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Definite Integration in Differential Fields

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Definite Integration in Differential Fields JOHANNES KEPLER UNIVERSITAT¨ LINZ JKU Technisch-Naturwissenschaftliche Fakult¨at Definite Integration in Differential Fields DISSERTATION zur Erlangung des akademischen Grades Doktor im Doktoratsstudium der Technischen Wissenschaften Eingereicht von: Dipl.-Ing. Clemens G. Raab Angefertigt am: Institut f¨urSymbolisches Rechnen (RISC) Beurteilung: Univ.-Prof. Dr. Peter Paule (Betreuung) Prof. Dr. Michael F. Singer Linz, August 2012 Eidesstattliche Erkl¨arung Ich erkl¨arean Eides statt, dass ich die vorliegende Dissertation selbstst¨andigund ohne fremde Hilfe verfasst, andere als die angegebenen Quellen und Hilfsmittel nicht benutzt bzw. die w¨ortlich oder sinngem¨aßentnommenen Stellen als solche kenntlich gemacht habe. Die vorliegende Dissertation ist mit dem elektronisch ¨ubermittelten Textdokument iden- tisch. Linz, August 2012 Clemens G. Raab Kurzzusammenfassung Das Ziel dieser Doktorarbeit ist die Weiterentwicklung von Computeralgebramethoden zur Berechnung von definiten Integralen. Eine Art den Wert eines definiten Integrals zu berechnen f¨uhrt ¨uber das Auswerten einer Stammfunktion des Integranden. Im neun- zehnten Jahrhundert war Joseph Liouville einer der ersten die die Struktur elementarer Stammfunktionen von elementaren Funktionen untersuchten. Im fr¨uhenzwanzigsten Jahr- hundert wurden Differentialk¨orper als algebraische Strukturen zur Modellierung der dif- ferentiellen Eigenschaften von Funktionen eingef¨uhrt.Mit deren Hilfe hat Robert H. Risch im Jahr 1969 einen vollst¨andigenAlgorithmus f¨urtranszendente elementare Integranden ver¨offentlicht. Seither wurde dieses Resultat von Michael F. Singer, Manuel Bronstein und einigen anderen auf bestimmte andere Klassen von Integranden erweitert. Andererseits k¨onnen,f¨urden Fall dass keine Stammfunktion in geeigneter Form verf¨ugbarist, basierend auf dem Prinzip der parametrischen Integration (oft creative telescoping genannt) lineare Relationen gefunden werden, welche vom Parameterintegral erf¨ulltwerden. Das Hauptresultat dieser Doktorarbeit erweitert das oben Erw¨ahnte zu einem vollst¨andi- gen Algorithmus f¨urelementare parametrische Integration einer bestimmten Funktionen- klasse, welche den Großteil der in der Praxis auftretenden speziellen Funktionen abdeckt, z.B. orthogonale Polynome, Polylogarithmen, Besselfunktionen, etc. Es wird auch eine Methode zur Modellierung dieser Funktionen mittels geeigneter Differentialk¨orper ange- geben. F¨urLiouville'sche Integranden weist dieser Algorithmus eine deutlich verbesserte Effizienz gegen¨uber dem entsprechenden von Singer et al. 1985 pr¨asentierten Algorith- mus auf. Zus¨atzlich wird auch eine Verallgemeinerung von Czichowskis Algorithmus zur Berechnung des logarithmischen Teils des Integrals dargelegt. Uberdies¨ werden auch teil- weise Erweiterungen des Integrationsalgorithmus auf weitere Funktionen behandelt. Als Teilprobleme des Integrationsalgorithmus m¨ussenauch L¨osungenbestimmten Typs von linearen gew¨ohnlichen Differentialgleichungen gefunden werden. Auch hierzu werden Beitr¨agegeleistet, wobei jene die sich mit der direkten L¨osungvon Differentialgleichungs- systemen befassen auf eine Zusammenarbeit mit Moulay A. Barkatou zur¨uckgehen. F¨urLiouville'sche Integranden wurde der Algorithmus in Form des Mathematica-Pakets Integrator implementiert. Teile davon k¨onnenauch mit allgemeineren Funktionen umge- hen. Diese Methoden k¨onnen auf einen Großteil der indefiniten wie definiten Integrale aus Integraltafeln angewandt werden. Zus¨atzlich wurden mit dem Paket auch interessante In- tegrale erfolgreich behandelt, die nicht in Tabellen aufscheinen bzw. bei welchen derzeitige Computeralgebrasysteme wie Mathematica oder Maple nicht zum Ziel f¨uhren. Außerdem zeigen wir wie Parameterintegrale aus der Arbeit anderer Forscher mit dem Paket gel¨ost werden, z.B. ein Integral aus der Untersuchung der Entropie bestimmter Prozesse. i Abstract The general goal of this thesis is to investigate and develop computer algebra tools for the simplification resp. evaluation of definite integrals. One way of finding the value ofadef- inite integral is via the evaluation of an antiderivative of the integrand. In the nineteenth century Joseph Liouville was among the first who analyzed the structure of elementary antiderivatives of elementary functions systematically. In the early twentieth century the algebraic structure of differential fields was introduced for modeling the differential properties of functions. Using this framework Robert H. Risch published a complete algorithm for transcendental elementary integrands in 1969. Since then this result has been extended to certain other classes of integrands as well by Michael F. Singer, Manuel Bronstein, and several others. On the other hand, if no antiderivative of suitable form is available, then linear relations that are satisfied by the parameter integral of interest may be found based on the principle of parametric integration (often called differentiating under the integral sign or creative telescoping). The main result of this thesis extends the results mentioned above to a complete algo- rithm for parametric elementary integration for a certain class of integrands covering a majority of the special functions appearing in practice such as orthogonal polynomials, polylogarithms, Bessel functions, etc. A general framework is provided to model those functions in terms of suitable differential fields. If the integrand is Liouvillian, thenthe present algorithm considerably improves the efficiency of the corresponding algorithm given by Singer et al. in 1985. Additionally, a generalization of Czichowski's algorithm for computing the logarithmic part of the integral is presented. Moreover, also partial generalizations to include other types of integrands are treated. As subproblems of the integration algorithm one also has to find solutions of linear or- dinary differential equations of a certain type. Some contributions are also madeto solve those problems in our setting, where the results directly dealing with systems of differential equations have been joint work with Moulay A. Barkatou. For the case of Liouvillian integrands we implemented the algorithm in form of our Mathematica package Integrator. Parts of the implementation also deal with more general functions. Our procedures can be applied to a significant amount of the entries in integral tables, both indefinite and definite integrals. In addition, our procedures have been successfully applied to interesting examples of integrals that do not appear in these tables or for which current standard computer algebra systems like Mathematica or Maple do not succeed. We also give examples of how parameter integrals coming from the work of other researchers can be solved with the software, e.g., an integral arising in analyzing the entropy of certain processes. ii Acknowledgements First and foremost I want to express my gratitude to my advisor Peter Paule who has been a mentor for me essentially for all my time at the university. We first met when I attended his courses on linear algebra in my first year. Many other courses and seminars followed throughout my studies as I greatly enjoyed his ability to give interesting and comprehensible lectures. I am very happy and thankful that he accepted me as his PhD student in the framework of the Doctoral Program (DK) \Computational Mathematics: Numerical Analysis and Symbolic Computation", which has been a very exciting envi- ronment due to its interdisciplinary orientation. He suggested this interesting topic to me, gave me a lot of freedom to pursue my own ideas and interests, and also provided guidance in several aspects of the work of a scientist. On the personal level it is valuable to have such a kind and understanding person as advisor who is full of enthusiasm and encouragement, which I appreciate a lot. Last but not least I am grateful for the financial support organized by my advisor. The research was funded by the Austrian Science Fund (FWF): grant no. W1214-N15, project DK6. The research was partially supported by the strategic program \Innovatives OO¨ 2010 plus" by the Upper Austrian Government. Manuel Kauers deserves special thanks for many helpful discussions where he gave an overview of existing algorithms in the context of this thesis and suggested some problems to look at. He also provided me with his experimental implementation of the procedures given in Bronstein's book [Bro], which our package Integrator now is based on. Being in the DK also provided me with many opportunities (along with generous financial support) to attend international conferences, get in touch with many researchers, and spend several months abroad, which I am very grateful for. In particular I profited a lot from the extensive research stays where I learned a lot each time. The following saying certainly applies: \We are like dwarfs standing on the shoulders of giants." Many thanks go to Michael F. Singer for sharing his rich knowledge and providing me with a multitude of pointers to the literature on several occasions. When I stayed at North Carolina State University (NCSU) for more than two months he was very kind and spent a lot of time explaining various results and discussing new ideas. He also made an effort to make my life there smooth and comfortable, which I am very grateful for. Special thanks go to Moulay A. Barkatou for accepting to work with me. When I stayed at the University of Limoges for six weeks in total he was so kind to introduce me to the notions and algorithms related to systems
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