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4. Complex Analysis, Rational and Meromorphic Asymptotics ANALYTIC COMBINATORICS P A R T T W O 4. Complex Analysis, Rational and Meromorphic Asymptotics http://ac.cs.princeton.edu ANALYTIC COMBINATORICS P A R T T W O 4. Complex Analysis, Rational and Meromorphic functions Analytic Combinatorics •Roadmap Philippe Flajolet and •Complex functions Robert Sedgewick OF •Rational functions •Analytic functions and complex integration CAMBRIDGE •Meromorphic functions http://ac.cs.princeton.edu II.4a.CARM.Roadmap Analytic combinatorics overview specification A. SYMBOLIC METHOD 1. OGFs 2. EGFs GF equation 3. MGFs B. COMPLEX ASYMPTOTICS SYMBOLIC METHOD 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 Construction G = ○ × SEQ( G ) Construction D = SET (CYC>1( Z )) 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 1. Use the symbolic method (lectures 1 and 2). Speci!cation D = SET (CYC>1( Z )) • Define a class of combinatorial objects. • Define a notion of size (and associated GF) • Use standard constructions to specify the structure. Symbolic transfer • 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!cation generating functions are treated as formal objects Symbolic transfer formal analytic 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 subexponential The location of a function’s singularities dictates factor the exponential growth of its coefficients. []()=θ() exponential Second principle of coefficient asymptotics growth factor The nature of a function’s singularities dictates the subexponential factor of the growth. Examples (preview): GF singularities exponential subexp. GF type location nature growth factor strings with ()= − rational /φ, /φˆ pole φ no 00 √ − − − derangements ()= meromorphic 1 pole 1N − − + √ square Catalan trees ()= − analytic / 4N root √ 11 ANALYTIC COMBINATORICS P A R T T W O 4. Complex Analysis, Rational and Meromorphic functions Analytic Combinatorics •Roadmap Philippe Flajolet and •Complex functions Robert Sedgewick OF •Rational functions •Analytic functions and complex integration CAMBRIDGE •Meromorphic functions http://ac.cs.princeton.edu II.4a.CARM.Roadmap ANALYTIC COMBINATORICS P A R T T W O 4. Complex Analysis, Rational and Meromorphic functions Analytic Combinatorics •Roadmap Philippe Flajolet and •Complex functions Robert Sedgewick OF •Rational functions •Analytic functions and complex integration CAMBRIDGE •Meromorphic functions http://ac.cs.princeton.edu II.4b.CARM.Complex Theory of complex functions Quintessential example of the power of abstraction. are complex 2 numbers Start by defining i to be the square root of −1 so that i = −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 real part z = x + iy ≡ |z| y imaginary part ≡ absolute value + | | ≡ conjugate ¯ = − (x, −y) represents z = x − iy Quick exercise: ¯ = | | 15 Basic operations 2 Natural approach: Use algebra, but convert i 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 z0 in Ω iff for z in an open disc in Ω centered at z0 it is representable by a power-series expansion ()= ( ) − ≥ Examples: = + + + + + ... is analytic for |z| < 1 . − + + + + + ... is analytic for |z| < ∞ . ≡ ! ! ! ! 17 Complex differentiation Definition. A function f (z ) defined in a region Ω is holomorphic or complex-differentiable at a ( + δ) () point z0 in Ω iff the limit ( )=lim − exists, for complex δ. δ δ → Note: Notationally the same as for reals, but much stronger—the value is independent of the way that δ approaches 0. Theorem. Basic Equivalence Theorem. For purposes of this lecture: A function is analytic in a region Ω iff it is complex-differentiable in Ω. Axiom 1. Useful facts: • If function is analytic (complex-differentiable) in Ω, it admits derivatives of any order in Ω. • We can differentiate a function via term-by-term differentiation of its series representation. • Taylor series expansions ala reals are effective. 18 Taylor's theorem immediately gives power series expansions for analytic functions. + + + + + ... ≡ ! ! ! ! sin + + ... ≡ ! − ! ! − ! cos + + ... ≡ − ! ! − ! = + + + + + ... − 19 Euler's formula Evaluate the exponential function at iθ θ (θ) (θ) (θ) (θ) θ = + + + + + ...+ + ... ! ! ! ! ! θ θ θ θ θ θ θ θ θ = + + + + + + ... ! − ! − ! ! ! − ! − ! ! ! 2 3 4 i = −1 i = −i i = 1 θ = cos θ + sin θ “Our jewel . one of the most remarkable, almost astounding, formulas in all of mathematics” Euler's formula — Richard Feynman, 1977 20 Polar coordinates Euler's formula gives another correspondence between complex numbers and points in the plane. r cosθ (x, y) θ = cos θ + sin θ r r sinθ θ Conversion functions defined for any complex number x + iy : • absolute value (modulus) = + • angle (argument) θ = arctan 21 ANALYTIC COMBINATORICS P A R T T W O 4. Complex Analysis, Rational and Meromorphic functions Analytic Combinatorics •Roadmap Philippe Flajolet and •Complex functions Robert Sedgewick OF •Rational functions •Analytic functions and complex integration CAMBRIDGE •Meromorphic functions http://ac.cs.princeton.edu II.4b.CARM.Complex ANALYTIC COMBINATORICS P A R T T W O 4. Complex Analysis, Rational and Meromorphic functions Analytic Combinatorics •Roadmap Philippe Flajolet and •Complex functions Robert Sedgewick OF •Rational functions •Analytic functions and complex integration CAMBRIDGE •Meromorphic functions http://ac.cs.princeton.edu II.4c.CARM.Rational Rational functions are complex functions that are the ratio of two polynomials. − + √ + + + ...+ − ()= ()= − ()= ()= ... − − − − − − ()= ()= ln + / ( )( ) ...( ) ()= − − − − − Approach: • Use partial fractions to expand into terms for which coefficient extraction is easy. • Focus on the largest term to approximate. [Same approach as for reals, but takes complex roots into account.] 24 Extracting coefficients from rational GFs Factor the denominator and use partial fractions to expand into sum of simple terms. Example 1. ()= Rational GF () (distinct roots) + ≡ − ≥ Factor denominator = ( )( ) − − Use partial fractions: ()= + Expansion must be of the form − − Cross multiply + = and solve for coefficients.
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