Game Development Reference
In-Depth Information
1
()
fx
=
2
(
)
1
+
x
2
′′
()
f
x
=
3
(
)
1
+
x
6
′′′
()
f
x
=
(D.16)
(
)
4
1
+
x
()
In general, the k -th derivative of
x is given by
f
()
1!
k
k
()
k
()
f
x
=
,
(D.17)
(
1
+
x
)
k
+
1
()
which when evaluated at
=
0
produces
k
() ( )
0
1
k
!
. Thus, the power se-
f
=−
k
x
()
ries for the function
x is given by
f
1
=− +
1
xx x
2
3
+−
1
+
x
()
kk
=−
1.
x
(D.18)
k
=
0
(
)
This series converges on the interval
. Integrating both sides, we arrive at
the following power series for the natural logarithm of 1
1, 1
+
on the same interval.
x
x xx
xx
2
3
4
(
)
ln1
+=− + − +−
234
1
()
kk
x
+
1
=
(D.19)
k
+
1
k
=
0
D.3 The Euler Formula
The Euler formula expresses the following relationship between the exponential
function and the sine and cosine functions.
ix
e
=
cos
xi x
+
sin
(D.20)
This can be verified by examining the power series of the function i e :
Search Nedrilad ::




Custom Search