Search Results for “buchstab function”

Source: Buchstab function

The Buchstab function (or Buchstab's function) is the unique continuous function

ω
:

R

≥
1

→

R

>
0

{\displaystyle \omega :\mathbb {R} _{\geq 1}\rightarrow \mathbb {R} _{>0}}

defined by the delay differential equation

ω
(
u
)
=

1
u

,

1
≤
u
≤
2
,

{\displaystyle \omega (u)={\frac {1}{u}},\qquad \qquad \qquad 1\leq u\leq 2,}

d

d
u

(
u
ω
(
u
)
)
=
ω
(
u
−
1
)
,

u
≥
2.

{\displaystyle {\frac {d}{du}}(u\omega (u))=\omega (u-1),\qquad u\geq 2.}

In the second equation, the derivative at u = 2 should be taken as u approaches 2 from the right. It is named after Alexander Buchstab, who wrote about it in 1937.