. The asymptotic behaviour of the power series and the asymptotic factor is balanced in such a way that the eigenfunction behaviour is apparent when the eigenvalue parameter admits an eigenvalue. This is the ingenious aspect of Jaffé’s ansatz. Although dealing with more fundamental mathematical issues, we fill a certain gap in understanding how the Jaffé approach to the CTCP works when applied to Fuchsian differential equations and their confluent and reduced cases. The considerations are split into the basic concept and the calculatory procedure of carrying it out.
5 - Applications and Examples
- Wolfgang Lay, Universität Stuttgart
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- Higher Special Functions
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- 23 May 2024, pp 190-274
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Summary
We now present examples of higher special functions. Since not all details are contained in the theoretical discussions of the field, it is these examples that mostly contribute to enable the reader to treat self-reliantly his or her concrete problem. Starting from the question of whether nature is linear or non-linear, the answer is not unambiguous. For weather phenomena, the underlying dynamics is undoubtedly non-linear. However, in the world of atoms and elementary particles, the processes become fundamentally linear. Mathematically, this is expressed by the fact that the underlying differential equation (the so-called Schrödinger equation) becomes linear in nature as soon as atomic processes are described. Hence, a substantial part of the examples here are devoted to solving boundary eigenvalue problems of the Schrödinger equation, the fundamental differential equation in the description of microscopic nature (quantum theory). The aim is autonomy to create new special functions by applying the methods developed. There may well be as yet unseen aspects, a variety of hitherto unknown special functions or even mathematically relevant new discoveries in the field.
Preface
- Wolfgang Lay, Universität Stuttgart
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- Higher Special Functions
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- 23 May 2024, pp ix-xii
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6 - Afterword
- Wolfgang Lay, Universität Stuttgart
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- Higher Special Functions
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- 23 May 2024, pp 275-280
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Acknowledgements
- Wolfgang Lay, Universität Stuttgart
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- Higher Special Functions
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- 16 May 2024
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- 23 May 2024, pp xiii-xiv
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1 - Introduction
- Wolfgang Lay, Universität Stuttgart
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- Higher Special Functions
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- 23 May 2024, pp 1-80
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Summary
Classical special functions are a traditional field of mathematics. As particular solutions of singular boundary eigenvalue problems of linear ordinary differential equations of second order, they are, by definition, functions that can be represented as the product of an asymptotic factor and a (finite or infinite) Taylor series. The coefficients of these series are, by definition, solutions of two-term recurrence relations, from which an algebraic boundary eigenvalue criterion can be formulated. This method is called the Sommerfeld polynomial method. Thus, one can say that the boundary eigenvalue condition is, by definition, algebraic in nature. It is the central message of this book that one can resolve this restriction methodically. The method developed for this also applies to problems that can be solved with classical methods. So, in order to present the newly developed method in light of what is known, and understand the new perspective more easily, the method is applied in this chapter to already known solutions. Accordingly, it is a ’phenomenological’ introduction, based on the ad hoc introduction of the relevant quantities.
2 - Singularities in Action
- Wolfgang Lay, Universität Stuttgart
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- Higher Special Functions
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- 23 May 2024, pp 81-113
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Summary
Singularities are central to treating the boundary eigenvalue problems in this book, both singularities of differential equations and those of their solutions. Poincaré was probably the first to recognise their importance and treat them conceptually, by introducing what he called the rank. However, I have chosen a slightly different definition, introducing the ’singularity’ s-rank. With this definition, the non-elementary regular singularity is standard, with s-rank 1. Given this concept, the singularities of our treated differential equations always have half-integer s-rank, because of the order (2) of the underlying differential equation. Moreover, regular and irregular singularities are distinguished, for s-rank larger than 1 or not. There are two types of regular singularities – s-rank 1 and s-rank 1/2 – the latter called elementary singularities. Among the irregular singularities are those having integer s-rank and odd half-integer s-rank. The irregular singularity whose s-rank is smallest is R = 3/2. The standard singularity is not – as with Poincaré – the elementary one, but the non-elementary regular singularity of the underlying differential equation with s-rank 1.
3 - Fuchsian Differential Equations: The Cornerstones
- Wolfgang Lay, Universität Stuttgart
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- 23 May 2024, pp 114-156
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Global properties now come into play. To distinguish Frobenius and Thomé solutions, we consider regular and irregular singularities of differential equations and hence Fuchsian differential equations and their confluent cases. The simplest singularity of a differential equation is a pole of at most first order in the coefficient before the first derivative and a pole of at most second order in the coefficient before the zeroth derivative. Such a singularity is called regular, otherwise it is irregular. A Fuchsian differential equation only has regular singularities; with irregular singularities it is a confluent case. We also need to consider the so-called form of a differential equation. A Fuchsian differential equation with one singularity is Laplace; two singularities is Euler; three singularities is Gauss; four singularities is Heun. While singular boundary eigenvalue problems of the Gauss differential equation and its confluent cases produce classical special functions, for the Heun differential equation and its confluent cases they produce higher special functions.
Appendix A - Standard Central Two-Point Connection Problem
- Wolfgang Lay, Universität Stuttgart
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Higher Special Functions
- A Theory of the Central Two-Point Connection Problem Based on a Singularity Approach
- Wolfgang Lay
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Higher special functions emerge from boundary eigenvalue problems of Fuchsian differential equations with more than three singularities. This detailed reference provides solutions for singular boundary eigenvalue problems of linear ordinary differential equations of second order, exploring previously unknown methods for finding higher special functions. Starting from the fact that it is the singularities of a differential equation that determine the local, as well as the global, behaviour of its solutions, the author develops methods that are both new and efficient and lead to functional relationships that were previously unknown. All the developments discussed are placed within their historical context, allowing the reader to trace the roots of the theory back through the work of many generations of great mathematicians. Particular attention is given to the work of George Cecil Jaffé, who laid the foundation with the calculation of the quantum mechanical energy levels of the hydrogen molecule ion.
Everyday Pain in Middle and Later Life: Associations with Daily and Momentary Present-Moment Awareness as One Key Facet of Mindfulness
- Theresa Pauly, Anna Nicol, Jennifer C. Lay, Maureen C. Ashe, Denis Gerstorf, Peter Graf, Wolfgang Linden, Kenneth M. Madden, Atiya Mahmood, Rachel A. Murphy, Christiane A. Hoppmann
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- Canadian Journal on Aging / La Revue canadienne du vieillissement / Volume 42 / Issue 4 / December 2023
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- 11 August 2023, pp. 621-630
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This study investigated everyday associations between one key facet of mindfulness (allocating attention to the present moment) and pain. In Study 1, 89 community-dwelling adults (33–88 years; Mage = 68.6) who had experienced a stroke provided 14 daily end-of-day present-moment awareness and pain ratings. In Study 2, 100 adults (50–85 years; Mage = 67.0 years) provided momentary present-moment awareness and pain ratings three times daily for 10 days. Multi-level models showed that higher trait present-moment awareness was linked with lower overall pain (both studies). In Study 1, participants reported less pain on days on which they indicated higher present-moment awareness. In Study 2, only individuals with no post-secondary education reported less pain in moments when they indicated higher present-moment awareness. Findings add to previous research using global retrospective pain measures by showing that present-moment awareness might correlate with reduced pain experiences, assessed close in time to when they occur.
List of Contributors
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- By Yehuda Adar, Javier Solana Álvarez, Camilla Baasch Andersen, Petra Butler, Luca G. Castellani, Martin Davies, Larry A. DiMatteo, Milena Djordjević, Sieg Eiselen, Hossam A. El-Saghir, Robert W. Emerson, Wolfgang Faber, Edoardo Ferrante, Harry M. Flechtner, Morton M. Fogt, Claire M. Germain, Helena Haapio, André Janssen, Sörren Kiene, Tadas Klimas, Stefan Kröll, Sonja A. Kruisinga, Kaon Lai, Ole Lando, Li Wei, Corinne Widmer Lüchinger, Ulrich Magnus, Virginia G. Maurer, Francesco G. Mazzotta, Olaf Meyer, Marie Stefanini Newman, Ann M. Olazábal, Vladimir Pavić, Pilar Perales Viscasillas, Sylvaine Poillot-Peruzzetto, Burghard Piltz, Jan Ramberg, Vikki Rogers, Ulrich G. Schroeter, Ingeborg Schwenzer, Jan M. Smits, Lisa Spagnolo, Aneta Spaic, Matthias Spilker, Marco Torsello, Bruno Zeller, Qi “George” Zhou
- Edited by Larry A. DiMatteo
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- International Sales Law
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- 18 December 2014
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- 17 February 2014, pp xxix-xlii
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Contributors
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- By Rose Teteki Abbey, K. C. Abraham, David Tuesday Adamo, LeRoy H. Aden, Efrain Agosto, Victor Aguilan, Gillian T. W. Ahlgren, Charanjit Kaur AjitSingh, Dorothy B E A Akoto, Giuseppe Alberigo, Daniel E. Albrecht, Ruth Albrecht, Daniel O. Aleshire, Urs Altermatt, Anand Amaladass, Michael Amaladoss, James N. Amanze, Lesley G. Anderson, Thomas C. Anderson, Victor Anderson, Hope S. Antone, María Pilar Aquino, Paula Arai, Victorio Araya Guillén, S. Wesley Ariarajah, Ellen T. Armour, Brett Gregory Armstrong, Atsuhiro Asano, Naim Stifan Ateek, Mahmoud Ayoub, John Alembillah Azumah, Mercedes L. García Bachmann, Irena Backus, J. Wayne Baker, Mieke Bal, Lewis V. Baldwin, William Barbieri, António Barbosa da Silva, David Basinger, Bolaji Olukemi Bateye, Oswald Bayer, Daniel H. Bays, Rosalie Beck, Nancy Elizabeth Bedford, Guy-Thomas Bedouelle, Chorbishop Seely Beggiani, Wolfgang Behringer, Christopher M. Bellitto, Byard Bennett, Harold V. Bennett, Teresa Berger, Miguel A. Bernad, Henley Bernard, Alan E. Bernstein, Jon L. Berquist, Johannes Beutler, Ana María Bidegain, Matthew P. Binkewicz, Jennifer Bird, Joseph Blenkinsopp, Dmytro Bondarenko, Paulo Bonfatti, Riet en Pim Bons-Storm, Jessica A. Boon, Marcus J. Borg, Mark Bosco, Peter C. Bouteneff, François Bovon, William D. Bowman, Paul S. Boyer, David Brakke, Richard E. Brantley, Marcus Braybrooke, Ian Breward, Ênio José da Costa Brito, Jewel Spears Brooker, Johannes Brosseder, Nicholas Canfield Read Brown, Robert F. Brown, Pamela K. Brubaker, Walter Brueggemann, Bishop Colin O. Buchanan, Stanley M. Burgess, Amy Nelson Burnett, J. Patout Burns, David B. Burrell, David Buttrick, James P. Byrd, Lavinia Byrne, Gerado Caetano, Marcos Caldas, Alkiviadis Calivas, William J. Callahan, Salvatore Calomino, Euan K. Cameron, William S. Campbell, Marcelo Ayres Camurça, Daniel F. Caner, Paul E. Capetz, Carlos F. Cardoza-Orlandi, Patrick W. Carey, Barbara Carvill, Hal Cauthron, Subhadra Mitra Channa, Mark D. Chapman, James H. Charlesworth, Kenneth R. Chase, Chen Zemin, Luciano Chianeque, Philip Chia Phin Yin, Francisca H. Chimhanda, Daniel Chiquete, John T. Chirban, Soobin Choi, Robert Choquette, Mita Choudhury, Gerald Christianson, John Chryssavgis, Sejong Chun, Esther Chung-Kim, Charles M. A. Clark, Elizabeth A. Clark, Sathianathan Clarke, Fred Cloud, John B. Cobb, W. Owen Cole, John A Coleman, John J. Collins, Sylvia Collins-Mayo, Paul K. Conkin, Beth A. Conklin, Sean Connolly, Demetrios J. Constantelos, Michael A. Conway, Paula M. Cooey, Austin Cooper, Michael L. Cooper-White, Pamela Cooper-White, L. William Countryman, Sérgio Coutinho, Pamela Couture, Shannon Craigo-Snell, James L. Crenshaw, David Crowner, Humberto Horacio Cucchetti, Lawrence S. Cunningham, Elizabeth Mason Currier, Emmanuel Cutrone, Mary L. Daniel, David D. Daniels, Robert Darden, Rolf Darge, Isaiah Dau, Jeffry C. Davis, Jane Dawson, Valentin Dedji, John W. de Gruchy, Paul DeHart, Wendy J. Deichmann Edwards, Miguel A. De La Torre, George E. Demacopoulos, Thomas de Mayo, Leah DeVun, Beatriz de Vasconcellos Dias, Dennis C. Dickerson, John M. Dillon, Luis Miguel Donatello, Igor Dorfmann-Lazarev, Susanna Drake, Jonathan A. Draper, N. Dreher Martin, Otto Dreydoppel, Angelyn Dries, A. J. Droge, Francis X. D'Sa, Marilyn Dunn, Nicole Wilkinson Duran, Rifaat Ebied, Mark J. Edwards, William H. Edwards, Leonard H. Ehrlich, Nancy L. Eiesland, Martin Elbel, J. Harold Ellens, Stephen Ellingson, Marvin M. Ellison, Robert Ellsberg, Jean Bethke Elshtain, Eldon Jay Epp, Peter C. Erb, Tassilo Erhardt, Maria Erling, Noel Leo Erskine, Gillian R. Evans, Virginia Fabella, Michael A. Fahey, Edward Farley, Margaret A. Farley, Wendy Farley, Robert Fastiggi, Seena Fazel, Duncan S. Ferguson, Helwar Figueroa, Paul Corby Finney, Kyriaki Karidoyanes FitzGerald, Thomas E. FitzGerald, John R. Fitzmier, Marie Therese Flanagan, Sabina Flanagan, Claude Flipo, Ronald B. Flowers, Carole Fontaine, David Ford, Mary Ford, Stephanie A. Ford, Jim Forest, William Franke, Robert M. Franklin, Ruth Franzén, Edward H. Friedman, Samuel Frouisou, Lorelei F. Fuchs, Jojo M. Fung, Inger Furseth, Richard R. Gaillardetz, Brandon Gallaher, China Galland, Mark Galli, Ismael García, Tharscisse Gatwa, Jean-Marie Gaudeul, Luis María Gavilanes del Castillo, Pavel L. Gavrilyuk, Volney P. Gay, Metropolitan Athanasios Geevargis, Kondothra M. George, Mary Gerhart, Simon Gikandi, Maurice Gilbert, Michael J. Gillgannon, Verónica Giménez Beliveau, Terryl Givens, Beth Glazier-McDonald, Philip Gleason, Menghun Goh, Brian Golding, Bishop Hilario M. Gomez, Michelle A. Gonzalez, Donald K. Gorrell, Roy Gottfried, Tamara Grdzelidze, Joel B. Green, Niels Henrik Gregersen, Cristina Grenholm, Herbert Griffiths, Eric W. Gritsch, Erich S. Gruen, Christoffer H. Grundmann, Paul H. Gundani, Jon P. Gunnemann, Petre Guran, Vidar L. Haanes, Jeremiah M. Hackett, Getatchew Haile, Douglas John Hall, Nicholas Hammond, Daphne Hampson, Jehu J. Hanciles, Barry Hankins, Jennifer Haraguchi, Stanley S. Harakas, Anthony John Harding, Conrad L. Harkins, J. William Harmless, Marjory Harper, Amir Harrak, Joel F. Harrington, Mark W. Harris, Susan Ashbrook Harvey, Van A. Harvey, R. Chris Hassel, Jione Havea, Daniel Hawk, Diana L. Hayes, Leslie Hayes, Priscilla Hayner, S. Mark Heim, Simo Heininen, Richard P. Heitzenrater, Eila Helander, David Hempton, Scott H. Hendrix, Jan-Olav Henriksen, Gina Hens-Piazza, Carter Heyward, Nicholas J. Higham, David Hilliard, Norman A. Hjelm, Peter C. Hodgson, Arthur Holder, M. Jan Holton, Dwight N. Hopkins, Ronnie Po-chia Hsia, Po-Ho Huang, James Hudnut-Beumler, Jennifer S. Hughes, Leonard M. Hummel, Mary E. Hunt, Laennec Hurbon, Mark Hutchinson, Susan E. Hylen, Mary Beth Ingham, H. Larry Ingle, Dale T. Irvin, Jon Isaak, Paul John Isaak, Ada María Isasi-Díaz, Hans Raun Iversen, Margaret C. Jacob, Arthur James, Maria Jansdotter-Samuelsson, David Jasper, Werner G. Jeanrond, Renée Jeffery, David Lyle Jeffrey, Theodore W. Jennings, David H. Jensen, Robin Margaret Jensen, David Jobling, Dale A. Johnson, Elizabeth A. Johnson, Maxwell E. Johnson, Sarah Johnson, Mark D. Johnston, F. Stanley Jones, James William Jones, John R. Jones, Alissa Jones Nelson, Inge Jonsson, Jan Joosten, Elizabeth Judd, Mulambya Peggy Kabonde, Robert Kaggwa, Sylvester Kahakwa, Isaac Kalimi, Ogbu U. Kalu, Eunice Kamaara, Wayne C. Kannaday, Musimbi Kanyoro, Veli-Matti Kärkkäinen, Frank Kaufmann, Léon Nguapitshi Kayongo, Richard Kearney, Alice A. Keefe, Ralph Keen, Catherine Keller, Anthony J. Kelly, Karen Kennelly, Kathi Lynn Kern, Fergus Kerr, Edward Kessler, George Kilcourse, Heup Young Kim, Kim Sung-Hae, Kim Yong-Bock, Kim Yung Suk, Richard King, Thomas M. King, Robert M. Kingdon, Ross Kinsler, Hans G. Kippenberg, Cheryl A. Kirk-Duggan, Clifton Kirkpatrick, Leonid Kishkovsky, Nadieszda Kizenko, Jeffrey Klaiber, Hans-Josef Klauck, Sidney Knight, Samuel Kobia, Robert Kolb, Karla Ann Koll, Heikki Kotila, Donald Kraybill, Philip D. W. Krey, Yves Krumenacker, Jeffrey Kah-Jin Kuan, Simanga R. Kumalo, Peter Kuzmic, Simon Shui-Man Kwan, Kwok Pui-lan, André LaCocque, Stephen E. Lahey, John Tsz Pang Lai, Emiel Lamberts, Armando Lampe, Craig Lampe, Beverly J. Lanzetta, Eve LaPlante, Lizette Larson-Miller, Ariel Bybee Laughton, Leonard Lawlor, Bentley Layton, Robin A. Leaver, Karen Lebacqz, Archie Chi Chung Lee, Marilyn J. Legge, Hervé LeGrand, D. L. LeMahieu, Raymond Lemieux, Bill J. Leonard, Ellen M. Leonard, Outi Leppä, Jean Lesaulnier, Nantawan Boonprasat Lewis, Henrietta Leyser, Alexei Lidov, Bernard Lightman, Paul Chang-Ha Lim, Carter Lindberg, Mark R. Lindsay, James R. Linville, James C. Livingston, Ann Loades, David Loades, Jean-Claude Loba-Mkole, Lo Lung Kwong, Wati Longchar, Eleazar López, David W. Lotz, Andrew Louth, Robin W. Lovin, William Luis, Frank D. Macchia, Diarmaid N. J. MacCulloch, Kirk R. MacGregor, Marjory A. MacLean, Donald MacLeod, Tomas S. Maddela, Inge Mager, Laurenti Magesa, David G. Maillu, Fortunato Mallimaci, Philip Mamalakis, Kä Mana, Ukachukwu Chris Manus, Herbert Robinson Marbury, Reuel Norman Marigza, Jacqueline Mariña, Antti Marjanen, Luiz C. L. Marques, Madipoane Masenya (ngwan'a Mphahlele), Caleb J. D. Maskell, Steve Mason, Thomas Massaro, Fernando Matamoros Ponce, András Máté-Tóth, Odair Pedroso Mateus, Dinis Matsolo, Fumitaka Matsuoka, John D'Arcy May, Yelena Mazour-Matusevich, Theodore Mbazumutima, John S. McClure, Christian McConnell, Lee Martin McDonald, Gary B. McGee, Thomas McGowan, Alister E. McGrath, Richard J. McGregor, John A. McGuckin, Maud Burnett McInerney, Elsie Anne McKee, Mary B. McKinley, James F. McMillan, Ernan McMullin, Kathleen E. McVey, M. Douglas Meeks, Monica Jyotsna Melanchthon, Ilie Melniciuc-Puica, Everett Mendoza, Raymond A. Mentzer, William W. Menzies, Ina Merdjanova, Franziska Metzger, Constant J. Mews, Marvin Meyer, Carol Meyers, Vasile Mihoc, Gunner Bjerg Mikkelsen, Maria Inêz de Castro Millen, Clyde Lee Miller, Bonnie J. Miller-McLemore, Alexander Mirkovic, Paul Misner, Nozomu Miyahira, R. W. L. Moberly, Gerald Moede, Aloo Osotsi Mojola, Sunanda Mongia, Rebeca Montemayor, James Moore, Roger E. Moore, Craig E. Morrison O.Carm, Jeffry H. Morrison, Keith Morrison, Wilson J. Moses, Tefetso Henry Mothibe, Mokgethi Motlhabi, Fulata Moyo, Henry Mugabe, Jesse Ndwiga Kanyua Mugambi, Peggy Mulambya-Kabonde, Robert Bruce Mullin, Pamela Mullins Reaves, Saskia Murk Jansen, Heleen L. Murre-Van den Berg, Augustine Musopole, Isaac M. T. Mwase, Philomena Mwaura, Cecilia Nahnfeldt, Anne Nasimiyu Wasike, Carmiña Navia Velasco, Thulani Ndlazi, Alexander Negrov, James B. Nelson, David G. Newcombe, Carol Newsom, Helen J. Nicholson, George W. E. Nickelsburg, Tatyana Nikolskaya, Damayanthi M. A. Niles, Bertil Nilsson, Nyambura Njoroge, Fidelis Nkomazana, Mary Beth Norton, Christian Nottmeier, Sonene Nyawo, Anthère Nzabatsinda, Edward T. Oakes, Gerald O'Collins, Daniel O'Connell, David W. Odell-Scott, Mercy Amba Oduyoye, Kathleen O'Grady, Oyeronke Olajubu, Thomas O'Loughlin, Dennis T. Olson, J. Steven O'Malley, Cephas N. Omenyo, Muriel Orevillo-Montenegro, César Augusto Ornellas Ramos, Agbonkhianmeghe E. Orobator, Kenan B. Osborne, Carolyn Osiek, Javier Otaola Montagne, Douglas F. Ottati, Anna May Say Pa, Irina Paert, Jerry G. Pankhurst, Aristotle Papanikolaou, Samuele F. Pardini, Stefano Parenti, Peter Paris, Sung Bae Park, Cristián G. Parker, Raquel Pastor, Joseph Pathrapankal, Daniel Patte, W. Brown Patterson, Clive Pearson, Keith F. Pecklers, Nancy Cardoso Pereira, David Horace Perkins, Pheme Perkins, Edward N. Peters, Rebecca Todd Peters, Bishop Yeznik Petrossian, Raymond Pfister, Peter C. Phan, Isabel Apawo Phiri, William S. F. Pickering, Derrick G. Pitard, William Elvis Plata, Zlatko Plese, John Plummer, James Newton Poling, Ronald Popivchak, Andrew Porter, Ute Possekel, James M. Powell, Enos Das Pradhan, Devadasan Premnath, Jaime Adrían Prieto Valladares, Anne Primavesi, Randall Prior, María Alicia Puente Lutteroth, Eduardo Guzmão Quadros, Albert Rabil, Laurent William Ramambason, Apolonio M. Ranche, Vololona Randriamanantena Andriamitandrina, Lawrence R. Rast, Paul L. Redditt, Adele Reinhartz, Rolf Rendtorff, Pål Repstad, James N. Rhodes, John K. Riches, Joerg Rieger, Sharon H. Ringe, Sandra Rios, Tyler Roberts, David M. Robinson, James M. Robinson, Joanne Maguire Robinson, Richard A. H. Robinson, Roy R. Robson, Jack B. Rogers, Maria Roginska, Sidney Rooy, Rev. Garnett Roper, Maria José Fontelas Rosado-Nunes, Andrew C. Ross, Stefan Rossbach, François Rossier, John D. Roth, John K. Roth, Phillip Rothwell, Richard E. Rubenstein, Rosemary Radford Ruether, Markku Ruotsila, John E. Rybolt, Risto Saarinen, John Saillant, Juan Sanchez, Wagner Lopes Sanchez, Hugo N. Santos, Gerhard Sauter, Gloria L. Schaab, Sandra M. Schneiders, Quentin J. Schultze, Fernando F. Segovia, Turid Karlsen Seim, Carsten Selch Jensen, Alan P. F. Sell, Frank C. Senn, Kent Davis Sensenig, Damían Setton, Bal Krishna Sharma, Carolyn J. Sharp, Thomas Sheehan, N. Gerald Shenk, Christian Sheppard, Charles Sherlock, Tabona Shoko, Walter B. Shurden, Marguerite Shuster, B. Mark Sietsema, Batara Sihombing, Neil Silberman, Clodomiro Siller, Samuel Silva-Gotay, Heikki Silvet, John K. Simmons, Hagith Sivan, James C. Skedros, Abraham Smith, Ashley A. Smith, Ted A. Smith, Daud Soesilo, Pia Søltoft, Choan-Seng (C. S.) Song, Kathryn Spink, Bryan Spinks, Eric O. Springsted, Nicolas Standaert, Brian Stanley, Glen H. Stassen, Karel Steenbrink, Stephen J. Stein, Andrea Sterk, Gregory E. Sterling, Columba Stewart, Jacques Stewart, Robert B. Stewart, Cynthia Stokes Brown, Ken Stone, Anne Stott, Elizabeth Stuart, Monya Stubbs, Marjorie Hewitt Suchocki, David Kwang-sun Suh, Scott W. Sunquist, Keith Suter, Douglas Sweeney, Charles H. Talbert, Shawqi N. Talia, Elsa Tamez, Joseph B. Tamney, Jonathan Y. Tan, Yak-Hwee Tan, Kathryn Tanner, Feiya Tao, Elizabeth S. Tapia, Aquiline Tarimo, Claire Taylor, Mark Lewis Taylor, Bishop Abba Samuel Wolde Tekestebirhan, Eugene TeSelle, M. Thomas Thangaraj, David R. Thomas, Andrew Thornley, Scott Thumma, Marcelo Timotheo da Costa, George E. “Tink” Tinker, Ola Tjørhom, Karen Jo Torjesen, Iain R. Torrance, Fernando Torres-Londoño, Archbishop Demetrios [Trakatellis], Marit Trelstad, Christine Trevett, Phyllis Trible, Johannes Tromp, Paul Turner, Robert G. Tuttle, Archbishop Desmond Tutu, Peter Tyler, Anders Tyrberg, Justin Ukpong, Javier Ulloa, Camillus Umoh, Kristi Upson-Saia, Martina Urban, Monica Uribe, Elochukwu Eugene Uzukwu, Richard Vaggione, Gabriel Vahanian, Paul Valliere, T. J. Van Bavel, Steven Vanderputten, Peter Van der Veer, Huub Van de Sandt, Louis Van Tongeren, Luke A. Veronis, Noel Villalba, Ramón Vinke, Tim Vivian, David Voas, Elena Volkova, Katharina von Kellenbach, Elina Vuola, Timothy Wadkins, Elaine M. Wainwright, Randi Jones Walker, Dewey D. Wallace, Jerry Walls, Michael J. Walsh, Philip Walters, Janet Walton, Jonathan L. Walton, Wang Xiaochao, Patricia A. Ward, David Harrington Watt, Herold D. Weiss, Laurence L. Welborn, Sharon D. Welch, Timothy Wengert, Traci C. West, Merold Westphal, David Wetherell, Barbara Wheeler, Carolinne White, Jean-Paul Wiest, Frans Wijsen, Terry L. Wilder, Felix Wilfred, Rebecca Wilkin, Daniel H. Williams, D. Newell Williams, Michael A. Williams, Vincent L. Wimbush, Gabriele Winkler, Anders Winroth, Lauri Emílio Wirth, James A. Wiseman, Ebba Witt-Brattström, Teofil Wojciechowski, John Wolffe, Kenman L. Wong, Wong Wai Ching, Linda Woodhead, Wendy M. Wright, Rose Wu, Keith E. Yandell, Gale A. Yee, Viktor Yelensky, Yeo Khiok-Khng, Gustav K. K. Yeung, Angela Yiu, Amos Yong, Yong Ting Jin, You Bin, Youhanna Nessim Youssef, Eliana Yunes, Robert Michael Zaller, Valarie H. Ziegler, Barbara Brown Zikmund, Joyce Ann Zimmerman, Aurora Zlotnik, Zhuo Xinping
- Edited by Daniel Patte, Vanderbilt University, Tennessee
-
- Book:
- The Cambridge Dictionary of Christianity
- Published online:
- 05 August 2012
- Print publication:
- 20 September 2010, pp xi-xliv
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- Chapter
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