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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