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Applied Mathematics, Calculus, and Differential Equations 215

Binomial series
Let x be a variable, let a be a real number, and let n be a positive
n
integer. Then the value of (x a) can be found by summing a
finite serieð known as the Binomial series:

(x a)
n
a n

na n 1 x

(n(n 1)/2!)a n 2 2
x

(n(n 1)(n 2)/3!)a n 3 3
x

(n(n 1)(n 2)(n 3)/4!)a n 4 4
x

...

x n

General Fourier series
A Fourier series representð a periodic function F having period
2L, such that the following equation holdð for some variable x,
some sequence a , a , a , ... a , ..., and some sequence b , b ,
3
2
n
1
2
1
b , ... b , ... :
3
n
F
a /2 a cos ( x/L) b sin ( x/L)
1
0
1

a cos (2 x/L) b sin (2 x/L)
2
2