File:Acosq1plotT.png

Graphic of real and imaginary parts of acosc1$\left( \mathrm e ^{ \mathrm i \pi/4 } x \right)$ versus $x$.

Define $\mathrm{acosq}_1(z)=\mathrm{acosc}_1\left( \mathrm e ^{ \mathrm i \pi/4 } z \right)$

Then $\Re(\mathrm{acosq}_1(x))$ and $\Im(\mathrm{acosq}_1(x))$ are plotted versus $x$ with thick blue and thick red lines.

The thin lines show the real and imaginary parts the cubic polynomial corresponding to the truncation of the Taylor expansion at zero;

\mathrm{expan}_1(z)=\frac{3 \pi}{2}

\frac{3 \pi z}{2}+\frac{3 \pi

  z^2}{2}+\frac{3}{16} \pi  \left(8+3 \pi
  ^2\right) z^3+O\left(z^4\right)$

\mathrm{expanq}_1(z)=:\mathrm{expan}_1(z \mathrm e^{\rm i \pi /4})$