176 CHAPTER. 5 • ELEMENTARY FUNCTIONS(b) z"' (a a real number) is given by the equation
z"' = r"' cos a9 + ir" sin a9,
where r > 0 and -7r < 9 $ 7r.
5. Let z., = (1 + i)" for n = 1, 2, .... Show that t he sequence {z.,} is a solution to
the difference equation z,, = 2z,._, - 2zn-2 for n 2: 3.- Verify
(a) Ident ity (5-24).
(b) Identity (5-25).
(c) Identity (5-26).
(d) Ident ity (5-27).
(e) Identity (5-29). - Does 1 raised to any power always equal l? Why or why not?
- Construct an example that shows t hat the principal value of (z 1 z2)i need not
equal the product of the principal values of zl and zl.
- If c is a complex number, the expression i^0 may be multivalued. Suppose all the
values of WI are identical. What are these values, and what can be said about t he
number c? Justify your assertions.
5.4 Trigonometric and Hyperbolic Functions
Based on the success we had in using power series to define the complex ex-
ponential, we have reason to believe that this approach will also be fruitful for
other elementary functions. The power series expansions for the real-valued sine
and cosine functions are(^00) x2n+l
sinx =E (-It( 2 )'and
n=O n + 1.
(^00) 2n
cosx= ~(-It (~n)!'
so it is natural to make the following definitions.
I Definition 5.5: sin z and cos z
oo z2n+I
sinz= L: (-It ( 2 )'
n= O n+l.00 n z2n
and cosz = L; (-1) ( 2 )''
n = O n ·With these definitions in place, we can now easily create the other complex
trigonometric functions, provided the denominators in the following expressions
are not zero.