HP 15c - Manual (Page 59)

Section 3: Calculating in Complex Mode 59
Definitions of Math Functions
The lists that follow define the operation of the HP-15C in Complex mode. In these
definitions, a complex number is denoted by z = x + iy (rectangular form) or z = re
i

(polar
form). Also 2 2
y x z
  .
Arithmetic Operations
(a + ib) ± (c + id) = (a ± c) + i(b ± d)
(a + ib)(c + id) = (ac − bd) + i(ad + bc)
z
2
= z × z
1/z = x / |z|
2
– iy / |z|
2
z
1
÷ z
2 =
z
1
× 1/z
2
Single Valued Functions
e
z
= e
x
(cos y + i sin y)
10
z
= e
z ln10
) (
2
1
sin iz iz e e
i
z
  
cos z = ½(e
iz
+ e
−iz
)
tan z = sin z / cos z
sinh z = ½(e
z
− e
−z
)
cosh z = ½(e
z
+ e
−z
)
tanh z = sinh z / cosh z
Multivalued Functions
In general, the inverse of a function f(z)—denoted by f
−1
(z) —has more than one value for
any argument z. For example, cos
−1
(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
−1
(z). In the discussion that follows, the single-valued inverse
function (restricted to the principal branch) is denoted by uppercase letters-such as
COS
−1
(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(

+ 2kπ)
for −

<

<

and k = 0, ±1, ±2, …. Then if n is a positive integer,
z
1/n
= r
1/n
e
i(

/ n+2k

/ n)
= r
/ n
e
i

/ n
e
i2k

/ n
.
Only k = 0,1, ... , n − 1 are necessary since e
i2kπ / n
repeats its values in cycles of n. The
equation defines the nth roots of z, and r
1/n
e
i

/ n
with −

<

<

is the principal branch of z
1/n
.
(A program listed on page 67 computes the nth roots of z.)