A Gaussian Method for the Square Root of Accretive Operators

A Approximation of β ⋆

Let Ψ s 0 ∈ E be the ellipse passing through the point 2 τ ⁢ i - 1 (cf. (4.11)). The value β ⋆ , such that the function Φ ( 2 ) ⁢ ( 1 + ρ ⁢ e - i ⁢ β ⁢ π ) possesses a local maximum for β > β ⋆ , is the one for which Ψ s 0 is also tangent to the curve χ ( 2 ) ⁢ ( Γ β ⋆ - ) at 2 τ ⁢ i - 1 (Figure 3). In order to compute β ⋆ , we consider the tangents at 2 τ ⁢ i - 1 to the ellipse and to the curve, and impose them to have the same slope. Before starting, we need to derive the semi-width 𝛾 and the semi-height 𝛿 of the ellipse. By geometrical evidence, we have that

and γ 2 = δ 2 + 1 . At this point, we remind that the slope of the tangent at 2 τ ⁢ i - 1 to the ellipse is

where we have used (A.1) and

which comes from (4.11). Now, it is not difficult to show that the angle between the tangent to the curve χ ( 2 ) ⁢ ( Γ β - ) at 2 τ ⁢ i - 1 and the line ℑ ⁡ ( z ) = 2 τ is given by 1 2 ⁢ ( π - 2 ⁢ β ⁢ π ) . Hence, in order to find an approximation of β ⋆ , by (A.2), we impose the condition

that leads to β ⋆ ∼ 1 4 as τ → + ∞ .