1Complex numbers

IA Vectors and Matrices

1.2 Complex exponential function

Exponentiation was originally defined for integer powers as repeated multiplica-

tion. This is then extended to rational powers using roots. We can also extend

this to any real number since real numbers can be approximated arbitrarily

accurately by rational numbers. However, what does it mean to take an exponent

of a complex number?

To do so, we use the Taylor series definition of the exponential function:

Definition (Exponential function). The exponential function is defined as

exp(z) = e

z

= 1 + z +

z

2

2!

+

z

3

3!

+ ··· =

∞

X

n=0

z

n

n!

.

This automatically allows taking exponents of arbitrary complex numbers.

Having defined exponentiation this way, we want to check that it satisfies the

usual properties, such as

exp

(

z

+

w

) =

exp

(

z

)

exp

(

w

). To prove this, we will

first need a helpful lemma.

Lemma.

∞

X

n=0

∞

X

m=0

a

mn

=

∞

X

r=0

r

X

m=0

a

r−m,m

Proof.

∞

X

n=0

∞

X

m=0

a

mn

= a

00

+ a

01

+ a

02

+ ···

+ a

10

+ a

11

+ a

12

+ ···

+ a

20

+ a

21

+ a

22

+ ···

= (a

00

) + (a

10

+ a

01

) + (a

20

+ a

11

+ a

02

) + ···

=

∞

X

r=0

r

X

m=0

a

r−m,m

This is not exactly a rigorous proof, since we should not hand-wave about

infinite sums so casually. But in fact, we did not even show that the definition of

exp

(

z

) is well defined for all numbers

z

, since the sum might diverge. All these

will be done in that IA Analysis I course.

Theorem. exp(z

1

) exp(z

2

) = exp(z

1

+ z

2

)

Proof.

exp(z

1

) exp(z

2

) =

∞

X

n=0

∞

X

m=0

z

m

1

m!

z

n

2

n!

=

∞

X

r=0

r

X

m=0

z

r−m

1

(r −m)!

z

m

2

m!

=

∞

X

r=0

1

r!

r

X

m=0

r!

(r −m)!m!

z

r−m

1

z

m

2

=

∞

X

r=0

(z

1

+ z

2

)

r

r!

Again, to define the sine and cosine functions, instead of referring to “angles”

(since it doesn’t make much sense to refer to complex “angles”), we again use a

series definition.

Definition (Sine and cosine functions). Define, for all z ∈ C,

sin z =

∞

X

n=0

(−1)

n

(2n + 1)!

z

2n+1

= z −

1

3!

z

3

+

1

5!

z

5

+ ···

cos z =

∞

X

n=0

(−1)

n

(2n)!

z

2n

= 1 −

1

2!

z

2

+

1

4!

z

4

+ ···

One very important result is the relationship between exp, sin and cos.

Theorem. e

iz

= cos z + i sin z.

Alternatively, since sin(−z) = −sin z and cos(−z) = cos z, we have

cos z =

e

iz

+ e

−iz

2

,

sin z =

e

iz

− e

−iz

2i

.

Proof.

e

iz

=

∞

X

n=0

i

n

n!

z

n

=

∞

X

n=0

i

2n

(2n)!

z

2n

+

∞

X

n=0

i

2n+1

(2n + 1)!

z

2n+1

=

∞

X

n=0

(−1)

n

(2n)!

z

2n

+ i

∞

X

n=0

(−1)

n

(2n + 1)!

z

2n+1

= cos z + i sin z

Thus we can write z = r(cos θ + i sin θ) = re

iθ

.