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inequality , quick question , (the context is cauchy theorem for fresnel integral) watch

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    please see attached, I am stuck on the second inequality.

    I have no idea where the 2/\pi has come from, I'm guessing it is a bound on sin \theta for \theta between \pi/4 and zero ? I know sin \theta \approx \theta for small \theta so we could bound it to \pi/4 since sin increases in this range  0, \pi /4 , however that would be a stronger approximation , loosing the actual integration over \theta which is clearly not what has been done.


    Many thanks in advance.
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    (Original post by xfootiecrazeesarax)
    please see attached, I am stuck on the second inequality.

    I have no idea where the 2/\pi has come from, I'm guessing it is a bound on sin \theta for \theta between \pi/4 and zero ? I know sin \theta \approx \theta for small \theta so we could bound it to \pi/4 since sin increases in this range  0, \pi /4 , however that would be a stronger approximation , loosing the actual integration over \theta which is clearly not what has been done.


    Many thanks in advance.
    Jordan's Lemma:  \frac{2}{\pi} < \frac{\sin{\theta}}{\theta} < 1 for  0 < \theta < \frac{\pi}{2}
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    (Original post by Blazy)
    Jordan's Lemma:  \frac{2}{\pi} < \frac{\sin{\theta}}{\theta} < 1 for  0 < \theta < \frac{\pi}{2}
    Nice. I was baffled by that - I had read the inequality as \frac{2}{\pi}-\theta and was trying to get some result out of Taylor's theorem + remainder term.

    That's not called Jordan's lemma, though, is it?
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    (Original post by atsruser)
    Nice. I was baffled by that - I had read the inequality as \frac{2}{\pi}-\theta and was trying to get some result out of Taylor's theorem + remainder term.

    That's not called Jordan's lemma, though, is it?
    I'm sure I've seen it called that as well, though strictly it's one of the steps in the proof of a bigger result that's more normally called Jordan's Lemma.

    (But in this context, the label seems reasonably appropriate, since the integral seems very similar if not identical in form to the "big" Jordan's Lemma result).
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    (Original post by atsruser)
    Nice. I was baffled by that - I had read the inequality as \frac{2}{\pi}-\theta and was trying to get some result out of Taylor's theorem + remainder term.

    That's not called Jordan's lemma, though, is it?
    (Original post by DFranklin)
    I'm sure I've seen it called that as well, though strictly it's one of the steps in the proof of a bigger result that's more normally called Jordan's Lemma. (But in this context, the label seems reasonably appropriate, since the integral seems very similar if not identical in form to the "big" Jordan's Lemma result).
    Yeah I was wondering that when I looked it up on Wikipedia but this was what I was taught in my 2nd year complex analysis course at uni.
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    http://mathworld.wolfram.com/JordansLemma.html

    Thanks I have just found this, so line (11) it is used in the proof of Jordan's lemma , as said above, but this line isn't actually proved.

    I just want the proof of this line.

    thanks
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    (Original post by xfootiecrazeesarax)
    http://mathworld.wolfram.com/JordansLemma.html

    Thanks I have just found this, so line (11) it is used in the proof of Jordan's lemma , as said above, but this line isn't actually proved.

    I just want the proof of this line.
    It follows from sin x being concave on [0,pi/2]. Look up inequalities for convex / concave functions (I know it as Jensen's inequality, but the wiki page is a bit OTT and doesn't help much I think).
 
 
 
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