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.