Changes between Version 5 and Version 6 of research/trig


Ignore:
Timestamp:
Oct 13, 2011, 4:42:46 PM (9 years ago)
Author:
sam
Comment:

use \max instead of \forall

Legend:

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  • research/trig

    v5 v6  
    1111{{{
    1212#!latex
    13 \[\big\vert\sin(x) - P(x)\big\vert \le E \qquad \forall x \in \bigg[-\frac{\pi}{2}, \frac{\pi}{2}\bigg]\]
     13\[\max_{x \in [-\pi/2, \pi/2]}{\big\vert\sin(x) - P(x)\big\vert} = E\]
    1414}}}
    1515
     
    1818{{{
    1919#!latex
    20 \[\big\vert\sin(x) - xQ(x^2)\big\vert \le E \qquad \forall x \in \bigg[-\frac{\pi}{2}, \frac{\pi}{2}\bigg]\]
     20\[\max_{x \in [-\pi/2, \pi/2]}{\big\vert\sin(x) - xQ(x^2)\big\vert} = E\]
    2121}}}
    2222
     
    2525{{{
    2626#!latex
    27 \[\big\lvert\sin(\sqrt{y}) - \sqrt{y}Q(y)\big\rvert \le E \qquad \forall y \in \bigg[0, \frac{\pi^2}{4}\bigg]\]
     27\[\max_{x \in [0, \pi^2/4]}{\big\lvert\sin(\sqrt{y}) - \sqrt{y}Q(y)\big\rvert} = E\]
    2828}}}
    2929
     
    3232{{{
    3333#!latex
    34 \[\bigg\lvert\frac{\sin(\sqrt{y})}{\sqrt{y}} - Q(y)\bigg\rvert \le \frac{E}{|\sqrt{y}|} \qquad \forall y \in \bigg[0, \frac{\pi^2}{4}\bigg]\]
     34\[\max_{x \in [0, \pi^2/4]}{\dfrac{\bigg\lvert\dfrac{\sin(\sqrt{y})}{\sqrt{y}} - Q(y)\bigg\rvert}{\dfrac{1}{|\sqrt{y}|}}} = E\]
    3535}}}
    3636
    37 If we want to force the asymptotic behaviour in x=0, we substitute Q(y) with 1+yR(y):
     37If we want to force the asymptotic behaviour at x=0, we substitute Q(y) with 1+yR(y):
    3838
    3939{{{
    4040#!latex
    41 \[\bigg\lvert\frac{\sin(\sqrt{y})}{\sqrt{y}} - 1 - yR(y)\bigg\rvert \le \frac{E}{|\sqrt{y}|} \qquad \forall y \in \bigg[0, \frac{\pi^2}{4}\bigg]\]
     41\[\max_{x \in [0, \pi^2/4]}{\dfrac{\bigg\lvert\dfrac{\sin(\sqrt{y})}{\sqrt{y}} - 1 - yR(y)\bigg\rvert}{\dfrac{1}{|\sqrt{y}|}}} = E\]
    4242}}}
    4343
     
    4646{{{
    4747#!latex
    48 \[\bigg\lvert\frac{\sin(\sqrt{y})-\sqrt{y}}{y\sqrt{y}} - R(y)\bigg\rvert \le \frac{E}{|y\sqrt{y}|} \qquad \forall y \in \bigg[0, \frac{\pi^2}{4}\bigg]\]
     48\[\max_{x \in [0, \pi^2/4]}{\dfrac{\bigg\lvert\dfrac{\sin(\sqrt{y})-\sqrt{y}}{y\sqrt{y}} - R(y)\bigg\rvert}{\dfrac{1}{|y\sqrt{y}|}}} = E\]
    4949}}}
    5050