Changes

Jump to: navigation, search

Sinc function

2 bytes added, 13:07, 4 September 2011
no edit summary
===First derivative===
''Computation at <math>x = 0</math>'':  {{point differentation incorrect}} Here, we need to compute the derivative using first principles, as a limit of a [[difference quotient]]:
<math>\operatorname{sinc}' 0 := \lim_{x \to 0} \frac{\operatorname{sinc} x - \operatorname{sinc} 0}{x - 0} = \lim_{x \to 0} \frac{(\sin x)/x - 1}{x} = \lim_{x \to 0} \frac{\sin x - x}{x^2}</math>
<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>
 
{{point differentation incorrect}}
''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:
3,033
edits

Navigation menu