Computer Algebra in Quantum Field Theory

1. Front Matter

   Pages i-xiv

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2. Harmonic Sums, Polylogarithms,Special Numbers, and Their Generalizations

   • Jakob Ablinger, Johannes Blümlein

   Pages 1-32

3. Multiple Zeta Values and Modular Forms in Quantum Field Theory

   • David Broadhurst

   Pages 33-73

4. Computer-Assisted Proofs of Some Identities for Bessel Functions of Fractional Order

   • Stefan Gerhold, Manuel Kauers, Christoph Koutschan, Peter Paule, Carsten Schneider, Burkhard Zimmermann

   Pages 75-96

5. Conformal Methods for Massless Feynman Integrals and Large N _f Methods

   • John A. Gracey

   Pages 97-118

6. The Holonomic Toolkit

   • Manuel Kauers

   Pages 119-144

7. Orthogonal Polynomials

   • Tom H. Koornwinder

   Pages 145-170

8. Creative Telescoping for Holonomic Functions

   • Christoph Koutschan

   Pages 171-194

9. Renormalization and Mellin Transforms

   • Dirk Kreimer, Erik Panzer

   Pages 195-223

10. Relativistic Coulomb Integrals and Zeilberger’s Holonomic Systems Approach. I

    • Peter Paule, Sergei K. Suslov

    Pages 225-241

11. Hypergeometric Functions in Mathematica ^®

    • Oleksandr Pavlyk

    Pages 243-258

12. Solving Linear Recurrence Equations with Polynomial Coefficients

    • Marko Petkovšek, Helena Zakrajšek

    Pages 259-284

13. Generalization of Risch’s Algorithm to Special Functions

    • Clemens G. Raab

    Pages 285-304

14. Multiple Hypergeometric Series: Appell Series and Beyond

    • Michael J. Schlosser

    Pages 305-324

15. Simplifying Multiple Sums in Difference Fields

    • Carsten Schneider

    Pages 325-360

16. Potential of FORM 4.0

    • Jos A. M. Vermaseren

    Pages 361-379

17. Feynman Graphs

    • Stefan Weinzierl

    Pages 381-406

18. Back Matter

    Pages 407-411

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The book focuses on advanced computer algebra methods and special functions that have striking applications in the context of quantum field theory. It presents the state of the art and new methods for (infinite) multiple sums, multiple integrals, in particular Feynman integrals, difference and differential equations in the format of survey articles. The presented techniques emerge from interdisciplinary fields: mathematics, computer science and theoretical physics; the articles are written by mathematicians and physicists with the goal that both groups can learn from the other field, including most recent developments. Besides that, the collection of articles also serves as an up-to-date handbook of available algorithms/software that are commonly used or might be useful in the fields of mathematics, physics or other sciences.

• Difference and Differential Equations
• Feynman Integrals
• Special Functions Applied to QFT
• Symbolic Summation and Integration

From the reviews:

“It is a collection of papers that grew out of a summer school course on integration, summation, and special functions in quantum field theory … . if you are interested in these sorts of special functions, and the computer algebra tools to manipulate them, whether or not your particular application is quantum field theory, then this book is an excellent description of the state of the art in computer algebra manipulation and proof.” (J. H. Davenport, Computing Reviews, April, 2014)

• RISC, Johannes Kepler University, Linz, Austria

  Carsten Schneider

• DESY, Zeuthen, Germany

  Johannes Blümlein

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