The zeros of the complementary error function (Q1014758)

The well-known complementary error function erfc (\(z\)) is defined by \[ \text{erfc (z)}=\frac{2}{\sqrt{\pi}}\int_z^\infty e^{-s^2}ds. \] It is shown that erfc (\(z\)) has no zeros in the sector \(3\pi/4\leq\arg\,z\leq 5\pi/4\). The authors establish this result by consideration of the two sectors \(3\pi/4\leq\arg\,z\leq\pi\) and \(\pi<\arg\,z\leq 5\pi/4\). In the first sector, they write \(z=-X+iY\), with \(X\geq 0\) and \(0\leq Y\leq X\), and decompose the integral into integrals taken along the straight line paths \((z,-X)\), \((-X,0)\) and \((0,\infty)\). They show that the real part of the decomposed integral is positive. Similar considerations with \(z=-X-iY\) are applied to the second sector.