Contour integrals, Keystrokes display – HP 15c User Manual

Page 73

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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

dz

z

f

)

(

, 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

dt

t

G

i

dt

t

G

dt

t

G

dz

z

f

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

dz

z

f

I

)

(

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 ba.

O4

004 44 4

Stores Re(ba) in R

4

.

´}

005- 42 30

O5

006- 44 5

Stores Im(ba) 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

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