Zeros of the partial sums of \(\cos(z)\) and \(\sin(z)\). III (Q972295)

Relations between the partial sums \(\exp_n (n z)\), \(\cos_n (n z)\), and \(\sin_n (n z)\) are studied, where \[ \begin{aligned} \exp_n (z) &= \sum_{k=0}^{n} \frac{z^k}{k !}\\ \cos_n (z) &= \sum_{k=0}^{n/2} \frac{(-1)^k z^{2k}}{(2k) !},\quad n\in 2{\mathbb N},\\ \sin_n (z) &= \sum_{k=0}^{(n-1)/2} \frac{(-1)^k z^{2k + 1}}{(2k+1) !},\quad n\in 2{\mathbb N} - 1.\end{aligned} \] On the base of these relations, the (asymptotical) zero distribution of these sums is described. The main result states that, for all \(n\geq 6\), the non-real zeros of \(\cos_n (n z)\) (for even \(n\)) and \(\sin_n (n z)\) (for odd \(n\)) in the open upper half plane lie in the set \[ \left\{z\in \Delta^{+}: \left|z - \frac{i}{3}\right| < \frac{2}{3}\right\}\setminus K, \] or in its reflection with respect to the real line, where \(K\) is a closed set in the unit disc inside the Szegö curve. For part II, cf. Numer. Math. 90, No. 2, 371--400 (2001; Zbl 0994.30008).