The zeros of the complementary error function. (English) Zbl 1167.33002

Elbert, Árpád; Laforgia, Andrea

Numer. Algorithms 49, No. 1-4, 153-157 (2008).

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.

Reviewer: R. B. Paris (Dundee)

MSC:

33B20

Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals)

Keywords:

zeros; complementary error function

References:

[1]	Tricomi, F.G.: Funzioni ipergeometriche confluenti. Consiglio Nazionale delle Ricerche, Monografie Matematiche, 1, Edizioni Cremonese, Roma (1954) · Zbl 0068.28005