Symbolic Integration and Special Functions

Symbolic Integration and Special Functions

Peter Paule (ORCID: 0000-0002-7264-7079)

Disciplines

Computer Sciences (10%); Mathematics (90%)

Keywords

    Symbolic Integration, Special Functions, Differential Algebra, Computer Algebra

Abstract Final report

This project is about Symbolic Integration. Symbolic Integration is that part of computer algebra that deals with the evaluation of integrals in closed form. While a quite satisfactory theory exists for the symbolic integration of elementary functions (i.e. functions that can be expressed with exponentials and logarithms), there are a lot of open problems concerning the integration of special functions. In the project, we want to address questions related to both the integration of elementary functions (e.g., definite integrals), and the integration of various types of special functions. Special Functions that we are interested in include in particular polylogarithms, holonomic functions, and functions defined by non-linear differential equations. Integration algorithms for special functions are of original interest in computer algebra, but they also have immediate impact on a variety of problems originating from a variety of contexts. We aim at devising algorithms that are able to do definite and indefinite integrals arising in applications that involving special functions.

This project is about Symbolic Integration. Symbolic Integration is that part of computer algebra that deals with the evaluation of integrals in closed form. While a quite satisfactory theory exists for the symbolic integration of elementary functions (i.e. functions that can be expressed with exponentials and logarithms), there are a lot of open problems concerning the integration of special functions. In the project, we want to address questions related to both the integration of elementary functions (e.g., definite integrals), and the integration of various types of special functions. Special Functions that we are interested in include in particular polylogarithms, holonomic functions, and functions defined by non-linear differential equations. Integration algorithms for special functions are of original interest in computer algebra, but they also have immediate impact on a variety of problems originating from a variety of contexts. We aim at devising algorithms that are able to do definite and indefinite integrals arising in applications that involving special functions.
Research institution(s)

Research Output

  • 655 Citations
  • 23 Publications
Publications
  • 2013
    Title Lattice Green functions of the higher-dimensional face-centered cubic lattices
    DOI 10.1088/1751-8113/46/12/125005
    Type Journal Article
    Author Koutschan C
    Journal Journal of Physics A: Mathematical and Theoretical
    Pages 125005
    Link Publication
  • 2012
    Title Multivariate linear recurrences and power series division
    DOI 10.1016/j.disc.2012.08.009
    Type Journal Article
    Author Hauser H
    Journal Discrete Mathematics
    Pages 3553-3560
    Link Publication
  • 2012
    Title Third order integrability conditions for homogeneous potentials of degree -1
    DOI 10.1063/1.4746691
    Type Journal Article
    Author Combot T
    Journal Journal of Mathematical Physics
    Pages 082704
    Link Publication
  • 2012
    Title The Andrews–Sellers family of partition congruences
    DOI 10.1016/j.aim.2012.02.026
    Type Journal Article
    Author Paule P
    Journal Advances in Mathematics
    Pages 819-838
    Link Publication
  • 2012
    Title Trading order for degree in creative telescoping
    DOI 10.1016/j.jsc.2012.02.002
    Type Journal Article
    Author Chen S
    Journal Journal of Symbolic Computation
    Pages 968-995
    Link Publication
  • 2012
    Title A symbolic summation approach to Feynman integral calculus
    DOI 10.1016/j.jsc.2011.12.044
    Type Journal Article
    Author Blümlein J
    Journal Journal of Symbolic Computation
    Pages 1267-1289
    Link Publication
  • 2012
    Title Twisting q-holonomic sequences by complex roots of unity
    DOI 10.1145/2442829.2442857
    Type Conference Proceeding Abstract
    Author Garoufalidis S
    Pages 179-186
    Link Publication
  • 2012
    Title Zeilberger's holonomic ansatz for Pfaffians
    DOI 10.1145/2442829.2442863
    Type Conference Proceeding Abstract
    Author Ishikawa M
    Pages 227-233
    Link Publication
  • 2011
    Title Proof of George Andrews’s and David Robbins’s q-TSPP conjecture
    DOI 10.1073/pnas.1019186108
    Type Journal Article
    Author Koutschan C
    Journal Proceedings of the National Academy of Sciences
    Pages 2196-2199
    Link Publication
  • 2011
    Title The O(as3) massive operator matrix elements of O(nf) for the structure function F2(x,Q2) and transversity
    DOI 10.1016/j.nuclphysb.2010.10.021
    Type Journal Article
    Author Ablinger J
    Journal Nuclear Physics B
    Pages 26-54
    Link Publication
  • 2011
    Title The sl3 Jones polynomial of the trefoil: A case study of q-holonomic sequences
    DOI 10.1016/j.aam.2011.04.001
    Type Journal Article
    Author Garoufalidis S
    Journal Advances in Applied Mathematics
    Pages 829-839
  • 2011
    Title Sparsity Optimized High Order Finite Element Functions on Simplices
    DOI 10.1007/978-3-7091-0794-2_2
    Type Book Chapter
    Author Beuchler S
    Publisher Springer Nature
    Pages 21-44
  • 2011
    Title Computer Algebra Meets Finite Elements: An Efficient Implementation for Maxwell’s Equations
    DOI 10.1007/978-3-7091-0794-2_6
    Type Book Chapter
    Author Koutschan C
    Publisher Springer Nature
    Pages 105-121
  • 2011
    Title Unfair permutations
    DOI 10.1016/j.ejc.2011.04.002
    Type Journal Article
    Author Prodinger H
    Journal European Journal of Combinatorics
    Pages 1282-1298
    Link Publication
  • 2013
    Title Relativistic Coulomb Integrals and Zeilberger’s Holonomic Systems Approach. I
    DOI 10.1007/978-3-7091-1616-6_9
    Type Book Chapter
    Author Paule P
    Publisher Springer Nature
    Pages 225-241
  • 2010
    Title Modern Summation Methods and the Computation of 2- and 3-loop Feynman Diagrams
    DOI 10.1016/j.nuclphysbps.2010.08.028
    Type Journal Article
    Author Ablinger J
    Journal Nuclear Physics B - Proceedings Supplements
    Pages 110-115
    Link Publication
  • 2020
    Title Surface Analysis of Biodegradable Mg-Alloys after Immersion in Simulated Body Fluid
    DOI 10.3390/ma13071740
    Type Journal Article
    Author Petrovic D
    Journal Materials
    Pages 1740
    Link Publication
  • 2009
    Title Determining the closed forms of the O(as3) anomalous dimensions and Wilson coefficients from Mellin moments by means of computer algebra
    DOI 10.1016/j.cpc.2009.06.020
    Type Journal Article
    Author Blümlein J
    Journal Computer Physics Communications
    Pages 2143-2165
    Link Publication
  • 2010
    Title Automatic Improvements of Wallis' Inequality
    DOI 10.1109/synasc.2010.89
    Type Conference Proceeding Abstract
    Author Paule P
    Pages 12-16
    Link Publication
  • 2009
    Title Automatic Classification of Restricted Lattice Walks
    DOI 10.46298/dmtcs.2724
    Type Journal Article
    Author Bostan A
    Journal Discrete Mathematics & Theoretical Computer Science
    Link Publication
  • 2012
    Title The Noncommutative A-Polynomial of (-2, 3, n) Pretzel Knots
    DOI 10.1080/10586458.2012.651409
    Type Journal Article
    Author Garoufalidis S
    Journal Experimental Mathematics
    Pages 241-251
    Link Publication
  • 2011
    Title Harmonic sums and polylogarithms generated by cyclotomic polynomials
    DOI 10.1063/1.3629472
    Type Journal Article
    Author Ablinger J
    Journal Journal of Mathematical Physics
    Pages 102301
    Link Publication
  • 2013
    Title Fast Summation Techniques for Sparse Shape Functions in Tetrahedral hp-FEM
    DOI 10.1007/978-3-642-35275-1_60
    Type Book Chapter
    Author Beuchler S
    Publisher Springer Nature
    Pages 511-518