Part BComplex 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

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N! for N=2,...,50

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Real analysis?

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(Catalan GF)

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CHAPTER

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Complex analysis: LOCAL vs GLOBAL

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Complex analysis: coeffs at 0 ~> elsewhere

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singularity

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Exponential growth of coefficients

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Exponential order of coeffs is computable:

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TRAINS

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Chapter 5Rational and Meromorphic Asymptotics

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Schur:21Wednesday, June 2, 2010

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Worked out Example: derangements

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SCHEMA

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Chapter 6Singularity Analysis

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(earlier: Darboux-Pólya; Tauberian thms)

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THEOREMS 1+2

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EGF

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Chapter 7Applications of Singularity

Analysis

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functional equation

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selves

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Exponential * nrational

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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.

APPLICATIONS of ALGEBRAIC FUNCTIONS

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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.

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Conclusion:SCHEMAS

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Chapter 8Saddle-point Asymptotics

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Modulus of an analytic function70Wednesday, June 2, 2010

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Cauchy coefficient integrals

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Saddle-point method:

= concentration + local quadratic approximation.

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The saddle-point theorem:

under conditions: concentration + local quadratic approximation

+ Hayman: admissible functions= closure properties

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Hardy & Ramanujan

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(double saddle-point)

Coalescence; e.g., maps

Gauss => Airy

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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

Conclusions: Saddle-point method

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