The pantograph equation is a linear functional differential equation of the form
 |
where 
is the derivative with respect to 
, 
and 
are constants and 
(Kato and McLeod 1971). For 
and 
, it is a delay
differential equation whose delay is proportional to 
. The name comes from the pantograph, the current collector
that contacts the overhead wire of an electric locomotive. A proportionally delayed
argument arises in modeling the motion of the wire and collector (Ockendon and Tayler
1971, Buhmann and Iserles 1993).
For 
, substituting a power
series about 
gives the solution with initial value 
as
 |
where the empty product is 1. The product is bounded in absolute value by 
for 
, so the power
series converges for every 
.
." Bull. Amer. Math. Soc. 77,
891-937, 1971.