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Getting Pi/2 Watch

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    Yesterday someone asked a question about everyone's favorite function,  f(x) = \frac{\sin(x)}{x} and I posed the challenge to get the result \displaystyle \int_{0}^{\infty} \frac{\sin(x)}{x} dx = \frac{\pi}{2} from the Gamma function. I thought this would be a good way to improve my frankly horrible LaTeX skills, I hope I can present Euler's brilliance.

    \displaystyle \Gamma(t) = \int_{0}^{\infty} x^{t-1} e^{-x} dx ,

    we now sub \ x=su^{n} , to get, \displaystyle \Gamma(t) = \int_{0}^{\infty} s^{t} n u^{nt-1} e^{-su^{n}} du ,

    removing constants we obtain,\displaystyle \frac{\Gamma(t)}{ns^{t}} = \int_{0}^{\infty} u^{nt-1} e^{-su^{n}} du

    we now consider the case that s is complex, which therefore gives us an equation for its conjugate,  \bar{s} , that being, \displaystyle \frac{\Gamma(t)}{ n\bar{s}^{t}} = \int_{0}^{\infty} u^{nt-1} e^{-\bar{s}u^{n}} du ,

    we then add/sub the two equations together, \displaystyle (\frac{\Gamma(t)}{s^{t}n} \pm \frac{\Gamma(t)}{\bar{s}^{t}n}) = \int_{0}^{\infty} u^{nt-1} (e^{-su^{n}} \pm e^{-\bar{s}u^{n}}) du ,

    then \displaystyle \frac{\Gamma(t)}{n}(\frac{1}{s^{  t}} \pm \frac{1}{\bar{s}^{t}}) = \int_{0}^{\infty} u^{nt-1} (e^{-su^{n}} \pm e^{-\bar{s}u^{n}}) du ,

    we then define \ s ,and. \bar{s} in polar and Cartesian form, so, \ s=a + ib , \bar{s}=a -ib , s=|s|e^{i \alpha} , \bar{s}=|s|e^{-i \alpha} ,

    applying these we get, \displaystyle \frac{\Gamma(t)}{n|s|^{t}}(e^{-i \alpha t} \pm e^{i \alpha t}) = \int_{0}^{\infty} u^{nt-1} e^{-au^{n}} (e^{-ibu^n} \pm e^{ibu^n}) du ,

    these reduce to, \displaystyle \frac{\Gamma(t)}{n|s|^{t}} \cos( \alpha t) = \int_{0}^{\infty} u^{nt-1} e^{-au^{n}}\cos(bu^{n}) du and \displaystyle \frac{\Gamma(t)}{n|s|^{t}} \sin( \alpha t) = \int_{0}^{\infty} u^{nt-1} e^{-au^{n}}\sin(bu^{n}) du , only the second of these we shall use.

    That being the second one. \displaystyle \frac{\Gamma(t)}{n|s|^{t}} \sin( \alpha t) = \int_{0}^{\infty} u^{nt-1} e^{-au^{n}}\sin(bu^{n}) du

    we now set \ a=0, b=1 ,and, n=1 thus meaning \ |s|=1 ,and, \alpha = \frac{\pi}{2} ,

    we must also use the Euler reflection formula,  \Gamma(t) \Gamma(1-t) = \frac{\pi}{\sin(\pi t)} ,

    using what we know, now we get a much more simple formula, \displaystyle \sin\left(\frac{\pi t}{2}\right) \frac{\pi t}{\sin(\pi t) \Gamma(1-t)} = \int_{0}^{\infty} u^{t-1}\sin(u) du ,

    now we take the limit as \ t \rightarrow 0 ,

    so we have \displaystyle \int_{0}^{\infty} u^{-1}\sin(u) du = \lim_{t\to 0}  \frac{\pi}{2} \frac{\sin(\frac{\pi}{2})}{\frac  {\pi t}{2}} \frac{\pi t}{\sin(\pi t) \Gamma(1-t)}  , the right hand side having had some \ t's , and,  \pi 's added, but all mathematically consistent.

    Evaluating the RHS we get \displaystlye \frac{\pi}{2} \frac{1}{\Gamma(1)}


    as we know \displaystlye \lim_{t\to 0} \frac{\sin(t)}{t} =1,
    from this we have, \displaystyle \int_{0}^{\infty} \frac{\sin(u)}{u} du = \frac{\pi}{2}  \Box

    (typo inevitable, please point out once seen)
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    Woah.
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    LaTeX tips: you'll want to use \sin, \cos, \trig instead of sin, cos, trig which are concantenations variables instead of functions.

    You'll want \displaystyle \left(\frac{\pi}{2}\right) instead of smaller brackets. You do this by using \left( and \right).

    Use \text{and} instead of and.

    Leave a space between the integrand and dx by using \, dx

    Great work and nice LaTex skills! Apologies for typo, typing on my phone is hard.
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    I used to understand all this ....
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    (Original post by TeeEm)
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    I used to understand all this ....
    do you miss that type of maths at all? or do you prefer the alevel/beginner undergrad content
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    (Original post by DylanJ42)
    do you miss that type of maths at all? or do you prefer the alevel/beginner undergrad content
    ask me in the morning ... my head is too fuzzy now!!
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    (Original post by TeeEm)
    ask me in the morning ... my head is too fuzzy now!!
    sure thing, get some sleep sir, it is getting late :yawn:
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    (Original post by DylanJ42)
    sure thing, get some sleep sir, it is getting late :yawn:
    goodnight !!
 
 
 
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