The Power of a Complex Number
If
is a complex number and
is a natural number, then we can take the power of complex number
is a complex number and
is a natural number, then we can take the power of complex number
is a natural number, then we can take the power of complex number
Properties of Complex Power
-
The magnitude is
-
The direction is
. Note the argument of the results must be reduced mod
. Note the argument of the results must be reduced mod
-
De Moivre formula:
for n as positive or negative integer. Some authors will make brief function
, such that De Moivre formula becomes
.
for n as positive or negative integer. Some authors will make brief function
, such that De Moivre formula becomes
.
, such that De Moivre formula becomes
.
-
Parallelogram equality
More general case, if we have two complex numbers
and
, then we can take the power of complex number and
, then we can take the power of complex number , then we can take the power of complex number
See Also : Complex Exponent , Complex Root , Complex Logarithm
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