6Complex power series
IA Analysis I
6.3 Hyperbolic trigonometric functions
Definition (Hyperbolic sine and cosine). We define
cosh z =
e
z
+ e
−z
2
= 1 +
z
2
2!
+
z
4
4!
+
z
6
6!
+ ···
sinh z =
e
z
− e
−z
2
= z +
z
3
3!
+
z
5
5!
+
z
7
7!
+ ···
Either from the definition or from differentiating the power series, we get
that
Proposition.
d
dz
cosh z = sinh z
d
dz
sinh z = cosh z
Also, by definition, we have
Proposition.
cosh iz = cos z
sinh iz = i sin z
Also,
Proposition.
cosh
2
z − sinh
2
z = 1,