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general power∗
pahio†
2013-03-21 18:06:58
The general power z µ , where z (6= 0) and µ are arbitrary complex numbers,
is defined via the complex exponential function and complex logarithm (denoted
here by “log”) of the base by setting
z µ := eµ log z = eµ(ln |z|+i arg z) .
The number z is the base of the power z µ and µ is its exponent.
Splitting the exponent µ = α+iβ in its real and imaginary parts one obtains
z µ = eα ln |z|−β arg z · ei(β ln |z|+α arg z) ,
and thus
|z µ | = eα ln |z|−β arg z ,
arg z µ = β ln |z|+α arg z.
This shows that both the modulus and the argument of the general power are in
general multivalued. The modulus is unique only if β = 0, i.e. if the exponent
µ = α is real; in this case we have
|z µ | = |z|µ ,
arg z µ = µ · arg z.
Let β 6= 0. If one lets the point z go round the origin anticlockwise, arg z
gets an addition 2π and hence the power z µ has been multiplied by a factor
having the modulus e−2πβ 6= 1, and we may say that z µ has come to a new
branch.
Examples
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1. z m , where m is a positive integer, coincides with the mth root of z.
2. 32 = e2 log 3 = e2(ln 3+2nπi) = 9(e2πi )2n = 9 ∀n ∈ Z.
π
π
3. ii = ei log i = ei(ln 1+ 2 i−2nπi) = e2nπ− 2 (with n = 0, ±1, ±2, . . .); all
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these values are positive real numbers, the simplest of them is √ π ≈
e
0.20788.
∗ hGeneralPoweri
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h1i hDefinitioni h30D30i
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4. (−1)i = e(2n+1)π (with n = 0, ±1, ±2, . . .) also are situated on the
positive real axis.
√
√
√
√
5. (−1) 2 = e 2 log (−1) = e 2i(π+2nπ) = ei(2n+1)π 2 (with n = 0, ±1, ±2, . . .);
all these are imaginary numbers (meaning here that their imaginary parts
are distinct from 0), situated on the circumference of the unit circle such
that all points of the circumference
are accumulation points of the se√
quence of the powers (−1) 2 (see this entry).
6. 21−i = 2e2nπ (cos ln 2 + i sin ln 2) (with n = 0, ±1, ±2, . . .), are situated
on the half line beginning from the origin with the argument ln 2 ≈ 0.69315
radians.
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