lajos/Shape of the Taylor stability region

The integral representation:

\[ |{\mathcal{T}}_p(\varrho e^{i\varphi})|^2-1=\left(\frac{\varrho ^{ p+1}}{p!}\right)^2 \left(I_c^2(p,\varrho ,\varphi )+I_s^2(p,\varrho ,\varphi )\right)-\frac{2 \varrho ^{p+1}}{p!} J(p,\varrho ,\varphi )+e^{2 \varrho \cos \varphi}-1, \]

where

\[ I_c(p,\varrho ,\varphi ):=\int_0^1 r^p e^{(1-r)\varrho \cos\varphi} \cos ((p+1) \varphi +(1-r)\varrho \sin \varphi) dr, \]

\[ I_s(p,\varrho ,\varphi ):=\int_0^1 r^p e^{(1-r)\varrho \cos\varphi} \sin ((p+1) \varphi +(1-r)\varrho \sin \varphi) dr, \]

\[ J(p,\varrho ,\varphi ):=\int_0^1 r^p e^{(2-r)\varrho \cos\varphi} \cos ((p+1) \varphi -r\varrho \sin \varphi) dr. \]

Their behavior of the integrals and why estimating them is difficult.

The region becomes disconnected, first for \(p=6\) (3 components). If \(p=11\), then it has 5 components.

The estimate in the right half-plane would be more complicated: the structure is more irregular, e.g., “bubbles”.

The slice along the imaginary axis: sawtooth structure, wavy boundary, disconnected components.

In the left half-plane: numerically a semi-disk shape, with bounding radius \(\sim \frac{p}{e}+\frac{\log (p)}{2 e}+\frac{1+\log (2 \pi )}{2 e}\). This is obtained from the series expansion of the incomplete gamma function (around \(+\infty\)) along the horzontal slice. But the bound \(\approx \frac{p}{e}+\frac{\log (p)}{2 e}+\frac{1+\log (2 \pi )}{2 e}+\frac{6}{10}\) is also confirmed numerically along the vertical (\(\varphi=\pi/2\)), the horizontal (\(\varphi=\pi\)) as well as the \(\varphi=3\pi/4\)-slice.