HP HP-15C - Arithmetic Operations; Single-Valued Functions; Multivalued Functions

SectionS: Calculating in Complex Mode 69
Arithmetic Operations
(a + ib)(c+id) = (ac - bd) + i(ad + be)
l/z = x/\z\-iy/\z\
zl ~=~ Z2 ~ zl * 1/22
Single-Valued Functions
ez = ex(cos y + i sin y )
sinz = — (eiz-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 = rei(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 el2kw/n repeats its values
in cycles of n. The equation defines the nth roots of z, and rl/nel6/n
with -TT < 9 ^ TT is the principal branch ofzl/n. (A program listed on
page 78 computes the nth roots of 2.)