Arithmetic Operations; Single-valued Functions; Multivalued Functions - HP -15C Advanced Functions Handbook

SectionS: Calculating in Complex Mode
69
Arithmetic Operations
(a + ib)(c+id) = (ac - bd) + i(ad + be)
l/z = x/\z\-iy/\z\
z
l ~=~
Z
2 ~
z
l * 1/22
Single-Valued Functions
e
z
= e
x
(cos y + i sin y )
sinz = — (e
iz
-e~
iz
)
tan z = sin 2/cos z
cosh z =
l
/z( e* + e~
z
)
tanh z = sinh z/cosh z
Multivalued Functions
In general, the inverse of a function f(z) — denoted by f '
l
( z ) — has
more than one value for any argument z. For example, cos~
l
(z) has
infinitely many values for each argument. But the HP-15C
calculates the single principal value, which lies in the part of the
range defined as the principal branch of. f~
l
(z). In the discussion
that follows, the single-valued inverse function (restricted to the
principal branch) is denoted by uppercase letters — such as
COS'^z) — to distinguish it from the multivalued inverse — cos~
1
(z).
For example, consider the nth roots of a complex number z. Write z
in polar form as z = re
i(e + 2kw)
for -TT < 6 ^ TT and k = 0, ±1, ±2, ... .
Then if n is a positive integer,
Only k = 0, 1, ..., n — 1 are necessary since e
l2kw/n
repeats its values
in cycles of n. The equation defines the nth roots of z, and
rl/n
e
l6/n
with -TT < 9 ^ TT is the principal branch ofz
l/n
. (A program listed on
page 78 computes the nth roots of 2.)  