HP HP-15C - Section 1 Getting Started; Keyboard Operation

16 Section 1: Using | SOLVE | Effectively
expressions for all but one variable in terms of the remaining
variable. By using these expressions, you can reduce the problem to
using [SOLVE | to find the root of a single equation. The values of the
other variables at the solution can then be calculated using the
derived expressions.
This is often useful for solving a complex equation for a complex
root. For such a problem, the complex equation can be expressed as
two real-valued equations—one for the real component and one for
the imaginary component—with two real variables—representing
the real and imaginary parts of the complex root.
For example, the complex equation z + 9 + 8e~z = 0 has no real roots
z, but it has infinitely many complex roots z — x + iy. This equation
can be expressed as two real equations
x + 9 + 8e~xcos y = 0
y — Seisin y = 0 .
The following manipulations can be used to eliminate y from the
equations. Because the sign of y doesn't matter in the equations,
assume y > 0, so that any solution (x,y) gives another solution
(x,-y). Rewrite the second equation as
x = ln(8(siny)/y),
which requires that sin y > 0, so that 2mr < y < (2n + l)rr for
integer n = 0, 1, ....
From the first equation
y = cos~1(-ex(x + 9)/8) + 2mr
= (2n + !)TT - cos~1(ex(x + 9)/8)
for n — 0,1,... Substitute this expression into the second equation,
, (2n + !)TT - cos-l(ex(x + 9)/8) \
+ lnl —^^^^^^ 1 = 0.
7 64 - (ex(x + 9))2 /