On the zeros of the partial sums of \(\cos(z)\) and \(\sin(z)\) (Q2564482)

This paper parallels for sine and cosine work done for the exponential function by \textit{A. J. Carpenter}, \textit{R. S. Varga} and \textit{J. Waldvolgel} [Rocky Mt. J. Math. 21, 99-120 (1991; Zbl 0734.30008)]. Let \(A:=\{z\in\mathbb{C}:|-ize|=1, |z|\leq 1,\text{ Im}(z)\geq 0\}\) and let \(C_\infty:=\{z:\overline z\in A\}\cup(-1/e,1/e)\). Let \(c_n(z)\) denote the \(n\)th partial sum for the Maclaurin series for \(\cos z\). G. Szegö, in a 1924 paper, showed that the set of points of accumulation of the zeros of \(c_n(nz)\) is equal to \(C_\infty\). Theorem 1 considers the set of zeros of \(c_n(nz)\) which do not lie within \(\delta\) or \(i\) or \(-i\), or within \(\varepsilon\) of the real line. It is shown that the distance from this subject of the zeros to \(C_\infty\) is \(O((\log n)/n\) as \(n\) increases to infinity. Next, a new set, \(A_n\) is defined. It depends on \(n\), and is a modification of \(C_\infty\). Theorem 2 shows that the distance from \(A_n\) to the set of zeros considered in Theorem 1 is \(O(1/n^2)\) as \(n\) approaches infinity. The paper includes very helpful figures, displaying the zeros of \(c_{40}(40z)\) in relation to \(C_\infty\) and to \(A_{40}\), and those of \(c_{60}(60z)\) in relation to \(C_\infty\) and to \(A_{60}\). The author makes use of methods of Varga and E. B. Saff as he gives detailed proofs of the theorems. He points out that the same results given in those theorems for \(c_n(nz)\) also hold for the zeros of the partial sums of \(\sin(nz)\), and provides figures showing the situation for \(n=41\) and \(n=61\).