A Formal Proof of Cauchy’s Residue Theorem

  • Wenda LiEmail author
  • Lawrence C. Paulson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9807)


We present a formalization of Cauchy’s residue theorem and two of its corollaries: the argument principle and Rouché’s theorem. These results have applications to verify algorithms in computer algebra and demonstrate Isabelle/HOL’s complex analysis library.


Holomorphic Function Complex Analysis Prime Number Theorem Argument Principle Quantifier Elimination 
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We are grateful to John Harrison for his insightful suggestions about mathematical formalization, and also to the anonymous reviewers for their useful comments on the first version of this paper. The first author was funded by the China Scholarship Council, via the CSC Cambridge Scholarship programme.


  1. 1.
    Ahlfors, L.V.: Complex Analysis: An Introduction to the Theory of Analytic Funtions of One Complex Variable. McGraw-Hill, New York (1966)Google Scholar
  2. 2.
    Avigad, J., Hölzl, J., Serafin, L.: A formally verified proof of the central limit theorem. CoRR abs/1405.7012 (2014)Google Scholar
  3. 3.
    Bak, J., Newman, D.: Complex Analysis. Springer, New York (2010)CrossRefzbMATHGoogle Scholar
  4. 4.
    Basu, S., Pollack, R., Roy, M.F.: Algorithms in Real Algebraic Geometry, vol. 10. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  5. 5.
    Boldo, S., Lelay, C., Melquiond, G.: Coquelicot: a user-friendly library of real analysis for Coq. Math. Comput. Sci. 9(1), 41–62 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brunel, A.: Non-constructive complex analysis in Coq. In: 18th International Workshop on Types for Proofs and Programs, TYPES 2011, Bergen, Norway, pp. 1–15, 8–11 September 2011Google Scholar
  7. 7.
    Bruno Brosowski, F.D.: An elementary proof of the Stone-Weierstrass theorem. Proc. Am. Math. Soc. 81(1), 89–92 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Caviness, B., Johnson, J.: Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer, New York (2012)Google Scholar
  9. 9.
    Conway, J.B.: Functions of One Complex Variable, vol. 11, 2nd edn. Springer, New York (1978)CrossRefGoogle Scholar
  10. 10.
    Cruz-Filipe, L., Geuvers, H., Wiedijk, F.: C-CoRN, the constructive Coq repository at Nijmegen. In: Asperti, A., Bancerek, G., Trybulec, A. (eds.) MKM 2004. LNCS, vol. 3119, pp. 88–103. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Hales, T.C., Adams, M., Bauer, G., Dang, D.T., Harrison, J., Hoang, T.L., Kaliszyk, C., Magron, V., McLaughlin, S., Nguyen, T.T., Nguyen, T.Q., Nipkow, T., Obua, S., Pleso, J., Rute, J., Solovyev, A., Ta, A.H.T., Tran, T.N., Trieu, D.T., Urban, J., Vu, K.K., Zumkeller, R.: A formal proof of the Kepler conjecture. arXiv:1501.02155 (2015)
  12. 12.
    Harrison, J.: Formalizing basic complex analysis. In: Matuszewski, R., Zalewska, A. (eds.) From Insight to Proof: Festschrift in Honour of Andrzej Trybulec, vol. 10(23), pp. 151–165. University of Białystok (2007)Google Scholar
  13. 13.
    Harrison, J.: Formalizing an analytic proof of the Prime Number Theorem (dedicated to Mike Gordon on the occasion of his 60th birthday). J. Autom. Reasoning 43, 243–261 (2009)CrossRefGoogle Scholar
  14. 14.
    Harrison, J.: The HOL light theory of Euclidean space. J. Autom. Reasoning 50, 173–190 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lang, S.: Complex Analysis. Springer, New York (1993)CrossRefzbMATHGoogle Scholar
  16. 16.
    Stein, E.M., Shakarchi, R.: Complex Analysis, vol. 2. Princeton University Press, Princeton (2010)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Computer LaboratoryUniversity of CambridgeCambridgeUK

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