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Sinc function

962 bytes added, 13:12, 4 September 2011
Differentation
<math>\lim_{x \to 0} \frac{\sin x - x}{x^2} \stackrel{*}{=} \lim_{x \to 0} \frac{\cos x - 1}{2x} \stackrel{*}{=} \lim_{x \to 0} \frac{-\sin x}{2} = \frac{-\sin 0}{2} = 0</math>
 
{{even function derivative at zero}}
''Computation at <math>x \ne 0</math>'': At any such point, we know that the function looks like <math>(\sin x)/x</math> in an [[open interval]] about the point, so we can use the [[quotient rule for differentiation]], which states that:
<math>\frac{d}{dx}(\operatorname{sinc}\ x) = \frac{x \cos x - \sin x}{x^2}</math>
 
Combining the two computations, we get:
 
<math>\operatorname{sinc}'(x) = \left\lbrace \begin{array}{rl} 0, & x = 0 \\ \frac{x \cos x - \sin x}{x^2}, & x \ne 0 \\\end{array}\right.</math>
 
==Taylor series and power series==
 
===Computation of power series===
 
We use the power series for the [[sine function]] (see [[sine function#Computation of power series]]):
 
<math>\! \sin x := x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots = \sum_{k=0}^\infty \frac{(-1)^kx^{2k+1}}{(2k + 1)!}</math>
 
Dividing both sides by <math>x</math> (valid when <math>x \ne 0</math>), we get:
 
<math>\! \frac{\sin x}{x} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + \dots = \sum_{k=0}^\infty \frac{(-1)^kx^{2k}}{(2k+1)!}</math>
 
We note that the power series ''also'' works at <math>x = 0</math> (because <math>\operatorname{sinc} \ 0 = 1</math>), hence it works globally, and is the power series for the sinc function.
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