[{"type":"thinking","thinking":"No direct Mathlib `logDeriv Gamma = digamma`. OK, so the identity-principle route for extending (G) to the real non-pole points. Let me look at how `logDeriv_Gammaℝ_eq` is proved (5e-ii) — maybe its proof generalizes easily, or maybe there's a differentiability lemma for Gammaℝ I can use for the identity principle.\n\nThe plan for `(G)-extended`: \n```\nlemma logDeriv_Gammaℝ_eq_of_not_neg_two_nat {s : ℂ} (hs : ∀ m : ℕ, s ≠ -2 * (m : ℂ)) :\n    logDeriv Complex.Gammaℝ s = -(Real.log Real.pi : ℂ) / 2 + (1/2 : ℂ) * Complex.digamma (s / 2)\n```\nvia the identity principle on U := {s | ∀ m, s ≠ −2m}: \n- U is open (complement of a discrete set — hmm, proof: the set {−2m} is locally finite... This could be annoying. Alternative: prove the identity DIRECTLY at each real point via a punctured-neighborhood continuity: both sides continuous at s₀ = −R (real, non-pole), equal on the punctured ball (im ≠ 0 points are dense in the ball) ⟹ equal at s₀. The continuity-of-equality at a point: f, g continuous at s₀, f = g on 𝓝[≠] s₀ (a dense punctured nbhd) ⟹ f s₀ = g s₀. This is `ContinuousAt.eq_of_eventuallyEq`? Hmm — more precisely: if f and g are continuous at s₀ and f =ᶠ[𝓝[≠] s₀] g, then f s₀ = g s₀. That's from tendsto uniqueness: Tendsto f (𝓝[≠] s₀) (𝓝 (f s₀)) (continuity restricted) and g similarly with f =ᶠ g ⟹ f s₀ = g s₀ via tendsto_nhds_unique — needs 𝓝[≠] s₀ to be non-trivial (NeBot — in ℂ every point is a limit point ✓ `neBot_nhdsWithin` or `ContinuousAt.eventuallyEq`...). \n\nActually — the cleanest might be: prove the identity holds on the punctured ball around −R (via (G), since those points have im ≠ 0... wait NO — a punctured ball around −R contains only... the ball around −R contains points with im = 0 (other real points ≠ −R — hmm real points near −R: R ∈ [4k+2.5, 4k+3]: the real points near −R are −R±ε: are THEY poles? −2m would need R±ε = 2m: only if R = 2m exactly — but dist(R/2,ℤ) ≥ 1/4 excludes R/2 ∈ ℤ — but nearby reals R±ε: R/2±ε/2 could hit an integer? dist(R/2,ℤ) ≥ 1/4 ⟹ within ±1/8 of R/2 no integer ✓ so for ε < 1/4: (R±ε)/2 ∉ ℤ ✓ so real points in ball(−R, 1/8) are all non-pole ✓). Hmm — but (G) applies only at im ≠ 0 points. The continuity-of-equality: f := logDeriv ΓR, g := −(logπ)/2+½ψ(s/2): both continuous at s₀ (−R): \n- f continuous: logDeriv ΓR continuous at s₀: ΓR differentiable at s₀ (s₀/2 ∉ −ℕ: dist(R/2,ℤ) ≥ 1/4 ⟹ R/2 ∉ ℤ ✓ — need differentiableAt_Gammaℝ: what discharges hd in the 5d applications? Let me grep) and ΓR(s₀) ≠ 0 (Gammaℝ_eq_zero_iff) ⟹ logDeriv continuous (DifferentiableAt.logDeriv? there's `DifferentiableAt.logDeriv`? logDeriv = deriv/f: continuous when f cont, f' cont?? deriv isn't continuous in general — but for ANALYTIC f, deriv is analytic ⟹ continuous ✓: (analyticAt_Gammaℝ s₀).deriv.continuousAt.div (analyticAt continuous) (nonzero). So need analyticAt_Gammaℝ at s₀.\n- g continuous: digamma(s/2) analytic at s₀/2 = −R/2 (−R/2 ∉ {0}∪−ℕ ✓ via analyticAt_digamma from LiFEntire) ⟹ continuous.\n- equal on ball(−R, 1/8) \\ {im = 0}?? no — on {im ≠ 0} ∩ ball: dense: 𝓝[≠] s₀ contains all im ≠ 0 nearby points... the equality f = g holds on {im ≠ 0} (by (G)) which is cofinite-dense: the set {s : s.im ≠ 0} ∈ 𝓝[≠] s₀? {im = 0} is the real line — its complement is dense and open: {im ≠ 0} is open dense ⟹ ∈ 𝓝[≠] s₀ for any s₀ (a punctured nbhd contains it — actually {im ≠ 0} ∪ {s₀} is a nbhd of s₀ since it's open dense... {im ≠ 0} ∪ {s₀} = ℂ ∖ ({im = 0} ∖ {s₀}) — open? {im=0}∖{s₀} is closed (real line minus a point — closed ✓): complement open ✓ contains s₀ ✓. So {im≠0} ∪ {s₀} ∈ 𝓝 s₀ ⟹ {im ≠ 0} ∈ 𝓝[≠] s₀ (nhdsWithin_mem: s ∈ 𝓝[≠] s₀ iff s ∪ {s₀} ∈ 𝓝 s₀ ✓ `nhdsWithin_mem_iff`?). Then f =ᶠ[𝓝[≠] s₀] g (eventually on that set ✓).\n- Then f s₀ = g s₀: from both continuous at s₀: ContinuousAt f s₀ : Tendsto f (𝓝 s₀) (𝓝 (f s₀)) ⟹ Tendsto f (𝓝[≠] s₀) (𝓝 (f s₀)) (tendsto_nhdsWithin... ContinuousAt.tendsto_nhdsWithin? there's `ContinuousAt.tendsto_nhdsWithin` or just mono: Tendsto.mono_left inf_le_left... nhdsWithin s₀ {s₀}ᶜ ≤ 𝓝 s₀ hmm: 𝓝[≠] s₀ = 𝓝 s₀ ⊓ 𝓟 {s₀}ᶜ ≤ 𝓝 s₀ ✓ Tendsto.mono h inf_le_left). Similarly g. Then tendsto_nhds_unique (neBot of 𝓝[≠] s₀ in ℂ — `instNeBotNhdsWithinCompl`? there's `(𝓝[≠] s₀).NeBot` instance in T2 spaces... `nhdsWithin_compl_neBot`? In ℂ: `Complex` is a perfect space: NeBot (𝓝[≠] x) ✓ instance exists) applied to the EventuallyEq.\n\nSo the s = −R identity is ~40 lines. And then the WHOLE case (iii) uses the identity at all points (im ≠ 0 via (G), im = 0 via the extension — but actually for im ≠ 0 I can ALSO just use the extended version — so the extension REPLACES (G) usage in case (iii) entirely if I prove the extended version for all non-pole s. Hmm — but the extension is pointwise continuity at each real point; for im ≠ 0 points (G) applies directly. So: `logDeriv_Gammaℝ_eq'` : ∀ s with (∀ m, s ≠ −2m) (im ≠ 0 OR real): just do two lemmas: use (G) for im ≠ 0; for s₀ real (im = 0) non-pole: the continuity extension `logDeriv_Gammaℝ_eq_of_im_eq_zero`. ~50 lines. Only needed at s₀ = −R (the only real point on the left arc — and right arc's +R: region (i) doesn't need (G) ✓).\n\nActually — simpler alternative for the im = 0 case: I realize the identity principle on the connected open set U = {s : ∀ m, s ≠ −2m} is NOT obviously less work (proving U open + both sides AnalyticOnNhd). The pointwise continuity route is more elementary. Let me go with the pointwise route.\n\nHmm wait — actually even simpler: for s₀ = −R, apply (G) at the sequence points and take the LIMIT of the inequality ‖bracket‖ ≤ C: no — the identity itself is what I need... The norm-continuity route: I need ‖logDeriv ΓR s₀ + logDeriv ΓR(1−s₀)‖ ≤ C: the function F(s) := ‖logDeriv ΓR s + logDeriv ΓR(1−s)‖ is continuous at s₀ (both logDeriv ΓR continuous at s₀ and at 1−s₀) and F(s) ≤ C for all s in the dense set D := {im ≠ 0} ∩ (the left arc... — actually for ALL im ≠ 0 points with the right re-range — the bound derivation used re s < −5/2 etc. — for nearby points s of s₀ = −R on the circle with im ≠ 0: those satisfy the case-(iii) conditions ✓ (continuity of re: nearby s have re < −5/2 ✓ — well, on the ARC: the bound was proved for all arc points with re < −5/2 and im ≠ 0: s₀ = −R has re = −R < −5/2 ✓: the arc points near s₀: a punctured BALL around s₀ ∩ {im ≠ 0} ∩ {circle}... the bound F ≤ C holds on {im ≠ 0} ∩ ball(s₀, δ)-ish? NO — the case-(iii) bound holds for s ON THE CIRCLE with re s < −5/2. For the continuity I need F ≤ C on a set accumulating at s₀ — the arc points near s₀ with im ≠ 0 form such a set: {s : ‖s‖ = R ∧ im s ≠ 0 ∧ dist s s₀ < δ}: nonempty for all δ (arc is 1-dimensional: pick s = R·e^{i(π±ε)} ✓). Tendsto along that set: F(s₀) = lim F(s) ≤ C ✓ — but constructing the explicit sequence sₙ = R·e^{i(π−1/(n+1))} is the concrete way:\n  sₙ := R * exp(I*(π − 1/(n+1))): \n  - ‖sₙ‖ = R·‖exp(I·x)‖ = R·1 = R (Complex.norm_exp_I_mul_ofReal... norm of exp(I·x) = 1: `Complex.norm_exp_I_mul_ofReal : ‖exp (I * x)‖ = 1`? there's Complex.abs_exp_ofReal_mul_I... `Complex.norm_exp_ofReal_mul_I : ‖Complex.exp (x * I)‖ = 1` ✓).\n  - im sₙ = R·sin(π − 1/(n+1)) = R·sin(1/(n+1)) > 0 ✓ (sin(1/(n+1)) > 0: 1/(n+1) ∈ (0, 1] ⊆ (0, π) ✓ Real.sin_pos_of_pos_of_lt_pi).\n  - sₙ → −R: re sₙ = R·cos(π−1/(n+1)) = −R·cos(1/(n+1)) → −R·1; im → R·0 = 0: Tendsto (fun n => R·exp(I(π−1/(n+1)))) = R·exp(I·π) = R·(−1) = −R (exp(Iπ) = −1: Complex.exp_pi_mul_I? `Complex.exp_pi_mul_I : exp (π * I) = -1` ✓). Continuity of exp/mul: Tendsto.comp ✓.\n  - Then F(sₙ) ≤ C ∀ n and F(sₙ) → F(s₀) (continuity at s₀) ⟹ F(s₀) ≤ C (le_of_tendsto_of_tendsto' hmm: `le_of_tendsto (hg : Tendsto g atTop (𝓝 b)) (h : ∀ᶠ n, g n ≤ b') : b ≤ b'` — the exact name: le_of_tendsto_of_tendsto' (hf : Tendsto f l (𝓝 a)) (hg : Tendsto g l (𝓝 b)) (h : f ≤ᶠ[l] g) : a ≤ b. With f := F∘sₙ, g := const C.)\n\n  OK ~40 lines with the sequence. Hmm — but WAIT: also the FE identity at s₀ = −R (logDeriv_zeta_fe) needs differentiableAt Gammaℝ at −R and 1+R — what discharges `DifferentiableAt ℂ Complex.Gammaℝ s`? Let me grep how 5d's applications of logDeriv_zeta_fe discharged hd.\n\nLet me look at how logDeriv_zeta_fe's hypotheses were discharged in 5d (LiReflect usage) and find differentiableAt/analyticAt Gammaℝ lemmas.","signature":""},{"type":"text","text":"Let me check how `DifferentiableAt Gammaℝ` was discharged in 5d and find the Gammaℝ differentiability API."},{"type":"tool_use","id":"Grep_149","name":"Grep","input":{"pattern":"(differentiableAt_Gammaℝ|analyticAt_Gammaℝ|DifferentiableAt.*Gammaℝ|Gammaℝ.*[Dd]ifferentiable)","path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1","output_mode":"content","-n":true,"head_limit":25},"extra_content":null}]