156 CHAPTElt 5 • ELEMENTARY FUNCTIONSClearly, this definition agrees with that of the real exponential function when
z is a real number. We now show that this complex exponential has two of
the key properties associated with its real counterpart and verify the identity
ei^9 =cos 8 + i sin 8, which, back in Chapter 1 (see Identity (1- 32 ) of Section 1.4)
we promised to establish.