Powers

Power1 regular

\[\begin{split}x^+ \approx \mathrm{pow}_{1,h}(x) := \begin{cases} \dfrac{\epsilon}{1-z} & x<0 \\[1em] \displaystyle \frac{a_0+z(a_1+z(a_2+z(a_3+a_4 z)))}{(1+z)^3} & \textrm{otherwise} \end{cases} \qquad z = \dfrac{x}{h}\end{split}\]

\[a_0 = \epsilon,\quad a_1 = 4\epsilon,\quad a_2 = 7\epsilon,\quad a_3 = 8\epsilon,\quad a_4 = 1-20\epsilon\]

continuous up to the 3rd derivatives. Condition: \(\epsilon<1/20\).

Power2 regular

\[\begin{split}(x^+)^2\approx \mathrm{pow}_{2,h}(x):= \begin{cases} \dfrac{\epsilon}{1-z} & x<0 \\[1em] \dfrac{\epsilon(1+z(3+4z(1+z))) + (1-12\epsilon)z^4}{(1+z)^2} & \textrm{otherwise} \end{cases}\end{split}\]

continuous up to the 3rd derivatives. Condition: \(\epsilon<1/12\).

Power4 regular

\[\begin{split}(x^+)^4 \approx \mathrm{pow}_{4,h}(x):= \begin{cases} \epsilon\exp(z) & x<0 \\ \epsilon+z(a+z(b+z(c+zd))) & \textrm{otherwise} \end{cases}\end{split}\]

\[a = \epsilon,\qquad b = (1/2)\epsilon,\qquad c = (1/6)\epsilon,\qquad d = 1-(5/3)\epsilon\]

\(\epsilon \leq 3/10\) for asymptotic \(x^4/2\)