  The location of roots of the characteristic equation of a linear delay
differential equation (DDE) determines the stability of the linear DDE.
However, by its transcendency, there is no general criterion on the contained
parameters for the stability. Here we mainly concentrate on the study of a
simple transcendental equation $(*)$ $z + a - w \mathrm{e}^{-\tau z} = 0$ with
coefficients of real $a$ and complex $w$ and a delay parameter $\tau > 0$ to
tackle this transcendency brought by delay. The consideration is twofold: (i)
to give the stability region in the parameter space for Eq.~$(*)$ by using the
critical delay and (ii) to compare this with a graphical method (so-called the
method of D-partitions) by combining with the delay sequence obtained by
conditions for purely imaginary roots. By (i), we obtain another proof of
Hayes' and Sakata's results, which reveals the nature of imaginary $w$ case in
Eq.~$(*)$. By (ii), we propose a method combining the analytic one and
geometric one. This combination is important because it will be helpful in
studying characteristic equations having higher-dimensional parameters.
