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In this program

Laplace's method and Watson's lemma
Method of stationary phase
Steepest descent
Saddle-point methods (introduction)
Asymptotics of special functions
Summation of divergent series: Borel summability (introduction)

Integral asymptotics

Saddle-point methods (introduction)

Asymptotic & perturbation methods · Standard syllabus

Topic

Saddle-point methods (introduction)

Practice by unit: Integral asymptotics

Graduate Asymptotic & Perturbation Methods — scope drawn from open asymptotics notes and typical US graduate asymptotics syllabi.

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