| Copyright Page | 5 |
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| Table of Contents | 6 |
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| Introduction | 12 |
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| Authors’ affiliations | 13 |
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| Complex elliptic pendulum | 15 |
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| 1 Introduction | 15 |
| 2 Previous results on complex classical mechanics | 16 |
| 3 Classical particle in a complex elliptic potential | 26 |
| 4 Summary and discussion | 29 |
| References | 30 |
| Parabolic attitude | 33 |
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| 1 Introduction | 33 |
| 2 Parabolic dynamics in one variable | 36 |
| 3 Germs tangent to the identity | 37 |
| 4 Semiattractive and quasi-parabolic germs | 40 |
| 4.1 Semiattractive germs | 40 |
| 4.2 Quasi-parabolic germs | 41 |
| 5 One-resonant germs | 43 |
| References | 45 |
| Power series with sum-product Taylor coefficients and their resurgence algebra | 48 |
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| 1 Introduction | 50 |
| 1.1 Power series with coeffcients of sum-product type | 50 |
| The notion of SP series | 50 |
| Special cases of SP series | 51 |
| Overview | 51 |
| 1.2 The outer/inner dichotomy and the ingress factor | 52 |
| The outer/inner dichotomy | 52 |
| A gratifying surprise: the mir-transform | 52 |
| The ingress factor and the cleansing of SP-series | 53 |
| 1.3 The four gates to the inner algebra | 53 |
| 2 Some resurgence background | 56 |
| 2.1 Resurgent functions and their three models | 56 |
| Resurgent algebras: the multiplicative structure | 58 |
| The upper/lower Borel-Laplace transforms | 59 |
| Monomials in all four models | 60 |
| The pros and cons of the upper/lower choices | 60 |
| 2.2 Alien derivations as a tool for Riemann surface description | 61 |
| Definition of the operators . and . | 62 |
| Lateral and median singularities | 63 |
| Compact description of Riemann surfaces | 64 |
| Strong versus weak resurgence | 64 |
| The pros and cons of the 2pi factor | 65 |
| 2.3 Retrieving the resurgence of a series from the resurgence of its Taylor coefficients | 66 |
| Retrieving closest singularities | 66 |
| Retrieving distant singularities | 67 |
| 3 The ingress factor | 68 |
| 3.1 Bernoulli numbers and polynomials | 68 |
| The Bernoulli numbers and polynomials | 68 |
| The Euler-MacLaurin formula | 70 |
| 3.2 Resurgence of the Gamma function | 70 |
| 3.3 Monomial/binomial/exponential factors | 72 |
| Monomial factors | 73 |
| Binomial factors | 74 |
| Exponential factors | 74 |
| 3.4 Resummability of the total ingress factor | 74 |
| 3.5 Parity relations | 75 |
| 4 Inner generators | 75 |
| 4.1 Some heuristics | 75 |
| 4.2 The long chain behind nir//mir | 80 |
| Details of the nine steps: | 81 |
| “Compact” and “layered” expansions of mir | 82 |
| 4.3 The nir transform | 82 |
| Integral expression of nir | 82 |
| 4.4 The reciprocation transform | 83 |
| 4.5 The mir transform | 86 |
| 4.6 Translocation of the nir transform | 89 |
| 4.7 Alternative factorisations of nir. The lir transform | 93 |
| 4.8 Application: kernel of the nir transform | 95 |
| 4.9 Comparing/extending/inverting nir and mir | 96 |
| 4.10 Parity relations | 98 |
| 5 Outer generators | 98 |
| 5.1 Some heuristics | 98 |
| 5.2 The short and long chains behind nur/mur | 99 |
| The short, four-link chain: | 99 |
| Details of the four steps: | 100 |
| The long, nine-link chain: | 101 |
| Details of the nine steps: | 101 |
| 5.3 The nur transform | 102 |
| 5.4 Expressing nur in terms of nir | 104 |
| 5.5 The mur transform | 105 |
| 5.6 Translocation of the nur transform | 106 |
| 5.7 Removal of the ingress factor | 106 |
| 5.8 Parity relations | 107 |
| 6 Inner generators and ordinary differential equations | 107 |
| 6.1 “Variable” and “covariant” differential equations | 107 |
| Existence and calculation of the variable ODEs | 110 |
| Existence and calculation of the covariant ODEs for f (0) = 0 | 110 |
| Existence and calculation of the covariant ODEs for f (0) = 0 | 111 |
| 6.2 ODEs for polynomial inputs f . Main statements | 112 |
