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.

118 The complex number and the operations of arithmetic. Bk. II. ch. I.

c) The power with an irrational exponent is understood to be the
limiting value of the series of numbers obtained by forming the
powers whose exponents are the rational numbers of the series that
defines the irrational number. If,., ',,* be the series of
defining numbers, (r (cos p -+ i sin 90))"t represents all those numbers,m 7n1 71"
whose modulus is the limiting value of the series r n, r', r"" ~ ~ ~*
and whose amplitudes are the limiting values of the series:
2(p + 2 k7) m' (p+2k) "(cp + 2 7)
l n,. m "
(=-=., 2,... 1, * *,... '...).
To each integer value of k belongs a different limiting value of the
amplitude, so that therefore for an irrational exponent there are
infinitely many numbers of the form:
(r (cos - sin (p)), =- r"{ cos pL (~ + 2k7 ) + i sin p,(q) + 2 k) },
i. e., ('P) "' I it '((pd-2 kn) r2 0 (-e') i - r^.". c,^:e('p+S2 9)
all with the same modulus ry.
d) A power with a negative exponent must as before signify
the reciprocal value of the power with a positive exponent. We have
(r (cos Sp + i sin lq))- t =- -- 1 =_-.._.
(9'(coscp+isinc))." v" { cos t(cp+2k/)-+i-sin y(q+2;k7) }
= y " c {cos (q - 2 7ki) - i sin t( (p + 2 ki )}
for every t; a result that can also be identified with the form of
De Moivre's theorem, since we can write this equation as follows:
(r (cos 9- i sin ))-ft= r-1- { cos ( — u(g -+ 27r)) -+isin ( — ((9 + 2k7-)) }.
73. The power with a complex exponent. (Exponential.)
The symbol e:'p, denoting a power with a real base and a purely
imaginary exponent, was defined in ~ 67 as the sum of the infinite
series
(i ___, (i CO (i CO
1 + - 1 + 2 >- + 13
whose real and imaginary constituents considered apart, form convergent infinite series with the values cos 9p and i sin 9p. In connexion
with this we define the power with the base e and complex exponent
x + iy as the value of the product of the exponential expression ex
by the exponential expression eiY; i. e. in a formula
ex+iy_ = ex. + iy - ex (cos y + i sin y)
( 1 -X + X-' + Xj + * *) (1+ -;- +-'- -Is -- * )
We shall see in next Chapter how this product of two infinite
series may be combined in a single infinite series. From the definition
follows the fundamental property of the exponential (~ 70):