Weierstrass and Hadamard products

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1 August 9, 3 Weierstrass and Hadamard products Paul Garrett garrett@mat.umn.edu ttp:// garrett/ [Tis document is ttp:// garrett/m/mfms/notes 3-4/b Hadamard products.pdf]. Weierstrass products. Poisson-Jensen formula 3. Hadamard products 4. Appendix: armonic functions Apart from factorization of polynomials, peraps te oldest product expression is Euler s sin πz πz = n= z n Granting tis, Euler equated te power series coefficients of z, evaluating ζ for te first time: Te Γ-function factors: e t t z dt t π 6 = n = Γz = n= z e γz n= + z e z/n n were te Euler-Masceroni constant γ is essentially defined by tis relation. Te integral Euler s converges for Rez >, wile te product Weierstrass converges for all complex z except non-positive integers. Granting tis, te Γ-function is visibly related to sine by Γz Γ z = z n= because te exponential factors are linear, and can cancel. z n = z sin πz π Linear exponential factors are exploited in Riemann s explicit formula [Riemann 859], derived from equality of te Euler product and Hadamard product [Hadamard 893] for te zeta function ζs = n n for s Res > : p prime p s = ζs = ea+bs s ρ s e ρs ρ n= + s e s/n n were te product expansion of Γ s is visible, corresponding to trivial zeros of ζs at negative even integers, and ρ ranges over all oter, non-trivial zeros, known to be in te critical strip < Res <. Te ard part of te proof below of Hadamard s teorem is adapted from [Alfors 953/966], wit various rearrangements. A somewat different argument is in [Lang 993]. We recall some standard folkloric? proofs of supporting facts about armonic functions, starting from scratc.

2 Paul Garrett: Weierstrass and Hadamard products August 9, 3. Weierstrass products Given a sequence of complex numbers z wit no accumulation point in C, we will construct an entire function wit zeros exactly te z. Tis is essentially elementary. [.] Basic construction Taylor-MacLaurin polynomials of log z will play a role: let p n z = z + z + z zn n We will sow tat tere is a sequence of integers n giving an absolutely convergent infinite product vanising precisely at te z, wit vanising at z = accommodated by a suitable leading factor z m : z m z z e pn z/z = z m z z z exp + z z z + z3 3z zn n z n Absolute convergence of log + a implies absolute convergence of te infinite product + a. Tus, we sow tat log z z + p n < z z Fix a large radius R, keep z < R, and ignore te finitely-many z wit z < R, so in te following we ave z/z <. Using te power series expansion of log, log z z p n z z n + Tus, we want a sequence of positive integers n suc tat n+ + z n + n z n + z/z n+ z/z z/z n+ n + z R z/z n+ n + < wit z < R Of course, te coice of n s must be compatible wit enlarging R, but tis is easily arranged. For example, n = succeeds: = z z <R z z + z R z z z <R + z Since {z } is discrete, te sum over z < R is finite, so we ave convergence, and convergence of te infinite product wit n = : z z e pz/z = z z z exp + z z z + z3 3z z z

3 Paul Garrett: Weierstrass and Hadamard products August 9, 3 [.] Canonical products and genus Given entire f wit zeros z and a zero of order m at, ratios ϕz = fz z m z e p n z/z z wit convergent infinite products are entire, and do not vanis. Non-vanising entire ϕ as an entire logaritm: z ϕ ζ dζ gz = log ϕz = ϕζ Tus, non-vanising entire ϕ is expressible as ϕz = e gz wit g entire Tus, te most general entire function wit prescribed zeros is of te form fz = e gz z m z z e p n z/z wit g entire Naturally, wit fixed f, altering te n necessitates a corresponding alteration in g. We are most interested in zeros {z } allowing a uniform integer giving convergence of te infinite product in an expression fz = e gz z m z z e p z/z = z m z z z exp + z z z + z3 3z z z Wen f admits a product expression wit a uniform, a product expression wit minimal uniform is a canonical product for f. Wen, furter, te leading factor e gz for f as gz a polynomial, te genus of f is te maximum of and te degree of g.. Poisson-Jensen formula Jensen s formula and te Poisson-Jensen formula are essential in te difficult alf of te Hadamard teorem below comparing genus of an entire function to its order of growt. Te logaritm uz = log fz of te absolute value fz of a non-vanising olomorpic function f on a neigborood of te unit disk is armonic, tat is, is anniilated by = x + y : expand log fz = x + i y x i y log fz + log f z Conveniently, te two-dimensional Laplacian is te product of te Caucy-Riemann operator and its conugate. Since log f is olomorpic and log f is anti-olomorpic, bot are anniilated by te product of te two linear operators. Tis verifies tat log fz is armonic. Tus, log fz satisfies te mean-value property for armonic functions: log f = u = ue iθ dθ = 3 log fe iθ dθ

4 Paul Garrett: Weierstrass and Hadamard products August 9, 3 Next, let f ave zeros ρ in z < but none on te unit circle. We manufacture a olomorpic function F from f but witout zeros in z <, and wit F = f on z =, by te standard ruse F z = fz ρ z z ρ Indeed, for z =, te numerator of eac factor as te same absolute value as te denominator: z ρ = z ρ = z ρ z = ρ z For simplicity, suppose no ρ is. Applying te mean-value identity to log F z gives log f log ρ = log F = log F e iθ dθ = log fe iθ dθ and ten te basic Jensen s formula log f log ρ = log fe iθ dθ for ρ < Te Poisson-Jensen formula is obtained by replacing by an arbitrary point z inside te unit disk. Tis is obtained by replacing f by f ϕ z, were ϕ z is a linear fractional transformation mapping z and stabilizing [] te unit disk: ϕ z = z z : w w + z zw + Tis replaces te zeros ρ by ϕ z ρ = ρ z zρ +. Instead of te mean-value property expressing f as an integral over te circle, use te Poisson formula see appendix for fz. Tis gives te basic Poisson-Jensen formula log fz ρ z log = zρ + log fe iθ z z e iθ dθ for z <, ρ < More generally, for olomorpic f on a neigborood of a disk of radius r > wit zeros ρ in tat disk, apply te previous to fr z wit zeros ρ /r in te unit disk: log fr z Replacing z by z/r gives [] log fz ρ /r z log = zρ /r + log ρ /r z/r zρ /r = + log fr e iθ log fre iθ To verify tat suc maps stabilize te unit disk, expand te natural expression: w + z zw + = zw + zw + w + z = zw + zw + w z z z e iθ dθ for z < z/r z/r e iθ dθ for z < r = zw + zw + zw + zw + w zw zw z = zw + z w > 4

5 Paul Garrett: Weierstrass and Hadamard products August 9, 3 wic rearranges sligtly to te general Poisson-Jensen formula log fz ρ z log = zρ /r + r log fre iθ r z z re iθ dθ for z < r, ρ < r Te case z = is te general Jensen formula for arbitrary radius r: log f log ρ = r log fre iθ dθ wit ρ < r 3. Hadamard products Te order of an entire function f is te smallest positive real λ, if it exists, suc tat, for every ε >, fz e z λ+ε for all sufficiently large z Te connection to infinite products is: [3..] Teorem: Hadamard Te genus and order λ are related by λ < +. In particular, one is finite if and only te oter is finite. Proof: First, te easier alf. For f of finite genus expressed as fz = e gz z m z z e p z/z = e gz z m z z z exp + z z z + z3 3z z z te leading exponential as polynomial g of degree at most, so e gz is of order at most. Te order of a product is at most te maximum of te orders of te factors, so it suffices to prove tat te order of te infinite product is at most +. Te assumption tat is te genus of f is equivalent to z + < We use tis to directly estimate te infinite product, sowing tat it as order of growt λ at most +. We need an estimate on F w = we p w applicable for all w, not merely for w <. We collect some inequalities. Tere is te basic log F w = log we p w e w / log F w + w for all w As before, for w <, log F w + w w w + w for w < Tis gives w log F w w + for w <. Adding to te latter te basic relation multiplied by w gives log F w w log F w + + w + for w < 5

6 Paul Garrett: Weierstrass and Hadamard products August 9, 3 In fact, te latter inequality also olds for w and log F w, from te basic relation. log F w < and w, from te basic relation, For log F w log F w + w w + w + for log F w < and w Now prove log F w w +, by induction on. For =, from log x x, log w w + w = w Assume log F w w. For w <, we reac te desired conclusion by log F w w log F w + + w + w w + + w + for w < For w and log F w >, from te basic relation log F w log F w + w w + w w + for w and log F w > For log F w and w, from te basic relation we already ave log F w log F w + w w w + for w and log F w < Tis proves log F w w + for all w. Estimate te infinite product: log z e p z/z = z log z e p z/z z + z < since / z + converges. Tus, suc an infinite product as growt order λ +. Now te difficult alf of te proof. Let λ < +. Jensen s formula will sow tat te zeros z are sufficiently spread out for convergence of / z +. Witout loss of generality, suppose f. From log f log z = r log fre iθ dθ wit z < r certainly log z <r/ ρ <r/ log ρ ε log f + r r λ+ε dθ r λ+ε for every ε > Wit νr te number of zeros inside te disk of radius r, tis gives lim r + νr = for all ε > rλ+ε Order te zeros by absolute value: z z... and for simplicity suppose no two ave te same size. Ten = ν z ε z λ+ε. Tus, z + ε = λ+ε + + λ+ε 6

7 Paul Garrett: Weierstrass and Hadamard products August 9, 3 Te latter converges for + λ+ε >, tat is, for λ + ε < +. Wen λ < +, tere is ε > making suc an equality old. It remains to sow tat te entire function gz in te leading exponential factor is of degree at most +, by sowing tat its + t derivative is. In te Poisson-Jensen formula log fz z z log = zz /r + r z <r log fre iθ r z z re iθ dθ for z < r application of x i y anniilates te anti-olomorpic parts, returning us to an equality of olomorpic functions, as follows. Te effect on te integrand is x i r z y z re iθ = z z re iθ z re iθ r z z re iθ z re iθ = z + zre iθ r + z z re iθ z re iθ re iθ = re iθ z Tus, f z fz z <r + z z Furter differentiation times in z gives z <r f z! = fz z z + z <r z <r z z z r = log fre iθ! z + z z r + + +! We claim tat te second sum and te integral go to as r +. Regarding te integral, Caucy s integral formula gives re iθ z re iθ dθ = + re iθ z re iθ dθ log fre iθ re iθ z re iθ dθ + Tus, letting M r be te maximum of f on te circle of radius r, and taking z < r/, up to sign te integral is log M r fre iθ Jensen s formula gives re iθ dθ z re iθ + r + M r log fre iθ dθ ε log fre iθ dθ log f Tus, for λ + ε < + te integral goes to as r +. For te second sum, again take z < r/, so for z < r r λ+ε r + z + z + + z z r + r z r z + r + r z + r + log fre iθ dθ We already sowed tat te number νr of z < r satisfies lim νr/r + =. Tus, tis sum goes to as r +. Tus, taking te limit, f =! f z z + 7

8 Paul Garrett: Weierstrass and Hadamard products August 9, 3 Returning to fz = e gz z z e p z/z, taking logaritmic derivative gives f f = g + + p z/z z z z and taking furter derivatives gives f = g + +! f z z + Since te t derivative of f /f is te latter sum, g + =, so g is a polynomial of degree at most. /// 4. Appendix: armonic functions We recall te mean-value property and Poisson s formula for armonic functions. Te Laplacian is = x + y and continuously twice-differentiable functions u wit u = are armonic. [4..] Teorem: Mean-value property For armonic u on a neigborood of te unit disk, u = ue iθ dθ Proof: Consider te rotation-averaged function vz = ue iθ z dθ for z Since te Laplacian is rotation-invariant, v is a rotation-invariant armonic function. In polar coordinates, for rotation-invariant functions vz = f z, te Laplacian is v = x + y f x + y = x x z f z + y y z f z = z f x z 3 f + x z f + z f y z 3 f + y z f = f + z f Te ordinary differential equation f + f /r = on an interval, R is an equation of Euler type, meaning expressible in te form r f + Brf + Cf = wit constants B, C. In general, suc equations are solved by letting fr = r λ, substituting, dividing troug by r λ, and solving te resulting indicial equation for λ: λλ + Aλ + B = Distinct roots λ, λ of te indicial equation produce linearly independent solutions r λ as in te case at and, a repeated root λ produces a second solution r λ log r. and r λ. However, Here, te indicial equation is λ =, so te general solution is a + b log r. 8

9 Paul Garrett: Weierstrass and Hadamard products August 9, 3 Wen b, te solution a + b log r blows up as r +. Since f = v = u is finite, it must be tat b =. Tus, a rotation-invariant armonic function on te disk is constant. Tus, its average over a circle is its central value. Tis proves te mean-value teorem for armonic functions. /// [4..] Remark: Te solutions a + b log r do indeed exaust te possible solutions: given f + f /r = on, R, r f = r f + f = r f /r + f = r Tus, r f is constant, and so on. Wit some computation, from mean-value property we will obtain [4..3] Teorem: Poisson s formula For u armonic on a neigborood of te unit disk, uz = ue iθ z z e iθ dθ for z < Proof: Composition wit olomorpic maps preserves armonic-ness. Wit ϕ z te linear fractional z transformation given by matrix ϕ z, te mean-value property for u ϕ z z gives uz = u ϕ z = u ϕ z e iθ dθ Linear fractional transformations stabilizing te unit disk map te unit circle to itself. Replace e iθ by e iθ = ϕ z e iθ uz = u ϕ z = ue iθ dθ Computing te cange of measure will yield te Poisson formula. Tis is computed by ie iθ θ θ = θ eiθ = ie iθ ze iθ + izeiθ e iθ z ze iθ = ieiθ ize iθ + ize iθ ie iθ z ze iθ = ieiθ z ze iθ Tus, giving te asserted integral. θ θ = e iθ ie iθ z ze iθ = zeiθ e iθ z eiθ z ze iθ = z z e iθ /// [Alfors 953/966] L. Alfors, Complex Analysis, McGraw-Hill, 953, second edition, 966. [Hadamard 893] J. Hadamard, Étude sur les Propriétés des Fonctions Entières et en Particulier d une Fonction Considérée par Riemann, J. Mat. Pures Appl. 893, 7-5. [Lang 993] S. Lang, Complex Analysis, Springer-Verlag, 993. [Riemann 859] B. Riemann, Über die Anzal der Primzalen unter ener gegebenen Gr sse, Monats. Akad. Berlin 859, [Wittaker-Watson 97] E.T. Wittaker, G.N. Watson, A Course of Modern Analysis, Cambridge University Press, 97, 4t edition, 95. 9