Graphics Programs Reference
In-Depth Information
Appendices
A1
Taylor Series
Function of a Single Variable
The Taylor series expansion of a function
f
(
x
) about the point
x
=
a
is the infinite
series
f
(
a
)
(
x
−
a
)
2
f
(
a
)
(
x
−
a
)
3
f
(
a
)(
x
f
(
x
)
=
f
(
a
)
+
−
a
)
+
+
+···
(A1)
2!
3!
In the specialcase
a
0 the series is also known as the
MacLaurin series
. It can be
shown that the Taylor series expansionis unique in the sense that notwofunctions
have identicalTaylor series.
A Taylor series is meaningfulonlyif all the derivatives of
f
(
x
)exist at
x
=
=
a
and
the series converges. Ingeneral, convergence occursonlyif
x
issufficiently close to
a
;
i.e., if
is infinite.
Anotherusefulform of the Taylor series is the expansionabout an arbitrary
valueof
x
:
|
x
−
a
| ≤
ε
, where
ε
iscalled the
radius of convergence
. Inmany cases
ε
f
(
x
)
h
2
f
(
x
)
h
3
f
(
x
)
h
f
(
x
+
h
)
=
f
(
x
)
+
+
2!
+
3!
+···
(A2)
Since it is not possible to evaluate all the termsofan infinite series, the effect of
truncating the series in Eq. (A2) isofgreat practical importance. Keeping the first
n
+
1 terms, wehave
f
(
x
)
h
2
f
(
n
)
(
x
)
h
n
f
(
x
)
h
+
=
+
+
2!
+···+
n
!
+
f
(
x
h
)
f
(
x
)
E
n
(A3)
where
E
n
is the
truncation error
(sum of the truncated terms). The boundson the
truncation errorare givenby
Taylor's theorem
:
h
n
+
1
(
n
f
(
n
+
1)
(
E
n
=
ξ
)
(A4)
+
1)!
411
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