Editing Imaginary unit (section)

=== Integer powers ===
The powers of {{mvar|i}} repeat in a cycle expressible with the following pattern, where {{mvar|n}} is any integer:

<math display=block>
i^{4n} = 1, \quad
i^{4n+1} = i, \quad
i^{4n+2} = -1, \quad
i^{4n+3} = -i.</math>

Thus, under multiplication, {{mvar|i}} is a generator of a [[cyclic group]] of order 4, a discrete subgroup of the continuous [[circle group]] of the unit complex numbers under multiplication.

Written as a special case of [[Euler's formula]] for an integer {{mvar|n}},

<math display=block>
i^n = {\exp}\bigl(\tfrac12\pi i\bigr)^n
    = {\exp}\bigl(\tfrac12 n \pi i\bigr)
    = {\cos}\bigl(\tfrac12 n\pi \bigr) + {i \sin}\bigl(\tfrac12 n\pi \bigr).
</math>

With a careful choice of [[branch cut]]s and [[principal value]]s, this last equation can also apply to arbitrary complex values of {{mvar|n}}, including cases like {{math|1=''n'' = ''i''}}.{{cn|date=March 2024}}