Plain-language statement
D(exp(cz))/exp(cz) = c/(2πi) for any coefficient c.
Exact Lean statement
theorem D_cexp_div (c : ℂ) (z : ℍ) :
D (fun w ↦ cexp (c * w)) z / cexp (c * z) = c / (2 * π * I)Formal artifact
Lean source
theorem D_cexp_div (c : ℂ) (z : ℍ) : D (fun w ↦ cexp (c * w)) z / cexp (c * z) = c / (2 * π * I) := by simp only [D] have h_deriv : deriv ((fun w : ℍ ↦ cexp (c * w)) ∘ ⇑ofComplex) (z : ℂ) = c * cexp (c * z) := ((eventuallyEq_coe_comp_ofComplex z.2).fun_comp (fun w ↦ cexp (c * w))).deriv_eq.trans ((Complex.hasDerivAt_exp (c * (z : ℂ))).scomp (z : ℂ) (by simpa using (hasDerivAt_id (z : ℂ)).const_mul c)).deriv rw [h_deriv]; field_simp [Complex.exp_ne_zero]- Project
- Sphere Packing in Dimension 8
- License
- Apache-2.0
- Commit
- acfc6204e65a
- Source
- SpherePacking/ModularForms/FG.lean:754-762
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Related declarations
Anti Der Pos
antiDerPos
Plain-language statement
If is a modular form where is positive for sufficiently large (i.e. constant term is positive) and the derivative is positive, then is also positive.
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.
Anti Serre Der Pos
antiSerreDerPos
Plain-language statement
Let be a holomorphic function where is real for all . Assume that Serre derivative is positive on the imaginary axis. If is positive for sufficiently large , then is positive for all .
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.
Closed Ball center subset upper Half Plane
closedBall_center_subset_upperHalfPlane
Plain-language statement
Closed ball centered at z with radius z.im/2 is contained in the upper half plane.
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.