Math 411: Honours Complex Variables - University of Alberta

holds over C. We obtain:
�
∞�
z
exp(z)exp(w) =
j=0
j
��
∞�
j!
k=0
∞� n� z
=
n=0 k=0
n−k
(n−k)!
∞� n�
�
1 n
=
n! k
n=0 k=0
n� (z +w)
=
n
n!
n=0
= exp(z +w).
wk �
k!
w k
k!
�
z n−k w k
15
by the Cauchy product formula, letting n = j +k,
We call exp: C → C the exponential function. The above property suggests
using the shorthand e z for exp(z). An interactive three-dimensional graph of
exp(z) is shown in Figure 3.1.
3. The sine and cosine functions on C are defined as
and
sin(z) :=
cos(z) :=
∞�
n=0
(−1) n z 2n+1
(2n+1)!
∞�
(−1)
n=0
n z2n
(2n)!
for z ∈ C. As for exp(z), we see that both sin(z) and cos(z) converge absolutely
for all z ∈ C. Moreover, we have for z ∈ C:
e iz ∞� (iz)
=
n=0
n
n!
∞� (iz)
=
n=0
2n
(2n)! +
∞� (iz)
n=0
2n+1
(2n+1)!
∞�
n z2n
= (−1)
(2n)! +i
∞�
(−1) n z2n+1 (2n+1)!
n=0
= cos(z)+isin(z).
Interactive three-dimensional graphs of the complex cosine and sine functions
are shown in Figures 3.2, and 3.3.
n=0