Contour Integrals - HP -15C Advanced Functions Handbook

SectionS: Calculating in Complex Mode 85
As n increases, the first guess A( n) comes ever closer to the desired
root 2. (When you're finished, press [T) | USER | to deactivate User
mode.)
Since all roots have negative real parts, the system is stable, but
the margin of stability (the smallest in magnitude among the real
parts, namely -0.1497) is small enough to cause concern if the
system must withstand much noise.
Contour Integrals
You can use \W\o evaluate the contour integral \f( z)dz, where C is a
curve in the complex plane.
First parameterize the curve C by z(t) = x(t) + iy(t) for ^ ^ t < t%.
Let G(t) =f(z(t))z'(t). Then
G(t)dt
ch rh
= 1 Re(G(t))dt + il lm(G(t))dt.
These integrals are precisely the type that [7T| evaluates in Complex
mode. Since G(t) is a complex function of a real variable t, [7F] will
sample G(t) on the interval t
l
^ t^t
2
and integrate Re(Cr(£))—the
value that your function returns to the real X-register. For the
imaginary part, integrate a function that evaluates G(t) and uses
[Res l m | to place Im (G( t)) into the real X-register.
The general-purpose program listed below evaluates the complex
integral
rb
I = J f(z)dz
along the straight line from a to b, where a and b are complex
numbers. The program assumes that your complex function sub-
routine is labeled "B" and evaluates the complex function f(z), and
that the limits a and b are in the complex Y- and X-registers,
respectively. The complex components of the integral / and the
uncertainty A/are returned in the X- and Y-registers.
Keystrokes Display
rsT||P/R| Program mode.
IT] CLEAR | PRGM| 000-