File:Arctra3tesT.jpg

Agreement of the primary approximation of function ArcTra with

$\mathrm{ArcTra}(z)\approx $ $\mathrm{app3}(z)=$ $ \displaystyle \ln(z) \left(1+ \frac{1}{z} \sum_{m=0}^M \frac{P_m(\ln(z))}{z^m}\right)$ $P_0(L)=-1$ $P_1(L)=1-L/2$ $P_2(L)=-1+3 L/2 -L^2/3$ $P_3(L)=1-3L+11L^2/6-L^3/4$ $P_4(L)=-1+5L-35L^2/6+25L^3/12-L^4/5$

For $M\!=\!4$, the agreement $A=A_3(x+\mathrm i y) $ is shown;

$\displaystyle \mathcal A_3(z)= $ $\displaystyle -\lg \left( \frac{ |\mathrm{tra}(\mathrm{app3}(z))-z| }       { |\mathrm{tra}(\mathrm{app3}(z))| + |z|} \right)$

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