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
:

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