Some Useful Mathematical Formulas - Digital Signal Processing: Fundamentals and Applications - page 828

Series of exponentials:

N 1
k ¼ 0 a k ¼

1 a N
1 a ; a s 1

(G.30)

N
k ¼ 0 a k ¼

1
1 a ; jaj < 1

(G.31)

N
k ¼ 0 ka k ¼

1
ð 1 aÞ

2 ; jaj < 1

(G.32)

2 pnk
N

( 01 n N 1
Nn ¼ 0 ; N

N 1
k ¼ 0 e

j

¼

(G.33)

L'Hospital's rule:

f ðxÞ
gðxÞ

results in the undetermined form 0

0 or N

If lim
x/a

, then

N

f 0 ðxÞ
g 0 ðxÞ

f ðxÞ
gðxÞ ¼ lim

lim
x/a

(G.34)

x/a

where f 0 ðxÞ¼ df ðxÞ
dx

and g 0 ðxÞ¼ dgðxÞ
dx

.

Solution of the quadratic equation:
For a quadratic equation expressed as

2

ax þ bx þ c ¼ 0

(G.35)

the solution is given by

p

b

2

x ¼ b

4 ac

(G.36)

2 a

Solution of simultaneous equations:
Simultaneous linear equations are listed below:

a 11 x 1 þ a 12 x 2 þ / þ a 1 n x n ¼ b 1
a 21 x 1 þ a 22 x 2 þ / þ a 2 n x n ¼ b 2
/
a n 1 x 1 þ a n 2 x 2 þ / þ a nn x n ¼ b n

(G.37)

The solution is given by Cramer's rule, that is
x 1 ¼ D 1

D ; x 2 ¼ D 2 D ; / ; x n ¼ D n

(G.38)

D