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  1. Mellin transforms and asymptotics: Harmonic sums Phillipe Flajolet, Xavier Gourdon, Philippe Dumas Die Theorie der reziprokenFunktionen und Integrale istein centrales Gebiet, welchesmanche anderen Gebiete derAnalysis miteinander verbindet. Hjalmar Mellin Reporter : Ilya Posov Saint-Petersburg State University Faculty of Mathematics and Mechanics Joint Advanced Student School 2004

  2. Contents • Mellin transform and its basic properties • Direct mapping • Converse mapping • Harmonic sums Joint Advanced Student School 2004

  3. Robert Hjalmar Mellin Born: 1854 in Liminka, Northern Ostrobothnia, FinlandDied: 1933 Joint Advanced Student School 2004

  4. Some terminology • Analytic function • Meromorphic function • Open strip Joint Advanced Student School 2004

  5. Mellin transform fundamental strip Joint Advanced Student School 2004

  6. Mellin transform example fundamental strip : Joint Advanced Student School 2004

  7. Gamma function fundamental strip : Joint Advanced Student School 2004

  8. Transform of step function Joint Advanced Student School 2004

  9. Basic properties (1/2) Joint Advanced Student School 2004

  10. Basic properties (2/2) Joint Advanced Student School 2004

  11. Zeta function Joint Advanced Student School 2004

  12. Some Mellin tranforms Joint Advanced Student School 2004

  13. Inversion Theorem 1 Joint Advanced Student School 2004

  14. Laurent expansion Joint Advanced Student School 2004

  15. Singular element Definition 1 Joint Advanced Student School 2004

  16. Singular expansion Definition 2 Joint Advanced Student School 2004

  17. Singular expansion of gamma function poles of gamma function Joint Advanced Student School 2004

  18. Direct mapping (1/3) Theorem 2 Joint Advanced Student School 2004

  19. Direct mapping (2/3) Joint Advanced Student School 2004

  20. Direct mapping (3/3) Joint Advanced Student School 2004

  21. Example 0 poles of gamma function Joint Advanced Student School 2004

  22. Example 1 Joint Advanced Student School 2004

  23. Example 2 Joint Advanced Student School 2004

  24. Example 3 (nonexplicit transform) Joint Advanced Student School 2004

  25. Converse mapping (1/3) Theorem 3 Joint Advanced Student School 2004

  26. Converse mapping (2/3) Theorem 3 (continue) Joint Advanced Student School 2004

  27. Converse mapping (3/3) Joint Advanced Student School 2004

  28. Example 4 Joint Advanced Student School 2004

  29. Example 5 Joint Advanced Student School 2004

  30. Harmonic sums Definition 3 (separation property) Joint Advanced Student School 2004

  31. Harmonic sum formula Joint Advanced Student School 2004

  32. Example 6 Joint Advanced Student School 2004

  33. Example 7 Simple poles in positive integers Double poles in 0 and -1 Joint Advanced Student School 2004

  34. Example 8 Joint Advanced Student School 2004

  35. Example 9 modified theta function Joint Advanced Student School 2004

  36. Example 10 Joint Advanced Student School 2004

  37. The end Joint Advanced Student School 2004