Section: 17 | Series |

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6 SERIES

6.1 FOURIER SERIES

If f(x) is a bounded periodic function of period 2L (i.e., f(x+2L)=f(x) ), and satisfies the Dirichlet conditions:
1. In any period $f (x)$ is continuous, except possibly for a finite number of jump discontinuities.
2. In any period $f (x)$ has only a finite number of maxima and minima,
then f(x) may be represented by the Fourier seriesf(x)=a02+∑n=1∞(ancosnπxL+bnsinnπxL) where an and bn are given below. This series will converge to f(x) at every point where f(x) is continuous, and to f(x+)+f(x−)2 (i.e., the average of the left-hand and right-hand limits) at every point where f(x) has a jump discontinuity. an=1L∫−LLf(x)cosnπxLdx, n=0,1,2,3,…,bn=1L∫−LLf(x)sinnπxLdx, n=1,2,3,… We may also write an=1L∫αα+2Lf(x)cosnπxLdx and bn=1L∫αα+2Lf(x)sinnπxLdx where α is any real number. Thus using α=0 : an=1L∫02Lf(x)cosnπxLdx, n=0,1,2,3,…,bn=1L∫02Lf(x)sinnπxLd, n=1,2,3,…
If in addition to the restrictions for (23), if $f (x)$ is an even function (i.e., $f (- x) = f (x)$ ), then the Fourier series reduces to $f (x) = \frac{a_{0}}{2} + \sum_{n = 1}^{\infty} a_{n} cos \frac{n π x}{L}$ That is, $b_{n} = 0$ . In this case, a simpler formula for $a_{n}$ is $a_{n} = \frac{2}{L} \int_{0}^{L} f (x) cos \frac{n π x}{L} d x, n = 0, 1, 2, 3, \dots$
If in addition to the restrictions for (23), if $f (x)$ is an odd function (i.e., $f (- x) = - f (x)$ ), then the Fourier series reduces to $f (x) = \sum_{n = 1}^{\infty} b_{n} sin \frac{n π x}{L}$ That is, $a_{n} = 0$ . In this case, a simpler formula for $b_{n}$ is $b_{n} = \frac{2}{L} \int_{0}^{L} f (x) sin \frac{n π x}{L} d x, n = 1, 2, 3, \dots$
Using the Euler relation $e^{i θ} = cos θ + i sin θ$ , we obtain the complex form of the Fourier series $f (x) = \frac{1}{2} \sum_{n = - \infty}^{n = + \infty} c_{n} e^{i ω_{n} x}$ where $c_{n} = \frac{1}{L} \int_{- L}^{L} f (x) e^{- i ω_{n} x} d x, n = 0, \pm 1, \pm 2, \pm 3, \dots$ with $ω_{n} = \frac{n π}{L}$ for $n = 0, \pm 1, \pm 2, \dots$ The set of coefficients $c_{n}$ is referred to as the Fourier spectrum.
If $f (x)$ has period $2 L$ , if it is expandable by a Fourier series, and if both sine and cosine terms are present, then equations (26) and (27) can be written as $f (x) = \frac{a_{0}}{2} + \sum_{n = 1}^{\infty} c_{n} sin (\frac{n π x}{L} + φ_{n}),$ where $a_{n} = c_{n} sin φ_{n}, b_{n} = c_{n} cos φ_{n}, c_{n} = \sqrt{a_{n}^{2} + b_{n}^{2}}, φ_{n} = arctan (\frac{a_{n}}{b_{n}})$ They can also be represented as $f (x) = \frac{a_{0}}{2} + \sum_{n = 1}^{\infty} c_{n} cos (\frac{n π x}{L} + φ_{n}),$ where $a_{n} = c_{n} cos ϕ_{n}, b_{n} = - c_{n} sin ϕ_{n}, c_{n} = \sqrt{a_{n}^{2} + b_{n}^{2}}, ϕ_{n} = arctan (- \frac{b_{n}}{a_{n}})$ and $φ_{n}$ is chosen so as to make $a_{n}$ , $b_{n}$ , and $c_{n}$ hold.
The following table of trigonometric identities is helpful when developing Fourier series. Note that $n$ is an integer. $n$ $n$ even $n$ odd $n / 2$ odd $n / 2$ even

$sin n π$ 0 0 0 0 0

$cos n π$ $(- 1)^{n}$ $+ 1$ $- 1$ $+ 1$ $+ 1$

$sin \frac{n π}{2}$ 0 $(- 1)^{(n - 1) / 2}$ 0 0

$cos \frac{n π}{2}$ $(- 1)^{n / 2}$ 0 $- 1$ $+ 1$

$sin \frac{n π}{4}$ $\frac{\sqrt{2}}{2} (- 1)^{(n^{2} + 4 n + 11) / 8}$ $(- 1)^{(n - 2) / 4}$ 0

Note also the useful formula for $sin \frac{n π}{2}$ and $cos \frac{n π}{2}$ : $\begin{matrix} sin \frac{n π}{2} & = \frac{(i)^{n + 1}}{2} [(- 1)^{n} - 1] \\ cos \frac{n π}{2} & = \frac{(i)^{n}}{2} [(- 1)^{n} + 1] \end{matrix}$

Auxiliary Formulas for Fourier Series

1.	$1 = \frac{4}{π} [sin \frac{π x}{k} + \frac{1}{3} sin \frac{3 π x}{k} + \frac{1}{5} sin \frac{5 π x}{k} + \dots]$	$[0 < x < k]$
2.	$x = \frac{2 k}{π} [sin \frac{π x}{k} - \frac{1}{2} sin \frac{2 π x}{k} + \frac{1}{3} sin \frac{3 π x}{k} - \dots]$	$[- k < x < k]$
3.	$x = \frac{k}{2} - \frac{4 k}{π^{2}} [cos \frac{π x}{k} + \frac{1}{3^{2}} cos \frac{3 π x}{k} + \frac{1}{5^{2}} cos \frac{5 π x}{k} + \dots]$	$[0 < x < k]$
4.	$x^{2} = \frac{2 k^{2}}{π^{3}} [(\frac{π^{2}}{1} - \frac{4}{1}) sin \frac{π x}{k} - \frac{π^{2}}{2} sin \frac{2 π x}{k} + (\frac{π^{2}}{3} - \frac{4}{3^{3}}) sin \frac{3 π x}{k} - \frac{π^{2}}{4} sin \frac{4 π x}{k} + (\frac{π^{2}}{5} - \frac{4}{5^{3}}) sin \frac{5 π x}{k} + \dots]$	$[0 < x < k]$
5.	$x^{2} = \frac{k^{2}}{3} - \frac{4 k^{2}}{π^{2}} [cos \frac{π x}{k} - \frac{1}{2^{2}} cos \frac{2 π x}{k} + \frac{1}{3^{2}} cos \frac{3 π x}{k} - \frac{1}{4^{2}} cos \frac{4 π x}{k} + \dots]$	$[- k < x < k]$

Fourier Expansions for Basic Periodic Functions

1.		$f (x) = \frac{4}{π} \sum_{n = 1, 3, 5 \dots} \frac{1}{n} sin \frac{n π x}{L}$
2.		$f (x) = \frac{2}{π} \sum_{n = 1}^{\infty} \frac{(- 1)^{n}}{n} (cos \frac{n π c}{L} - 1) sin \frac{n π x}{L}$
3.		$f (x) = \frac{c}{L} + \frac{2}{π} \sum_{n = 1}^{\infty} \frac{(- 1)^{n}}{n} sin \frac{n π c}{L} cos \frac{n π x}{L}$
4.		$f (x) = \frac{2}{L} \sum_{n = 1}^{\infty} sin \frac{n π}{2} \frac{sin (\frac{1}{2} n π c / L)}{\frac{1}{2} n π c / L} sin \frac{n π x}{L}$
5.		$f (x) = \frac{2}{π} \sum_{n = 1}^{\infty} \frac{(- 1)^{n + 1}}{n} sin \frac{n π x}{L}$
6.		$f (x) = \frac{1}{2} - \frac{4}{π^{2}} \sum_{n = 1, 3, 5, \dots} \frac{1}{n^{2}} cos \frac{n π x}{L}$
7.		$f (x) = \frac{8}{π^{2}} \sum_{n = 1, 3, 5, \dots} \frac{(- 1)^{(n - 1) / 2}}{n^{2}} sin \frac{n π x}{L}$
8.		$f (x) = \frac{1}{2} - \frac{1}{π} \sum_{n = 1}^{\infty} \frac{1}{n} sin \frac{n π x}{L}$
9.		$f (x) = \frac{1}{2} (1 + a) + \frac{2}{π^{2} (1 - a)} \sum_{n = 1}^{\infty} \frac{1}{n^{2}} [(- 1)^{n} cos n π a - 1] cos \frac{n π x}{L}; (a = \frac{c}{2 L})$
10.		$f (x) = \frac{2}{π} \sum_{n = 1}^{\infty} \frac{(- 1)^{n - 1}}{n} [1 + \frac{sin n π a}{n π (1 - a)}] sin \frac{n π x}{L}; (a = \frac{c}{2 L})$
11.		$f (x) = \frac{1}{2} - \frac{4}{π^{2} (1 - 2 a)} \sum_{n = 1, 3, 5, \dots} \frac{1}{n^{2}} cos n π a cos \frac{n π x}{L}; (a = \frac{c}{2 L})$
12.		$f (x) = \frac{2}{π} \sum_{n = 1}^{\infty} \frac{(- 1)^{n}}{n} [1 + \frac{1 + (- 1)^{n}}{n π (1 - 2 a)} sin n π a] sin \frac{n π x}{L}; (a = \frac{c}{2 L})$
13.		$f (x) = \frac{4}{π} \sum_{n = 1}^{\infty} \frac{1}{n} sin \frac{n π}{4} sin n π a sin \frac{n π x}{L}; (a = \frac{c}{2 L})$
14.		$f (x) = \frac{9}{π^{2}} \sum_{n = 1}^{\infty} \frac{1}{n^{2}} sin \frac{n π}{3} sin \frac{n π x}{L}; (a = \frac{c}{2 L})$
15.		$f (x) = \frac{32}{3 π^{2}} \sum_{n = 1}^{\infty} \frac{1}{n^{2}} sin \frac{n π}{4} sin \frac{n π x}{L}; (a = \frac{c}{2 L})$
16.		$f (x) = \frac{1}{π} + \frac{1}{2} sin ω t - \frac{2}{π} \sum_{n = 2, 4, 6, \dots} \frac{1}{n^{2} - 1} cos n ω t$

6.2 BINOMIAL SERIES

The expression in parentheses following certain of the series indicates the region of convergence. If not otherwise indicated it is to be understood that the series converges for all finite values of $x$ .

1.	$(x + y)^{n} = x^{n} + {n x}^{n - 1} y + \frac{n (n - 1)}{2!} x^{n - 2} y^{2} + \frac{n (n - 1) (n - 2)}{3!} x^{n - 3} y^{3} + \dots$	$(y^{2} < x^{2})$
2.	$(1 \pm x)^{n} = 1 \pm n x + \frac{n (n - 1) x^{2}}{2!} \pm \frac{n (n - 1) (n - 2) x^{3}}{3!} + \dots$	$(x^{2} < 1)$
3.	$(1 \pm x)^{- n} = 1 \mp n x + \frac{n (n + 1) x^{2}}{2!} \mp \frac{n (n + 1) (n + 2) x^{3}}{3!} + \dots$	$(x^{2} < 1)$
4.	$(1 \pm x)^{- 1} = 1 \mp x + x^{2} \mp x^{3} + x^{4} \mp x^{5} + \dots$	$(x^{2} < 1)$
5.	$(1 \pm x)^{- 2} = 1 \mp 2 x + 3 x^{2} \mp 4 x^{3} + 5 x^{4} \mp 6 x^{5} + \dots$	$(x^{2} < 1)$

6.3 REVERSION OF SERIES

Let a series be represented by $y = a_{1} x + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4} + a_{5} x^{5} + a_{6} x^{6} + \dots$ with $a_{1} \neq 0$ . The coefficients of the series $x = A_{1} y + A_{2} y^{2} + A_{3} y^{3} + A_{4} y^{4} + A_{5} y^{5} + \dots$ are $\begin{matrix} A_{1} & = \frac{1}{a_{1}} A_{2} = - \frac{a_{2}}{a_{1}^{3}} A_{3} = \frac{1}{a_{1}^{5}} (2 a_{2}^{2} - a_{1} a_{3}) A_{4} = \frac{1}{a_{1}^{7}} (5 a_{1} a_{2} a_{3} - a_{1}^{2} a_{4} - 5 a_{2}^{3}) \\ A_{5} & = \frac{1}{a_{1}^{9}} (6 a_{1}^{2} a_{2} a_{4} + 3 a_{1}^{2} a_{3}^{2} + 14 a_{2}^{4} - a_{1}^{3} a_{5} - 21 a_{1} a_{2}^{2} a_{3}) \end{matrix}$

6.4 TAYLOR SERIES

$f (x) = f (a) + (x - a) f^{'} (a) + \frac{(x - a)^{2}}{2!} f^{''} (a) + \frac{(x - a)^{3}}{3!} f^{'''} (a) + \dots + \frac{(x - a)^{n}}{n!} f^{(n)} (a) + \dots (Taylor"s Series, increment form)$
$\begin{array}{l} f (x + h) & = & f (x) + h f^{'} (x) + \frac{h^{2}}{2!} f^{″} (x) + \frac{h^{3}}{3!} f^{‴} (x) + \dots \\ = & f (h) + x f^{'} (h) + \frac{x^{2}}{2!} f^{″} (h) + \frac{x^{3}}{3!} f^{‴} (h) + \dots \end{array}$
If $f (x)$ is a function possessing derivatives of all orders throughout the interval $[a, b]$ then the Taylor's series with remainder term is (for $a \leq x \leq b$ ) $f (x) = f (a) + (x - a) f^{'} (a) + \frac{(x - a)^{2}}{2!} f^{''} (a) + \dots + (x - a)^{n - 1} \frac{f^{(n - 1)} (a)}{(n - 1)!} + R_{n}$ where $R_{n} = \frac{f^{(n)} [a + θ \cdot (x - a)]}{n!} (x - a)^{n}, 0 < θ < 1 .$ Special cases: There are values $0 < θ < 1$ and $a < X < b$ , such that $\begin{matrix} f (b) & = f (a) + (b - a) f^{'} (a) + \frac{(b - a)^{2}}{2!} f^{''} (a) + \dots + \frac{(b - a)^{n - 1}}{(n - 1)!} f^{(n - 1)} (a) + \frac{(b - a)^{n}}{n!} f^{(n)} (X) \\ f (a + h) & = f (a) + {h f}^{'} (a) + \frac{h^{2}}{2!} f^{''} (a) + \dots + \frac{h^{n - 1}}{(n - 1)!} f^{(n - 1)} (a) + \frac{h^{n}}{n!} f^{(n)} (a + θ h) \end{matrix}$
Taylor's series for a function of two variables: Define the notation: $\begin{matrix} (h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y}) f (x, y) & = h \frac{\partial f (x, y)}{\partial x} + k \frac{\partial f (x, y)}{\partial y}; \\ {(h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y})}^{2} f (x, y) & = h^{2} \frac{\partial^{2} f (x, y)}{\partial x^{2}} + 2 h k \frac{\partial^{2} f (x, y)}{\partial x \partial y} + k^{2} \frac{\partial^{2} f (x, y)}{\partial y^{2}} \end{matrix}$ with an extnsion to higher powers. If ${{(h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y})}^{n} f (x, y) |}_{x = a}^{y = b}$ where the bar and subscripts means that after differentiation we are to replace $x$ by $a$ and $y$ by $b$ , then $f (a + h, b + k) = f (a, b) + {(h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y}) f (x, y) |}_{x = a}^{y = b} + \dots + \frac{1}{n!} {{(h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y})}^{n} f (x, y) |}_{x = a}^{y = b} + \dots$
Maclaurin series (use $a = 0$ in equation (28)) $f (x) = f (0) + {x f}^{'} (0) + \frac{x^{2}}{2!} f^{''} (0) + \frac{x^{3}}{3!} f^{'''} (0) + \dots + x^{n - 1} \frac{f^{(n - 1)} (0)}{(n - 1)!} + R_{n}$ where $R_{n} = \frac{x^{n} f^{(n)} (θ x)}{n!}, 0 < θ < 1 .$

6.5 EXPONENTIAL SERIES

$\begin{matrix} e & = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \dots \\ e^{x} & = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!} + \dots \\ a^{x} & = 1 + x log_{e} a + \frac{(x log_{e} a)^{2}}{2!} + \frac{(x log_{e} a)^{3}}{3!} + \dots \\ e^{x} & = e^{a} [1 + (x - a) + \frac{(x - a)^{2}}{2!} + \frac{(x - a)^{3}}{3!} + \dots] \end{matrix}$

6.6 LOGARITHMIC SERIES

$\begin{matrix} log_{e} (1 + x) & = & x - \frac{1}{2} x^{2} + \frac{1}{3} x^{3} - \frac{1}{4} x^{4} + \dots & (- 1 < x \leq 1) \\ log_{e} (n + 1) - log_{e} (n - 1) & = & 2 [\frac{1}{n} + \frac{1}{3 n^{3}} + \frac{1}{5 n^{5}} + \dots] \\ log_{e} \frac{1 + x}{1 - x} & = & 2 [x + \frac{x^{3}}{3} + \frac{x^{5}}{5} + \dots + \frac{x^{2 n - 1}}{2 n - 1} + \dots] & (- 1 < x < 1) \\ log_{e} x & = & log_{e} a + \frac{(x - a)}{a} - \frac{(x - a)^{2}}{2 a^{2}} + \frac{(x - a)^{3}}{3 a^{3}} - + \dots & (0 < x \leq 2 a) \end{matrix}$

6.7 TRIGONOMETRIC SERIES

Let $B_{n}$ represent the $n^{th}$ Bernoulli number and let $E_{n}$ represent the $n^{th}$ Euler number.

•	$sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + \dots$	(all real values of $x$ )
•	$cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + \dots$	(all real values of $x$ )
•	$tan x = x + \frac{x^{3}}{3} + \frac{2 x^{5}}{15} + \frac{17 x^{7}}{315} + \frac{62 x^{9}}{2835} + \dots + \frac{(- 1)^{n - 1} 2^{2 n} (2^{2 n} - 1) B_{2 n}}{(2 n)!} x^{2 n - 1} + \dots$	$(x^{2} < \frac{π^{2}}{4})$
•	$cot x = \frac{1}{x} - \frac{x}{3} - \frac{x^{3}}{45} - \frac{2 x^{5}}{945} - \frac{x^{7}}{4725} - \dots - \frac{(- 1)^{n + 1} 2^{2 n}}{(2 n)!} B_{2 n} x^{2 n - 1} - \dots$	$(x^{2} < π^{2})$
•	$sec x = 1 + \frac{x^{2}}{2} + \frac{5}{24} x^{4} + \frac{61}{720} x^{6} + \frac{277}{8064} x^{8} + \dots + \frac{(- 1)^{n}}{(2 n)!} E_{2 n} x^{2 n} + \dots$	$(x^{2} < \frac{π^{2}}{4})$
•	$csc x = \frac{1}{x} + \frac{x}{6} + \frac{7}{360} x^{3} + \frac{31}{15, 120} x^{5} + \frac{127}{604, 800} x^{7} + \dots + \frac{(- 1)^{n + 1} 2 (2^{2 n - 1} - 1)}{(2 n)!} B_{2 n} x^{2 n - 1} + \dots$	$(x^{2} < π^{2})$
•	$sin^{- 1} x = x + \frac{x^{3}}{2 \cdot 3} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 5} x^{5} + \frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6 \cdot 7} x^{7} + \dots$	$(x^{2} < 1, - \frac{π}{2} < sin^{- 1} x < \frac{π}{2})$
•	$cos^{- 1} x = \frac{π}{2} - (x + \frac{x^{3}}{2 \cdot 3} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 5} x^{5} + \frac{1 \cdot 3 \cdot 5 x^{7}}{2 \cdot 4 \cdot 6 \cdot 7} + \dots)$	$(x^{2} < 1, 0 < cos^{- 1} x < π)$
•	$tan^{- 1} x = x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7} + \dots$	$(x^{2} < 1)$
•	$tan^{- 1} x = \frac{π}{2} - \frac{1}{x} + \frac{1}{3 x^{3}} - \frac{1}{5 x^{5}} + \frac{1}{7 x^{7}} - \dots$	$(x > 1)$
•	$tan^{- 1} x = - \frac{π}{2} - \frac{1}{x} + \frac{1}{3 x^{3}} - \frac{1}{5 x^{5}} + \frac{1}{7 x^{7}} - \dots$	$(x < - 1)$
•	$cot^{- 1} x = \frac{π}{2} - x + \frac{x^{3}}{3} - \frac{x^{5}}{5} + \frac{x^{7}}{7} - \dots$	( $x^{2} < 1$ )

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