User Manual

Section 3: Calculating in Complex Mode 73
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 f to evaluate the contour integral

C
dzzf )(
, where C is a curve in the complex
plane.
First parameterize the curve C by z(t)= x(t) + i y(t) for t
1
≤ t ≤ t
2
. Let G(t)=f(z(t))z’(t). Then

 


2
1
2
1
2
1
))(Im())(Re(
)()(
t
t
t
t
C
t
t
dttGidttG
dttGdzzf
These integrals are precisely the type that f evaluates in Complex mode. Since G(t) is a
complex function of a real variable t, f will sample G(t) on the interval t
1
≤ t ≤ t
2
and
integrate Re(G(t))—the value that your function returns to the real X-register. For the
imaginary part, integrate a function that evaluates G(t) and uses } to place Im(G(t))
into the real X-register.
The general-purpose program listed below evaluates the complex integral


b
a
dzzfI )(
along the straight line from a to b, where a and b are complex numbers. The program
assumes that your complex function subroutine 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 I and the uncertainty ΔI are returned in the X- and
Y-registers.
Keystrokes
Display
|¥
Program mode.
´CLEARM
000-
´bA
001-42,21,11
®
002- 34
-
003- 30
Calculates b – a.
O4
004 44 4
Stores Re(b – a) in R
4
.
´}
005- 42 30
O5
006- 44 5
Stores Im(b – a) in R
5
.
|K
007- 43 36
Recalls a.
O6
008- 44 6
Stores Re(a) in R
6
.
´}
009- 42 30
O7
010- 44 7
Stores Im(a) in R
7
.
0
011- 0