16 1. The Complex Numbers
For a given pair J, K, we let N = J + K. Then uN is the sum of all terms ajbk
for which j + k N, while sJ tK is the sum of those terms ajbk for which j J
and k K. Thus,
|uN sJ tK | {|ajbk| : j + k N, and either j J or k K}


j=J+1
|aj|

k=0
|bk| +

j=0
|aj|

k=K+1
|bk|.
(1.3.4)
The sum
∑∞
j=J+1
|aj| converges to zero as J because the series
∑∞
j=0
aj
converges absolutely, while
∑∞
k=K+1
|bk| converges to 0 as K because
∑∞
k=0
bk
converges absolutely.
To complete the proof, we choose for each non-negative integer N a J and K
with J + K = N, and we do this in such a way that as N the corresponding
J and K both also tend to ∞. For example, we could choose J = K = N/2 if N is
even, and J = K + 1 = N/2 + 1/2 if N is odd. Then, from (1.3.4) it is clear that
lim |uN sJ sK | = 0, which implies lim uN = (lim sJ )(lim sK).
The Exponential of an Imaginary Number. If z = x + iy, then the law of
exponents (Theorem 1.3.2) implies that
ez
=
ex eiy
.
We understand the behavior of
ex
for real x from calculus. It is 1 at 0, is everywhere
positive, increasing and concave upward; it rapidly approaches +∞ as x and
rapidly approaches 0 as x −∞. What about
eiy
the exponential of a purely
imaginary number? Here there are some surprises.
By the law of exponents,
eiy e−iy
=
e0
= 1. Thus, 1/
eiy
=
e−iy.
But also, for
any z C, ez =
ez
(Exercise 1.3.1) and, in particular, eiy =
e−iy
= 1/
eiy.
This
implies that |
eiy |2
= 1 and, consequently, |
eiy
| = 1. Thus,
eiy
is always a point on
the circle of radius 1 centered at 0 (we call this the unit circle).
A more explicit description of the number
eiy
comes from examination of its
power series definition. If we group together the even numbered terms (terms with
n of the form n = 2k) and the odd numbered terms (terms with n of the form
n = 2k + 1) in this power series, we derive the identity
eiy
=

n=0
(iy)n
n!
=

k=0
(−1)k
y2k
(2k)!
+ i

k=0
(−1)k
y2k+1
(2k + 1)!
= cos y + i sin y.
Thus, we have proved the following theorem, which is known as Euler’s Identity.
Theorem 1.3.4. The identity
eiy
= cos y + i sin y holds for all y R.
This shows that, not only is eiy a point on the unit circle, it is the point which
is reached by rotating through an angle y (measured in radians) from the initial
point (1, 0).
Example 1.3.5. Express the complex numbers e2πi, eπi and eπi/2 in standard
form.
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