Calculus: Analytic Geometry and Calculus, with Vectors

628 Series

20 Without pretending to give a reasonable introduction to complex numbers,
we take a hasty look at some remarkable formulas that involve these numbers.
Let i, the so-called imaginary unit, be a number for which i2 = -1. It is possi-
ble to set forth rules for operating with complex numbers of the form x + iy,
where x and y are real. For example, if z = x + iy, then
z` = (x + iy)2 = x2 + 2ixy + i2y2 = (x2 - y22) + 2ixy.
A complex number x + iy for which x and y are real and y = 0 is identified
with the real number x, so the set of real numbers is a subset of the set of complex
numbers. The series in the right members of the formulas

21 T! T! 3-1 +

2 4 6

(2) cosz=l -2i+zl-i-6i+

zs zb z7

(3) sinz=z-3i+si-Zj+

are power series in z. With the agreement that a sequence xi + iyi, x2 + iy2i

of complex numbers converges to L, + iL2 if lim x = L, and lim y = L2,

it is possible to construct a theory of series of complex numbers. It can then be

shown that, for each z, the series in (1), (2), and (3) converge to complex num-

bers which are real only in special cases and which can be denoted by ez, cos z,

and sin z even when they are not real. Thus ez, cos z, and sin z are defined for

each complex z and the formulas (1), (2), (3) are valid for each z. If w is a com-

plex number, which may be real but is not necessarily so, we can put z = iw

in (1) to obtain

(4)

etw=1+0+(1 +0+ lw4 +0+...

+0+iw+0+3 ?+ 0+(25? + ...

Since i2 = -1, is = -i, i' = 1, ib = i, ib = -1, , this formula can be

put in the form

( 4 6 3 b 7

Since (2) and (3) show that the series in parentheses converge to cos w and sin w,

we obtain the first of the four Euler formulas

(6) ei'° = cos w + i sin w

(7) e1'° =cosw-isinw

(8)

ei,° + ei,° eiD - e-iw

cos w = 2 sin w = 2i

Since (2) and (3) show that cos (-w) = cos w and sin (-w) sin w, we

can replace w by -w in (6) to obtain (7). Adding and subtracting (6) and (7)

enables us to solve for cos w and sin w to obtain (8). Thus we have proved the

four Euler formulas. These formulas are sometimes said to be the most remark-

able formulas in mathematics. They have very many important applications,