4. Complex Analysis, Rational and Meromorphic Asymptotics - PowerPoint PPT Presentation

A N A L Y T I C C O M B I N A T O R I C S P A R T T W O 4. Complex Analysis, Rational and Meromorphic Asymptotics http://ac.cs.princeton.edu

A N A L Y T I C C O M B I N A T O R I C S P A R T T W O 4. Complex Analysis, Rational and Meromorphic functions Analytic •Roadmap Combinatorics •Complex functions Philippe Flajolet and •Rational functions Robert Sedgewick OF •Analytic functions and complex integration •Meromorphic functions CAMBRIDGE http://ac.cs.princeton.edu II.4a.CARM.Roadmap

⬅ Analytic combinatorics overview specification A. SYMBOLIC METHOD 1. OGFs 2. EGFs GF equation 3. MGFs SYMBOLIC METHOD B. COMPLEX ASYMPTOTICS asymptotic 4. Rational & Meromorphic estimate 5. Applications of R&M COMPLEX ASYMPTOTICS 6. Singularity Analysis desired 7. Applications of SA result ! 8. Saddle point 3

Starting point The symbolic method supplies generating functions that vary widely in nature. � � ( � ) = � + � + � � + . . . + � � − � � ( � ) = � − � √ � ( � ) = � + � − � � � � ( � ) = � − � � − � − � � − . . . − � � � − � � � � � � � � ( � ) = � � ( � ) = � ( � ) = � � + � � / � � − � ln ( � − � )( � − � � ) . . . ( � − �� ) � − � Next step: Derive asymptotic estimates of coefficients. [ � � ] � ( � ) ∼ � � − � � [ � � ] � ( � ) ∼ � [ � � ] � ( � ) ∼ [ � � ] � � ( � ) = � β � � (ln � ) � + � � � �� � √ [ � � ] � � ( � ) ∼ � � � − � / � � − � / � [ � � ] � ( � ) ∼ � � / � − [ � � ] � ( � ) = ln � √ � ! � �� Classical approach: Develop explicit expressions for coefficients, then approximate Analytic combinatorics approach: Direct approximations. 4

Starting point Catalan trees Derangements D = SET ( CYC > 1 ( Z )) G = ○ × SEQ ( G ) Construction Construction � � − � − � � ln � ( � ) = � ( � ) = � OGF equation EGF equation � − � ( � ) = � − � √ � − � � � ( � ) = � + Explicit form of OGF Explicit form of EGF � − � � � � ( − � ) � � ( � ) = − � � �� ��� � � � ( − � � ) � � � � ( � ) = Expansion Expansion � ! � � � ≥ � � ≥ � � ≥ � � � = � � � � − � ( − � ) � � � � = � Explicit form of coefficients Explicit form of coefficients � � − � � ! � ≤ � ≤ � � � ∼ � � − � � � ∼ � − � Approximation Approximation √ �� � � − � − � � / � − � � / � ( � + � + � � / � ! + . . . + � � / � !) � Problem: Explicit forms can be unwieldy (or unavailable). � − � Opportunity: Relationship between asymptotic result and GF . 5

Analytic combinatorics overview To analyze properties of a large combinatorial structure: Ex. Derangements Speci fi cation 1. Use the symbolic method (lectures 1 and 2). D = SET ( CYC > 1 ( Z )) • Define a class of combinatorial objects. • Define a notion of size (and associated GF) Symbolic transfer • Use standard constructions to specify the structure. • Use a symbolic transfer theorem. Result: A direct derivation of a GF equation. � ( � ) = � − � GF equation � − � 2. Use complex asymptotics (starting with this lecture). Analytic transfer • Start with GF equation. • Use an analytic transfer theorem. � � ∼ � Result: Asymptotic estimates of the desired properties. � Asymptotics 6

A shift in point of view Speci fi cation generating functions are treated as formal objects Symbolic transfer analytic formal object! object! GF equation GF Analytic transfer generating functions are treated as analytic objects Asymptotics 7

GFs as analytic objects (real) Q. What happens when we assign real values to a GF? coefficients are positive so f(x) is positive (0, 1) � � ( � ) = � − � � (1, − 1) singularity continuation A. We can use a series representation (in a certain interval) that allows us to extract coefficients. � [ � � ] � ( � ) = � � � − � � = � + � � + � � � + � � � + . . . ��� � ≤ � < � / � Useful concepts: � � ( � ) = � + � � + �� � � + . . . Differentiation: Compute derivative term-by-term where series is valid. Singularities: Points at which series ceases to be valid. Continuation: � ( � ) = − � Use functional representation even where series may diverge. 8

GFs as analytic objects (complex) Q. What happens when we assign complex values to a GF? stay tuned for � ( � ) = � − � interpretation of plot � − � singularity A. We can use a series representation (in a certain domain) that allows us to extract coefficients. Same useful concepts: Differentiation: Compute derivative term-by-term where series is valid. Singularities: Points at which series ceases to be valid. Continuation: Use functional representation even where series may diverge. 9

GFs as analytic objects (complex) Q. What happens when we assign complex values to a GF? stay tuned for � ( � ) = � − � interpretation of plot � − � singularity A. A surprise! Serendipity is not an accident Singularities provide full information on growth of GF coefficients! “Singularities provide a royal road to coefficient asymptotics.” 10

General form of coefficients of combinatorial GFs First principle of coefficient asymptotics The location of a function’s singularities dictates subexponential factor the exponential growth of its coefficients. [ � � ] � ( � ) = � � θ ( � ) Second principle of coefficient asymptotics exponential growth factor The nature of a function’s singularities dictates the subexponential factor of the growth. Examples (preview): singularities singularities GF GF exponential exponential subexp. subexp. GF GF type growth factor location nature � � − � � strings with φ � � / φ , � / ˆ � � ( � ) = rational φ pole no 00 � − � � − � � √ � � ( � ) = � − � � − � 1 1 N derangements meromorphic pole � − � � √ � ( � ) = � + � − � � � / � square Catalan trees analytic 4 N � root � √ �� � 11

A N A L Y T I C C O M B I N A T O R I C S P A R T T W O 4. Complex Analysis, Rational and Meromorphic functions Analytic •Roadmap Combinatorics •Complex functions Philippe Flajolet and •Rational functions Robert Sedgewick OF •Analytic functions and complex integration •Meromorphic functions CAMBRIDGE http://ac.cs.princeton.edu II.4a.CARM.Roadmap

A N A L Y T I C C O M B I N A T O R I C S P A R T T W O 4. Complex Analysis, Rational and Meromorphic functions Analytic •Roadmap Combinatorics •Complex functions Philippe Flajolet and •Rational functions Robert Sedgewick OF •Analytic functions and complex integration •Meromorphic functions CAMBRIDGE http://ac.cs.princeton.edu II.4b.CARM.Complex

Theory of complex functions Quintessential example of the power of abstraction. are complex numbers Start by defining i to be the square root of − 1 so that i 2 = − 1 real ? Continue by exploring natural definitions of basic operations • Addition • Multiplication • Division 1 + i • Exponentiation • Functions • Differentiation • Integration 14

Standard conventions Correspondence with points in the plane � = � + �� ( x, y ) represents x � � � � z = x + iy real part |z| y � � � � imaginary part � | � | ≡ � � + � � absolute value � = � − �� ¯ conjugate ( x, − y ) represents � = | � | � z = x − iy � ¯ Quick exercise: 15

Basic operations Natural approach: Use algebra, but convert i 2 to − 1 whenever it occurs Addition ( � + �� ) + ( � + �� ) = ( � + � ) + ( � + � ) � Multiplication ( � + �� ) ∗ ( � + �� ) = �� + ��� + ��� + ��� � = ( �� − �� ) + ( �� + �� ) � Division � + �� = � − �� � � � � = ¯ � � + � � | � | Exponentiation? 16

Analytic functions Definition. A function f ( z ) defined in Ω is analytic at a point z 0 in Ω iff for z in an open disc in � � ( � − � � ) � � ( � ) = � Ω centered at z 0 it is representable by a power-series expansion � ≥ � Examples: � � − � = � + � + � � + � � + � � + . . . is analytic for | z | < 1 . � ! + � � � ! + � � � ! + � � � � ≡ � + � � ! + . . . is analytic for | z | < ∞ . 17