Lecture 17: Mittag-Leffler's Theorem and Series Convergence

This file formalizes results from Lecture 17:

• Lemma 1: A convergence test for series of the form ∑ aₙvₙ using summation by parts.
• Theorem 1: The Mittag-Leffler Theorem on existence of meromorphic functions with prescribed poles and principal parts.

Main results

• summation_by_parts_convergence: If the partial sums Aₙ = ∑_{k=0}^{n} aₖ are bounded, vₙ → 0, and ∑ ‖vₙ - vₙ₊₁‖ < ∞, then ∑ aₙvₙ converges.

• MittagLeffler.mittag_leffler: Given a sequence b : ℕ → ℂ with ‖b n‖ → ∞ and polynomials P n without constant term, there exists a meromorphic function on ℂ with poles at just the points b n and singular parts P n (1/(z - b n)). The most general such function can be written as g(z) + Σₙ [Pₙ(1/(z - bₙ)) - pₙ(z)] where g is entire and pₙ are correcting polynomials.

Implementation notes

The existence of correcting polynomials with the required Taylor remainder decay bounds is proved using the Cauchy power series and the geometric approximation lemma for power series (HasFPowerSeriesOnBall.uniform_geometric_approx). For each ν with b ν ≠ 0, the singular part Pᵥ(1/(z-bᵥ)) is analytic at z=0 with a power series converging on a ball of radius ‖bᵥ‖/2. The Taylor polynomial (partial sum) of sufficiently high degree approximates the function to within 2⁻ᵛ on the smaller ball of radius ‖bᵥ‖/4. The corrected series then converges uniformly on compact subsets by the Weierstrass M-test.

References

• Lecture 17 of the Complex Variables course

theorem summation_by_parts_convergence (a v : ℕ → ℂ) (hA : ∃ (M : ℝ), ∀ (n : ℕ), ‖∑ k ∈ Finset.range (n + 1), a k‖ ≤ M) (hv : Filter.Tendsto v Filter.atTop (nhds 0)) (hvs : Summable fun (n : ℕ) => ‖v n - v (n + 1)‖) : ∃ (l : ℂ), Filter.Tendsto (fun (N : ℕ) => ∑ i ∈ Finset.range N, a i * v i) Filter.atTop (nhds l)

Lemma 1, Lecture 17 (Summation by parts test for convergence). If the partial sums Aₙ = ∑_{k=0}^{n} aₖ are bounded (in norm), vₙ → 0, and ∑ ‖vₙ - vₙ₊₁‖ < ∞, then ∑ aₙvₙ converges (the partial sums have a limit).

The proof uses Abel's summation by parts formula: ∑_{i=0}^{N} aᵢvᵢ = vₙAₙ + ∑_{i=0}^{N-1} Aᵢ(vᵢ - vᵢ₊₁) The first term tends to zero (bounded × vanishing), and the second series converges absolutely since ‖Aᵢ(vᵢ - vᵢ₊₁)‖ ≤ M‖vᵢ - vᵢ₊₁‖ and ∑‖vᵢ - vᵢ₊₁‖ < ∞.

Mittag-Leffler's Theorem

noncomputable def MittagLeffler.singularPart (P : Polynomial ℂ) (b z : ℂ) : ℂ

The singular part function: z ↦ P(1/(z - b)).

def MittagLeffler.HasNoConstantTerm (P : Polynomial ℂ) : Prop

A polynomial has no constant term if P(0) = 0.

def MittagLeffler.HasSingularPartAt (f : ℂ → ℂ) (P : Polynomial ℂ) (b : ℂ) : Prop

f has singular part P(1/(z-b)) at b.

theorem MittagLeffler.singularPart_analyticAt (P : Polynomial ℂ) (b z₀ : ℂ) (hz : z₀ ≠ b) : AnalyticAt ℂ (singularPart P b) z₀

The singular part z ↦ P(1/(z - b)) is analytic at any point z₀ ≠ b.

theorem MittagLeffler.partialSum_eq_polynomial_eval (ps : FormalMultilinearSeries ℂ ℂ ℂ) (N : ℕ) (z : ℂ) : ps.partialSum N z = Polynomial.eval z (∑ k ∈ Finset.range N, Polynomial.C (ps.coeff k) * Polynomial.X ^ k)

In the 1D case (ℂ → ℂ), partial sums of a formal power series equal polynomial evaluation.

theorem MittagLeffler.exists_correcting_poly_single (P : Polynomial ℂ) (b : ℂ) (hb : b ≠ 0) (ε : ℝ) (hε : 0 < ε) : ∃ (p : Polynomial ℂ), ∀ (z : ℂ), ‖z‖ ≤ ‖b‖ / 4 → ‖singularPart P b z - Polynomial.eval z p‖ ≤ ε

For b ≠ 0, there exists a polynomial approximating singularPart P b to within ε on the ball of radius ‖b‖/4 around 0. The polynomial is a partial sum of the Cauchy power series of singularPart P b centered at 0.

theorem MittagLeffler.exists_correcting_polynomials (b : ℕ → ℂ) (P : ℕ → Polynomial ℂ) (hb : Filter.Tendsto (fun (n : ℕ) => ‖b n‖) Filter.atTop Filter.atTop) (hP : ∀ (n : ℕ), HasNoConstantTerm (P n)) : ∃ (p : ℕ → Polynomial ℂ), ∀ (ν : ℕ) (z : ℂ), ‖z‖ ≤ ‖b ν‖ / 4 → ‖singularPart (P ν) (b ν) z - Polynomial.eval z (p ν)‖ ≤ 2⁻¹ ^ ν

Taylor remainder bound for correcting polynomials (Theorem 8, p.125 and formula (29), p.126 from the textbook).

For each ν, Pᵥ(1/(z - bᵥ)) is analytic on {z : ‖z‖ < ‖bᵥ‖}. By the Cauchy power series and geometric approximation (HasFPowerSeriesOnBall.uniform_geometric_approx), choosing the truncation degree nᵥ large enough, we obtain a polynomial pᵥ (partial sum of the power series of Pᵥ(1/(z-bᵥ)) at z=0) satisfying ‖Pᵥ(1/(z-bᵥ)) - pᵥ(z)‖ ≤ 2⁻ᵛ for ‖z‖ ≤ ‖bᵥ‖/4.

theorem MittagLeffler.singularPart_meromorphicAt (P : Polynomial ℂ) (b : ℂ) : MeromorphicAt (singularPart P b) b

The singular part z ↦ P(1/(z - b)) is meromorphic at its pole b.

theorem MittagLeffler.polynomial_eval_analyticAt (p : Polynomial ℂ) (z₀ : ℂ) : AnalyticAt ℂ (fun (z : ℂ) => Polynomial.eval z p) z₀

Polynomial evaluation z ↦ eval z p is analytic at every point.

theorem MittagLeffler.correctedTerm_analyticAt (P : Polynomial ℂ) (b : ℂ) (p : Polynomial ℂ) (z₀ : ℂ) (hz : z₀ ≠ b) : AnalyticAt ℂ (fun (z : ℂ) => singularPart P b z - Polynomial.eval z p) z₀

Each corrected term Pᵥ(1/(z-bᵥ)) - pᵥ(z) is analytic at points z₀ ≠ bᵥ.

theorem MittagLeffler.summable_correctedTerms (b : ℕ → ℂ) (P p : ℕ → Polynomial ℂ) (hbound : ∀ (ν : ℕ) (z : ℂ), ‖z‖ ≤ ‖b ν‖ / 4 → ‖singularPart (P ν) (b ν) z - Polynomial.eval z (p ν)‖ ≤ 2⁻¹ ^ ν) (hb : Filter.Tendsto (fun (n : ℕ) => ‖b n‖) Filter.atTop Filter.atTop) (z : ℂ) : Summable fun (ν : ℕ) => singularPart (P ν) (b ν) z - Polynomial.eval z (p ν)

The corrected terms are summable at each point.

theorem MittagLeffler.h_analyticAt_away (b : ℕ → ℂ) (P p : ℕ → Polynomial ℂ) (hbound : ∀ (ν : ℕ) (z : ℂ), ‖z‖ ≤ ‖b ν‖ / 4 → ‖singularPart (P ν) (b ν) z - Polynomial.eval z (p ν)‖ ≤ 2⁻¹ ^ ν) (hb : Filter.Tendsto (fun (n : ℕ) => ‖b n‖) Filter.atTop Filter.atTop) (z₀ : ℂ) (hz₀ : ∀ (n : ℕ), z₀ ≠ b n) : AnalyticAt ℂ (fun (z : ℂ) => ∑' (ν : ℕ), (singularPart (P ν) (b ν) z - Polynomial.eval z (p ν))) z₀

h is analytic away from all poles (Lecture 17 proof, Step 4).

For z₀ away from all poles, split h(z) = Σ_{ν<N} + Σ_{ν≥N} where N is chosen so that all poles bᵥ with ν ≥ N satisfy ‖bᵥ‖/4 > ‖z₀‖. The finite sum is analytic at z₀ (each term has its pole at bᵥ ≠ z₀). The tail converges uniformly on B(0, ‖z₀‖+1) by the Weierstrass M-test (bound by 2⁻ᵛ), so its sum is holomorphic there by the Weierstrass theorem on uniform limits of holomorphic functions.

theorem MittagLeffler.remaining_tsum_analyticAt (b : ℕ → ℂ) (P p : ℕ → Polynomial ℂ) (hbound : ∀ (ν : ℕ) (z : ℂ), ‖z‖ ≤ ‖b ν‖ / 4 → ‖singularPart (P ν) (b ν) z - Polynomial.eval z (p ν)‖ ≤ 2⁻¹ ^ ν) (hb : Filter.Tendsto (fun (n : ℕ) => ‖b n‖) Filter.atTop Filter.atTop) (hinj : Function.Injective b) (n : ℕ) : AnalyticAt ℂ (fun (z : ℂ) => ∑' (ν : ℕ), if ν = n then 0 else singularPart (P ν) (b ν) z - Polynomial.eval z (p ν)) (b n)

The "if-tsum" ∑' ν, (if ν = n then 0 else f ν z) (remaining corrected terms with the nth term zeroed out) is analytic at b n when b is injective.

This uses the same Weierstrass M-test argument as h_analyticAt_away, noting that each term f ν with ν ≠ n is analytic at b n (since b is injective, so b n ≠ b ν), and the bound 2⁻ᵛ still applies.

noncomputable def MittagLeffler.hFunc (b : ℕ → ℂ) (P p : ℕ → Polynomial ℂ) (z : ℂ) : ℂ

The function h(z) = ∑' ν, (Pᵥ(1/(z-bᵥ)) - pᵥ(z)).

theorem MittagLeffler.mittag_leffler (b : ℕ → ℂ) (P : ℕ → Polynomial ℂ) (hb : Filter.Tendsto (fun (n : ℕ) => ‖b n‖) Filter.atTop Filter.atTop) (hP : ∀ (n : ℕ), HasNoConstantTerm (P n)) (hinj : Function.Injective b) : ∃ (h : ℂ → ℂ), Meromorphic h ∧ (∀ (n : ℕ), HasSingularPartAt h (P n) (b n)) ∧ ∀ (f : ℂ → ℂ), (∀ (n : ℕ), HasSingularPartAt f (P n) (b n)) → (∀ z ∉ Set.range b, AnalyticAt ℂ f z) → ∀ (z : ℂ), AnalyticAt ℂ (fun (w : ℂ) => f w - h w) z

Theorem 1, Lecture 17 (Mittag-Leffler's Theorem).

Let {bᵥ} be a sequence in ℂ with ‖bᵥ‖ → ∞ (and the bᵥ distinct), and let Pᵥ(ζ) be polynomials without constant term. Then there exist functions f meromorphic in ℂ with poles at exactly the points bᵥ and corresponding singular parts Pᵥ(1/(z - bᵥ)).

The most general such meromorphic function is f(z) = g(z) + ∑ᵥ [Pᵥ(1/(z - bᵥ)) - pᵥ(z)] where g is holomorphic everywhere and the pᵥ are correcting polynomials.

The proof constructs h(z) = ∑ᵥ [Pᵥ(1/(z-bᵥ)) - pᵥ(z)] using correcting polynomials pᵥ with ‖Pᵥ(1/(z-bᵥ)) - pᵥ(z)‖ ≤ 2⁻ᵛ for ‖z‖ ≤ ‖bᵥ‖/4, then verifies: (a) h is meromorphic on ℂ (analytic away from all poles by the Weierstrass M-test, meromorphic at each pole by splitting off the finite singular term), (b) h has singular part Pₙ(1/(z-bₙ)) at each bₙ (the tsum minus the singular part reduces to an analytic correction plus a remaining tsum that is analytic), (c) any other meromorphic function with the same singular parts differs from h by an entire function (at poles, the singular parts cancel; away from poles, both are analytic).