An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 71-72. Power with a real exponent. 117
multiples of 2z belong to the same number 9,p. There are therefore
only n such numbers; they belong to the values k = 0, 1, 2... n - 1.
For, in the first place, among the different forms for 4', all that have
a negative sign for k can evidently be brought to be forms with a
positive sign by addition of integer multiples of 2 r; then again,
for 7- = (n - 1) + k', we have 2-r - =2r --
Every complex number has therefore n distinct nlh roots; these
are included in the form:
n. -. - -;- - + 2 7p f /7t,.. 2p + 71 J
/r (cos p+ i sip =rn(cos C 2lX + iSin ) i. e..(p p+2 kz
r (/r ) (2k, or, Vre=i r. n; (k 0, 1, 2... n- 1).
n
Every positive or negative real number also has n distinct roots;
of these however in case of a positive number for which 'p = 0,
when n is odd, only one is real: k = 0; when n is even, two are real:
7 = 0, k = 2n; in case of a negative number for which p = r,
when n is odd, one is real: 7- -I (n - 1); when n is even, they are
all imaginary or complex.
n,- 2 c.. 27kt n (2 k + 1) (2 71 + 1)
- 1 = Cos -- -- i in i- i == cos + i- sill
n nn n n
n(4k+-1)/ (4k +).. (47+-1)7 /. (43)7 7r
/4-i C== $cos - +- - S l, / cos -i sin,
2+COS ) lll 3i ' 2 it O - 2 n
(k7=0, 1, 2...n — 1).
Accordingly by the first definition for an arbitrary fractional
exponent we have:
(r (cos cp + i sin 9p))n = Y/r"t (cos mn p +- i sin in )
-f n (t C + 2 k 4- + i sill,? mp 4+ 2 7G 7r)
== r (cos p ---2- +r s —[isin
3n n
(k=-0, 1, 2...- 1).
By the second:
(1. /eos rP + i *in *p~ ^^7 I~ir( -/ p4- 2/I;'7r, q+ 4-2 k'n! I?
(r (cos +-i sin ()) -= -ricos + i Sill — ))
= rn (cos + 2 k- + i sin rP + 2 kn 7r
(k'- 0, 1,2...n-1).
But the last expressions on the right are identical in both equations;
for, since m and n are relatively prime, each number 0, in,
2m,... (n - 1) m, divided by n, leaves a different remainder; hence
expresses, it may be in altered order, values differing from
those in the upper line only by multiples of 2zr. The value belonging
to k = 0 is styled the simplest value among the roots.