Game Development Reference
In-Depth Information
D.2 Power Series
Equation (D.10) can be used to derive power series expansions for common func-
tions by using
=
0
. Because the exponential function
e is equal to its own de-
c
0
rivative and
e
=
1
, its power series is given by
xxx
2
3
4
e
x
=+ +
1
x
+
+
+
2!
3!
4!
k
x
k
=
.
(D.11)
!
k
=
0
For the sine function, we first observe the following.
()
()
f
x
=
sin
x
f
0
=
0
()
()
fx
=
cos
x
f
0
=
1
(D.12)
′′
()
′′
()
f
x
=−
sin
x f
0
=
0
()
()
f
′′′
x
=−
cos
xf
′′′
0
=−
1
The power series for the sine function is thus given by
3
5
7
xxx
sin
xx
=− +
+−
3! 5! 7!
1 .
21!
()
(
k
21
k
+
x
=
(D.13)
)
k
+
k
=
0
Similarly, the power series for the cosine function is given by
2
4
6
xxx
cos
x
=−
1
+
+−
2!
4!
6!
()
()
k
2
k
1 .
2!
x
=
(D.14)
k
k
=
0
Another interesting function is
1
fx
()
=
(D.15)
1
+
x
(
)
(
)
because it is the derivative of
ln1 x
+
on the interval
−∞
1,
. The first few de-
()
rivatives of
x are the following.
f
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