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Astronomical Applications Department, U.S. Naval Observatory - Sensitivity Integral (Page 3)

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Astronomical Applications Department, U.S. Naval Observatory - Sensitivity Integral
evaluation.
1. Definite Integral
Here is the integration kernel (the diffraction intensity envelope):
:=
integrand
0
(
)
sin s x
2
(
)
-
(
)
sin r x
(
)
sin r g x
2
Here is the whole integrand:
:=
integrand
1
(
)
convert
,
[
]
(
)
seq
,
a
k
x
k
=
k
..
0
4
+ integrand
0
:=
integrand
1
(
)
+
+
+
+
a
0
a
1
x
a
2
x
2
a
3
x
3
a
4
x
4
(
)
sin s x
2
(
)
-
(
)
sin r x
(
)
sin r g x
2
Let's integrate just the kernel to get an idea of what we're up against.
d
1
2
integrand
0
x
(
)
cost %
+
+
+
+
399 additions
7156 multiplications
26 divisions
896 functions
928 subscripts
Well now that's unpleasant. Even worse, attempting the full integral causes terminal constipation.
Apparently, we'll have to approach the problem with brain engaged.
2. Indefinite Integral
Let's see what Maple will do with the indefinite integral.
=
(
)
S , , ,
r s g T
d
integrand
1
x
=
(
)
S , , ,
r s g T
d
(
)
+
+
+
+
a
0
a
1
x
a
2
x
2
a
3
x
3
a
4
x
4
(
)
sin s x
2
(
)
-
(
)
sin r x
(
)
sin r g x
2
x
:=
indef
(
)
value %
(
)
cost
(
)
rhs %
+
+
+
619 additions
195 functions
1761 multiplications
70 divisions
That was easier than anticipated. We can do a bit more simplification of this expression.
;
:=
indef
(
)
collect
,
,
indef [
]
,
,
,
sin cos
(
)
seq
,
a
i
=
i
..
0
4
x
factor
(
)
cost
(
)
rhs %
Page 3

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