Part B Complex Asymptotics * Chapter 4: Complex Analysis * Chapter 5: Rational and Meromorphic Asymptotics * Chapter 6: Singularity Analysis of GFs *Chapter 7: Applications of Singularity Analysis *Chapter 8: Saddle-point Methods 1
N! for N=2,...,50 2
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CHAPTER 6
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TRAINS 20
Chapter 5 Rational and Meromorphic Asymptotics 21
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Worked out Example 3: derangements 29
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Chapter 6 Singularity Analysis 34
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EGF (exponential GF) 46
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Chapter 7 Applications of Singularity Analysis 53
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functional equation 58
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Exponential * n^rational 65
APPLICATIONS of ALGEBRAIC FUNCTIONS Trees with a finite set of node degrees Excursions with finite set of steps [Lalley, BaFl] Maps embedded into the plane [Tutte,...] Gimenez-Noy: Planar graphs Context-free structures = Drmota-Lalley-Woods Thm. 66
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Singularity Analysis applies to non-linear ODEs = models of “logarithmic trees” the holonomic framework = solutions of linear ODEs with rational coefficients. 71
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Conclusion: SCHEMAS 74
Chapter 8 Saddle-point Asymptotics 75
Modulus of an analytic function 76
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Cauchy coefficient integrals 78
Saddle-point bounds 79
Saddle-point method: = concentration + local quadratic approximation. 80
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The saddle-point theorem: under conditions: concentration + local quadratic approximation 83
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Hardy & Ramanujan 85
(double saddle-point) 86
Conclusions: Saddle-point method Applies to many entire functions: involutions, set partitions, etc. Applies to function with violent growth at singularity(-ies): integer partitions Applies to coefficients of large order in large powers 87
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