The least period of a periodic function, periodic sequence, or periodic point is the smallest positive period allowed in the corresponding setting, when such a minimum exists.
For a periodic function 
of a real variable, the least
period is the smallest real number 
satisfying
 |
(1)
|
for all real numbers 
.
It is also called the fundamental period (Adkins and Davidson 2016, p. 725).
For example, the sine and cosine
functions have least period 
. A constant
function has every positive real number as a period, so it has no least period.
For a periodic sequence 
,
the least period is the smallest positive integer 
satisfying
 |
(2)
|
for every index 
. The term fundamental period is
also used in this setting (Ellis 2007, p. 38). For example, the sequence
1, 2, 1, 2, 1, 2, ... has least period 2. A sequence
whose terms are all equal has least period 1.
For a periodic point 
of a function 
,
the least period is the smallest positive integer 
satisfying
 |
(3)
|
where 
denotes 
-fold
iteration. In this setting, the least period is also
called the exact period. For the function 
, every nonzero point
has least period 2, whereas 
has least period 1.
