Appendix: Some Useful Formulae
Sum of Integers and Their Powers
n
"k = n(n + 1)
2
k=1
n
2
"k = n(n + 1)(2n + 1)
6
k=1
n
3
3
"k = îÅ‚n(n + 1)Å‚Å‚
ïÅ‚ śł
2
ðÅ‚ ûÅ‚
k=1
n
4
"k = n(n + 2)(2n + 1)(3n2 + 3n - 1)
30
k=1
n
"(2k - 1) = n2
k=1
n
-
-
=
"(2k - 1)2 n(4n2 1)
3
k=1
n
= -
"(2k - 1)3 n2(2n2 1)
k=1
n
=
"k(k + 1)2 n(n + 1)(n + 2)(3n + 5)
12
k=1
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© 2000 by CRC Press LLC
© 2001 by CRC Press LLC
Arithmetic Series
n-1
"(a + kr) = n [2a + (n - 1)r]
2
k=0
Geometric Series
n
-
k-1
"aq = a(qn 1) q `" 1
q - 1
k=1
Arithmo-Geometric Series
n-1
= +
"(a + kr)qk a - [a + (n - 1)r]qn rq(1 - qn-1) q `" 1
(1 - q) (1 - q)2
k=0
Taylor s Series
"
(k)
f(x + a) = (x)
"f ak
k!
k=0
îÅ‚ Å‚Å‚
"f "f 1 "2 f "2 f "2 f
2
f(x + a, y + b) = f(x, y) + a + b + +&
ïÅ‚a + b2 + 2ab śł
"x "y 2! "x2 "y2 "x"y
ðÅ‚ ûÅ‚
© 2001 by CRC Press LLC
Trigonometric Functional Relations
1 1
sin(x) Ä… sin(y) = 2 sinîÅ‚ (x Ä… y)Å‚Å‚ cosîÅ‚ (x m y)Å‚Å‚
ïÅ‚2 śł ïÅ‚2 śł
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
1 1
cos(x) + cos(y) = 2 cosîÅ‚ (x + y)Å‚Å‚ cosîÅ‚ (x - y)Å‚Å‚
ïÅ‚2 śł ïÅ‚2 śł
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
1 1
cos(x) - cos(y) = 2 sinîÅ‚ (x + y)Å‚Å‚ sinîÅ‚ (y - x)Å‚Å‚
ïÅ‚2 śł ïÅ‚2 śł
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
1 1
ëÅ‚
sinëÅ‚ xöÅ‚ =Ä… (1 - cos(x))
íÅ‚ Å‚Å‚
2 2
1 1
cosëÅ‚ xöÅ‚ =Ä… (1 + cos(x))
ìÅ‚ ÷Å‚
íÅ‚ Å‚Å‚
2 2
sin(2x) = 2 sin(x)cos(x)
sin(3x) = 3 sin(x) - 4 sin3(x)
sin(4x) = cos(x)[4 sin(x) - 8 sin3(x)]
cos(2x) = 2 cos2(x) - 1
cos(3x) = 4 cos3(x) - 3 cos(x)
cos(4x) = 8 cos4(x) - 8 cos2(x) + 1
)
Relation of Trigonometric and Hyperbolic Functions
sin(x) =- j sinh(jx)
cos(x) = cosh(jx)
1
tan(x) = tanh(jx)
j
© 2001 by CRC Press LLC
Expansion of Elementary Functions in Power Series
"
xk
ex =
"
k!
k=0
"
x2k+1
k
sin(x) =
"(-1) (2k + 1)!
k=0
"
x2k
k
cos(x) =
"(-1) (2k)!
k=0
"
2k+1
sinh(x) =
"(2x
k + 1)!
k=0
"
2k
cosh(x) =
"(x
2k)!
k=0
© 2001 by CRC Press LLC