English
For complex numbers z and w with the stated nondegeneracy condition, the norm of z^w equals the real part of w applied to the log-magnitude of z divided by exp of Arg z times Im w.
Русский
Для комплексных z и w при условии некорректности нормa z^w равна отношению экспоненты и лог-модуля z.
LaTeX
$$$$ \\| z^w \\| = \\| z \\|^{\\Re w} \\/ \\exp( \\operatorname{Arg}(z) \\cdot \\Im w ). $$$$
Lean4
/-- For a functor `F : C ⥤ D` and an object `d : D`, we obtain a contravariant functor from the
category of costructured arrows `F.op.obj c ⟶ op d` to the category of structured arrows
`d ⟶ F.obj c`.
-/
@[simps]
def toStructuredArrow' (F : C ⥤ D) (d : D) : (CostructuredArrow F.op (op d))ᵒᵖ ⥤ StructuredArrow d F
where
obj X := @StructuredArrow.mk _ _ _ _ _ (unop X.unop.left) F X.unop.hom.unop
map
f :=
StructuredArrow.homMk f.unop.left.unop
(by
dsimp
rw [← Quiver.Hom.unop_op (F.map f.unop.left.unop), ← unop_comp, ← F.op_map, f.unop.w, Functor.const_obj_map]
simp)
