large x and small x expansion for gamma-like function











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Find two approximations for the integral ($x>0$)
begin{equation}
I(x) = frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}}
e^{x cos^2(theta)}dtheta
end{equation}

one for small $x$ (keeping up to linear order in $x$) and one for large values of $x$ (keeping only the leading order term).





For the small x expansion, I tried using a Maclaurin series for the function in the exponential, and it seemed to work...I am unsure what to do for large x or how to check my answer. I know this is likely related to the Gamma/Gaussian function, but am fairly novice at problems of this form. Any help deeply appreciated!










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  • 1




    For large $x$ check out en.wikipedia.org/wiki/Laplace%27s_method
    – Yuriy S
    Nov 17 at 21:48












  • Thanks @Yuriy. I haven't been exposed to this method, so I think I'd like to solve using asymptotic series. Still working through a few trials, but am interested in easier ways to solve problems like these outside of my current situation.
    – niagarajohn
    Nov 17 at 21:52






  • 1




    Laplace's method is the standard one for such situations, and it's really simple. But of course, you can choose to try some other way
    – Yuriy S
    Nov 17 at 21:53










  • Seconding @Yuriy's suggestion.
    – Antonio Vargas
    Nov 18 at 3:13










  • My attempt for small x: begin{align} I(x) &= frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}} e^{x cos^2(theta)}dtheta\ &= frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}} dtheta [1 + xcos^2{theta} + order{x^2}]\ &= frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}} dtheta [1 + frac{x}{2}(1+cos{2theta}) + order{x^2}]\ &approx frac{1}{2pi} [pi + frac{x}{2}(pi)] end{align} Therefore, begin{align} &boxed{ I(x)approx frac{1}{2} + frac{x}{4} text{for small $x$}} end{align}
    – niagarajohn
    Nov 18 at 14:53

















up vote
2
down vote

favorite
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Find two approximations for the integral ($x>0$)
begin{equation}
I(x) = frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}}
e^{x cos^2(theta)}dtheta
end{equation}

one for small $x$ (keeping up to linear order in $x$) and one for large values of $x$ (keeping only the leading order term).





For the small x expansion, I tried using a Maclaurin series for the function in the exponential, and it seemed to work...I am unsure what to do for large x or how to check my answer. I know this is likely related to the Gamma/Gaussian function, but am fairly novice at problems of this form. Any help deeply appreciated!










share|cite|improve this question




















  • 1




    For large $x$ check out en.wikipedia.org/wiki/Laplace%27s_method
    – Yuriy S
    Nov 17 at 21:48












  • Thanks @Yuriy. I haven't been exposed to this method, so I think I'd like to solve using asymptotic series. Still working through a few trials, but am interested in easier ways to solve problems like these outside of my current situation.
    – niagarajohn
    Nov 17 at 21:52






  • 1




    Laplace's method is the standard one for such situations, and it's really simple. But of course, you can choose to try some other way
    – Yuriy S
    Nov 17 at 21:53










  • Seconding @Yuriy's suggestion.
    – Antonio Vargas
    Nov 18 at 3:13










  • My attempt for small x: begin{align} I(x) &= frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}} e^{x cos^2(theta)}dtheta\ &= frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}} dtheta [1 + xcos^2{theta} + order{x^2}]\ &= frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}} dtheta [1 + frac{x}{2}(1+cos{2theta}) + order{x^2}]\ &approx frac{1}{2pi} [pi + frac{x}{2}(pi)] end{align} Therefore, begin{align} &boxed{ I(x)approx frac{1}{2} + frac{x}{4} text{for small $x$}} end{align}
    – niagarajohn
    Nov 18 at 14:53















up vote
2
down vote

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up vote
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Find two approximations for the integral ($x>0$)
begin{equation}
I(x) = frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}}
e^{x cos^2(theta)}dtheta
end{equation}

one for small $x$ (keeping up to linear order in $x$) and one for large values of $x$ (keeping only the leading order term).





For the small x expansion, I tried using a Maclaurin series for the function in the exponential, and it seemed to work...I am unsure what to do for large x or how to check my answer. I know this is likely related to the Gamma/Gaussian function, but am fairly novice at problems of this form. Any help deeply appreciated!










share|cite|improve this question















Find two approximations for the integral ($x>0$)
begin{equation}
I(x) = frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}}
e^{x cos^2(theta)}dtheta
end{equation}

one for small $x$ (keeping up to linear order in $x$) and one for large values of $x$ (keeping only the leading order term).





For the small x expansion, I tried using a Maclaurin series for the function in the exponential, and it seemed to work...I am unsure what to do for large x or how to check my answer. I know this is likely related to the Gamma/Gaussian function, but am fairly novice at problems of this form. Any help deeply appreciated!







sequences-and-series complex-analysis asymptotics gamma-function






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edited Nov 22 at 0:52

























asked Nov 17 at 21:44









niagarajohn

186




186








  • 1




    For large $x$ check out en.wikipedia.org/wiki/Laplace%27s_method
    – Yuriy S
    Nov 17 at 21:48












  • Thanks @Yuriy. I haven't been exposed to this method, so I think I'd like to solve using asymptotic series. Still working through a few trials, but am interested in easier ways to solve problems like these outside of my current situation.
    – niagarajohn
    Nov 17 at 21:52






  • 1




    Laplace's method is the standard one for such situations, and it's really simple. But of course, you can choose to try some other way
    – Yuriy S
    Nov 17 at 21:53










  • Seconding @Yuriy's suggestion.
    – Antonio Vargas
    Nov 18 at 3:13










  • My attempt for small x: begin{align} I(x) &= frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}} e^{x cos^2(theta)}dtheta\ &= frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}} dtheta [1 + xcos^2{theta} + order{x^2}]\ &= frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}} dtheta [1 + frac{x}{2}(1+cos{2theta}) + order{x^2}]\ &approx frac{1}{2pi} [pi + frac{x}{2}(pi)] end{align} Therefore, begin{align} &boxed{ I(x)approx frac{1}{2} + frac{x}{4} text{for small $x$}} end{align}
    – niagarajohn
    Nov 18 at 14:53
















  • 1




    For large $x$ check out en.wikipedia.org/wiki/Laplace%27s_method
    – Yuriy S
    Nov 17 at 21:48












  • Thanks @Yuriy. I haven't been exposed to this method, so I think I'd like to solve using asymptotic series. Still working through a few trials, but am interested in easier ways to solve problems like these outside of my current situation.
    – niagarajohn
    Nov 17 at 21:52






  • 1




    Laplace's method is the standard one for such situations, and it's really simple. But of course, you can choose to try some other way
    – Yuriy S
    Nov 17 at 21:53










  • Seconding @Yuriy's suggestion.
    – Antonio Vargas
    Nov 18 at 3:13










  • My attempt for small x: begin{align} I(x) &= frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}} e^{x cos^2(theta)}dtheta\ &= frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}} dtheta [1 + xcos^2{theta} + order{x^2}]\ &= frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}} dtheta [1 + frac{x}{2}(1+cos{2theta}) + order{x^2}]\ &approx frac{1}{2pi} [pi + frac{x}{2}(pi)] end{align} Therefore, begin{align} &boxed{ I(x)approx frac{1}{2} + frac{x}{4} text{for small $x$}} end{align}
    – niagarajohn
    Nov 18 at 14:53










1




1




For large $x$ check out en.wikipedia.org/wiki/Laplace%27s_method
– Yuriy S
Nov 17 at 21:48






For large $x$ check out en.wikipedia.org/wiki/Laplace%27s_method
– Yuriy S
Nov 17 at 21:48














Thanks @Yuriy. I haven't been exposed to this method, so I think I'd like to solve using asymptotic series. Still working through a few trials, but am interested in easier ways to solve problems like these outside of my current situation.
– niagarajohn
Nov 17 at 21:52




Thanks @Yuriy. I haven't been exposed to this method, so I think I'd like to solve using asymptotic series. Still working through a few trials, but am interested in easier ways to solve problems like these outside of my current situation.
– niagarajohn
Nov 17 at 21:52




1




1




Laplace's method is the standard one for such situations, and it's really simple. But of course, you can choose to try some other way
– Yuriy S
Nov 17 at 21:53




Laplace's method is the standard one for such situations, and it's really simple. But of course, you can choose to try some other way
– Yuriy S
Nov 17 at 21:53












Seconding @Yuriy's suggestion.
– Antonio Vargas
Nov 18 at 3:13




Seconding @Yuriy's suggestion.
– Antonio Vargas
Nov 18 at 3:13












My attempt for small x: begin{align} I(x) &= frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}} e^{x cos^2(theta)}dtheta\ &= frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}} dtheta [1 + xcos^2{theta} + order{x^2}]\ &= frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}} dtheta [1 + frac{x}{2}(1+cos{2theta}) + order{x^2}]\ &approx frac{1}{2pi} [pi + frac{x}{2}(pi)] end{align} Therefore, begin{align} &boxed{ I(x)approx frac{1}{2} + frac{x}{4} text{for small $x$}} end{align}
– niagarajohn
Nov 18 at 14:53






My attempt for small x: begin{align} I(x) &= frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}} e^{x cos^2(theta)}dtheta\ &= frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}} dtheta [1 + xcos^2{theta} + order{x^2}]\ &= frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}} dtheta [1 + frac{x}{2}(1+cos{2theta}) + order{x^2}]\ &approx frac{1}{2pi} [pi + frac{x}{2}(pi)] end{align} Therefore, begin{align} &boxed{ I(x)approx frac{1}{2} + frac{x}{4} text{for small $x$}} end{align}
– niagarajohn
Nov 18 at 14:53












2 Answers
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For large $x$, Laplace's method seems like the best option. Or its counterpart, the Watson's lemma.



Let's transform the integral first:



$$ I(x) = frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}}
e^{x cos^2(theta)}dtheta =frac{1}{pi}int_0^{frac{pi}{2}}
e^{x cos^2(theta)}dtheta=frac{e^{x}}{pi} int_0^{frac{pi}{2}}
e^{-x sin^2(theta)}dtheta= \ =frac{e^{x}}{pi} int_0^1
frac{e^{-x s^2}}{sqrt{1-s^2}} d s=frac{e^{x}}{2pi} int_0^1
frac{e^{-x t}}{sqrt{t}sqrt{1-t}} d t$$



Now for the latter integral, the main contribution for $x to infty$ will be given by $t to0$. And the form of it allows us to use the Watson's lemma directly (see the Wikipedia link).



Using the notation from the article, we have:



$$phi(t)=frac{1}{sqrt{t}sqrt{1-t}}=t^{-1/2} g(t)$$



Where $g(t)$ can be expanded into Taylor series around $0$. Additionally, $int_0^1 |phi(t)| dt=pi<infty$. So the lemma conditions are satisfied.



Then we can represent:



$$int_0^1
frac{e^{-x t}}{sqrt{t}sqrt{1-t}} d t asymp sum_{n=0}^infty frac{Gamma(n+1/2) g^{(n)}(0)}{n! ~x^{n+1/2}} $$






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    up vote
    -1
    down vote













    How about this:
    $$I(x)=frac{1}{2pi}int_{-pi/2}^{pi/2}e^{xcos^2theta}dtheta=frac{1}{2pi}int_{-pi/2}^{pi/2}e^{x(1-sin^2theta)}dtheta$$
    $$theta=pi/2-phi$$
    $$dtheta=-dphi$$
    $$I(x)=frac{1}{2pi}int_0^pi e^{xsin^2phi}dphi$$



    EDIT:



    taking the other approach, I have:
    $$I'(x)=frac{1}{2pi}int_{-pi/2}^{pi/2}cos^2theta e^{xcos^2theta}dtheta$$
    then using $u=costheta$ you can obtain:
    $$I'(x)=frac1piint_0^1frac{u^2}{sqrt{1-u^2}}e^{xu^2}du$$
    now using $v=sqrt{1-u^2}$ we can get:
    $$I'(x)=frac1piint_0^1sqrt{1-v^2}e^{x(1-v^2)}dv$$
    which can be re-written as:
    $$I'(x)=frac1piint_0^1sqrt{1-v^2}e^{-(sqrt{x}v)^2}dv$$
    if you continue and differentiate once again you get:
    $$I''(x)=frac{1}{2pi x^2}int_0^xbeta e^beta dbeta$$
    so:
    $$I(x)=frac1{2pi}iintfrac{(x-1)e^x+1}{x^2}dxdx$$






    share|cite|improve this answer



















    • 1




      How does this help find the asymptotic?
      – Yuriy S
      Nov 17 at 21:58










    • You could try to get a term for $I^2(x)$ then use polar coordinates
      – Henry Lee
      Nov 17 at 21:59






    • 1




      I think you are onto something here. I've seen large and small x expansions for the erf function and this could be the trick to get it into that form. Will take a look!
      – niagarajohn
      Nov 17 at 23:26










    • ^That didn't work...
      – niagarajohn
      Nov 18 at 14:56










    • Ok try and find $I’(x)$ then make a substitution, find a term for $I’(x)$ in terms of x then integrate this
      – Henry Lee
      Nov 18 at 15:35











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    2 Answers
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    2 Answers
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    active

    oldest

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    up vote
    1
    down vote













    For large $x$, Laplace's method seems like the best option. Or its counterpart, the Watson's lemma.



    Let's transform the integral first:



    $$ I(x) = frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}}
    e^{x cos^2(theta)}dtheta =frac{1}{pi}int_0^{frac{pi}{2}}
    e^{x cos^2(theta)}dtheta=frac{e^{x}}{pi} int_0^{frac{pi}{2}}
    e^{-x sin^2(theta)}dtheta= \ =frac{e^{x}}{pi} int_0^1
    frac{e^{-x s^2}}{sqrt{1-s^2}} d s=frac{e^{x}}{2pi} int_0^1
    frac{e^{-x t}}{sqrt{t}sqrt{1-t}} d t$$



    Now for the latter integral, the main contribution for $x to infty$ will be given by $t to0$. And the form of it allows us to use the Watson's lemma directly (see the Wikipedia link).



    Using the notation from the article, we have:



    $$phi(t)=frac{1}{sqrt{t}sqrt{1-t}}=t^{-1/2} g(t)$$



    Where $g(t)$ can be expanded into Taylor series around $0$. Additionally, $int_0^1 |phi(t)| dt=pi<infty$. So the lemma conditions are satisfied.



    Then we can represent:



    $$int_0^1
    frac{e^{-x t}}{sqrt{t}sqrt{1-t}} d t asymp sum_{n=0}^infty frac{Gamma(n+1/2) g^{(n)}(0)}{n! ~x^{n+1/2}} $$






    share|cite|improve this answer



























      up vote
      1
      down vote













      For large $x$, Laplace's method seems like the best option. Or its counterpart, the Watson's lemma.



      Let's transform the integral first:



      $$ I(x) = frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}}
      e^{x cos^2(theta)}dtheta =frac{1}{pi}int_0^{frac{pi}{2}}
      e^{x cos^2(theta)}dtheta=frac{e^{x}}{pi} int_0^{frac{pi}{2}}
      e^{-x sin^2(theta)}dtheta= \ =frac{e^{x}}{pi} int_0^1
      frac{e^{-x s^2}}{sqrt{1-s^2}} d s=frac{e^{x}}{2pi} int_0^1
      frac{e^{-x t}}{sqrt{t}sqrt{1-t}} d t$$



      Now for the latter integral, the main contribution for $x to infty$ will be given by $t to0$. And the form of it allows us to use the Watson's lemma directly (see the Wikipedia link).



      Using the notation from the article, we have:



      $$phi(t)=frac{1}{sqrt{t}sqrt{1-t}}=t^{-1/2} g(t)$$



      Where $g(t)$ can be expanded into Taylor series around $0$. Additionally, $int_0^1 |phi(t)| dt=pi<infty$. So the lemma conditions are satisfied.



      Then we can represent:



      $$int_0^1
      frac{e^{-x t}}{sqrt{t}sqrt{1-t}} d t asymp sum_{n=0}^infty frac{Gamma(n+1/2) g^{(n)}(0)}{n! ~x^{n+1/2}} $$






      share|cite|improve this answer

























        up vote
        1
        down vote










        up vote
        1
        down vote









        For large $x$, Laplace's method seems like the best option. Or its counterpart, the Watson's lemma.



        Let's transform the integral first:



        $$ I(x) = frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}}
        e^{x cos^2(theta)}dtheta =frac{1}{pi}int_0^{frac{pi}{2}}
        e^{x cos^2(theta)}dtheta=frac{e^{x}}{pi} int_0^{frac{pi}{2}}
        e^{-x sin^2(theta)}dtheta= \ =frac{e^{x}}{pi} int_0^1
        frac{e^{-x s^2}}{sqrt{1-s^2}} d s=frac{e^{x}}{2pi} int_0^1
        frac{e^{-x t}}{sqrt{t}sqrt{1-t}} d t$$



        Now for the latter integral, the main contribution for $x to infty$ will be given by $t to0$. And the form of it allows us to use the Watson's lemma directly (see the Wikipedia link).



        Using the notation from the article, we have:



        $$phi(t)=frac{1}{sqrt{t}sqrt{1-t}}=t^{-1/2} g(t)$$



        Where $g(t)$ can be expanded into Taylor series around $0$. Additionally, $int_0^1 |phi(t)| dt=pi<infty$. So the lemma conditions are satisfied.



        Then we can represent:



        $$int_0^1
        frac{e^{-x t}}{sqrt{t}sqrt{1-t}} d t asymp sum_{n=0}^infty frac{Gamma(n+1/2) g^{(n)}(0)}{n! ~x^{n+1/2}} $$






        share|cite|improve this answer














        For large $x$, Laplace's method seems like the best option. Or its counterpart, the Watson's lemma.



        Let's transform the integral first:



        $$ I(x) = frac{1}{2pi}int_{-frac{pi}{2}}^{frac{pi}{2}}
        e^{x cos^2(theta)}dtheta =frac{1}{pi}int_0^{frac{pi}{2}}
        e^{x cos^2(theta)}dtheta=frac{e^{x}}{pi} int_0^{frac{pi}{2}}
        e^{-x sin^2(theta)}dtheta= \ =frac{e^{x}}{pi} int_0^1
        frac{e^{-x s^2}}{sqrt{1-s^2}} d s=frac{e^{x}}{2pi} int_0^1
        frac{e^{-x t}}{sqrt{t}sqrt{1-t}} d t$$



        Now for the latter integral, the main contribution for $x to infty$ will be given by $t to0$. And the form of it allows us to use the Watson's lemma directly (see the Wikipedia link).



        Using the notation from the article, we have:



        $$phi(t)=frac{1}{sqrt{t}sqrt{1-t}}=t^{-1/2} g(t)$$



        Where $g(t)$ can be expanded into Taylor series around $0$. Additionally, $int_0^1 |phi(t)| dt=pi<infty$. So the lemma conditions are satisfied.



        Then we can represent:



        $$int_0^1
        frac{e^{-x t}}{sqrt{t}sqrt{1-t}} d t asymp sum_{n=0}^infty frac{Gamma(n+1/2) g^{(n)}(0)}{n! ~x^{n+1/2}} $$







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Nov 19 at 11:31

























        answered Nov 19 at 10:59









        Yuriy S

        15.3k433115




        15.3k433115






















            up vote
            -1
            down vote













            How about this:
            $$I(x)=frac{1}{2pi}int_{-pi/2}^{pi/2}e^{xcos^2theta}dtheta=frac{1}{2pi}int_{-pi/2}^{pi/2}e^{x(1-sin^2theta)}dtheta$$
            $$theta=pi/2-phi$$
            $$dtheta=-dphi$$
            $$I(x)=frac{1}{2pi}int_0^pi e^{xsin^2phi}dphi$$



            EDIT:



            taking the other approach, I have:
            $$I'(x)=frac{1}{2pi}int_{-pi/2}^{pi/2}cos^2theta e^{xcos^2theta}dtheta$$
            then using $u=costheta$ you can obtain:
            $$I'(x)=frac1piint_0^1frac{u^2}{sqrt{1-u^2}}e^{xu^2}du$$
            now using $v=sqrt{1-u^2}$ we can get:
            $$I'(x)=frac1piint_0^1sqrt{1-v^2}e^{x(1-v^2)}dv$$
            which can be re-written as:
            $$I'(x)=frac1piint_0^1sqrt{1-v^2}e^{-(sqrt{x}v)^2}dv$$
            if you continue and differentiate once again you get:
            $$I''(x)=frac{1}{2pi x^2}int_0^xbeta e^beta dbeta$$
            so:
            $$I(x)=frac1{2pi}iintfrac{(x-1)e^x+1}{x^2}dxdx$$






            share|cite|improve this answer



















            • 1




              How does this help find the asymptotic?
              – Yuriy S
              Nov 17 at 21:58










            • You could try to get a term for $I^2(x)$ then use polar coordinates
              – Henry Lee
              Nov 17 at 21:59






            • 1




              I think you are onto something here. I've seen large and small x expansions for the erf function and this could be the trick to get it into that form. Will take a look!
              – niagarajohn
              Nov 17 at 23:26










            • ^That didn't work...
              – niagarajohn
              Nov 18 at 14:56










            • Ok try and find $I’(x)$ then make a substitution, find a term for $I’(x)$ in terms of x then integrate this
              – Henry Lee
              Nov 18 at 15:35















            up vote
            -1
            down vote













            How about this:
            $$I(x)=frac{1}{2pi}int_{-pi/2}^{pi/2}e^{xcos^2theta}dtheta=frac{1}{2pi}int_{-pi/2}^{pi/2}e^{x(1-sin^2theta)}dtheta$$
            $$theta=pi/2-phi$$
            $$dtheta=-dphi$$
            $$I(x)=frac{1}{2pi}int_0^pi e^{xsin^2phi}dphi$$



            EDIT:



            taking the other approach, I have:
            $$I'(x)=frac{1}{2pi}int_{-pi/2}^{pi/2}cos^2theta e^{xcos^2theta}dtheta$$
            then using $u=costheta$ you can obtain:
            $$I'(x)=frac1piint_0^1frac{u^2}{sqrt{1-u^2}}e^{xu^2}du$$
            now using $v=sqrt{1-u^2}$ we can get:
            $$I'(x)=frac1piint_0^1sqrt{1-v^2}e^{x(1-v^2)}dv$$
            which can be re-written as:
            $$I'(x)=frac1piint_0^1sqrt{1-v^2}e^{-(sqrt{x}v)^2}dv$$
            if you continue and differentiate once again you get:
            $$I''(x)=frac{1}{2pi x^2}int_0^xbeta e^beta dbeta$$
            so:
            $$I(x)=frac1{2pi}iintfrac{(x-1)e^x+1}{x^2}dxdx$$






            share|cite|improve this answer



















            • 1




              How does this help find the asymptotic?
              – Yuriy S
              Nov 17 at 21:58










            • You could try to get a term for $I^2(x)$ then use polar coordinates
              – Henry Lee
              Nov 17 at 21:59






            • 1




              I think you are onto something here. I've seen large and small x expansions for the erf function and this could be the trick to get it into that form. Will take a look!
              – niagarajohn
              Nov 17 at 23:26










            • ^That didn't work...
              – niagarajohn
              Nov 18 at 14:56










            • Ok try and find $I’(x)$ then make a substitution, find a term for $I’(x)$ in terms of x then integrate this
              – Henry Lee
              Nov 18 at 15:35













            up vote
            -1
            down vote










            up vote
            -1
            down vote









            How about this:
            $$I(x)=frac{1}{2pi}int_{-pi/2}^{pi/2}e^{xcos^2theta}dtheta=frac{1}{2pi}int_{-pi/2}^{pi/2}e^{x(1-sin^2theta)}dtheta$$
            $$theta=pi/2-phi$$
            $$dtheta=-dphi$$
            $$I(x)=frac{1}{2pi}int_0^pi e^{xsin^2phi}dphi$$



            EDIT:



            taking the other approach, I have:
            $$I'(x)=frac{1}{2pi}int_{-pi/2}^{pi/2}cos^2theta e^{xcos^2theta}dtheta$$
            then using $u=costheta$ you can obtain:
            $$I'(x)=frac1piint_0^1frac{u^2}{sqrt{1-u^2}}e^{xu^2}du$$
            now using $v=sqrt{1-u^2}$ we can get:
            $$I'(x)=frac1piint_0^1sqrt{1-v^2}e^{x(1-v^2)}dv$$
            which can be re-written as:
            $$I'(x)=frac1piint_0^1sqrt{1-v^2}e^{-(sqrt{x}v)^2}dv$$
            if you continue and differentiate once again you get:
            $$I''(x)=frac{1}{2pi x^2}int_0^xbeta e^beta dbeta$$
            so:
            $$I(x)=frac1{2pi}iintfrac{(x-1)e^x+1}{x^2}dxdx$$






            share|cite|improve this answer














            How about this:
            $$I(x)=frac{1}{2pi}int_{-pi/2}^{pi/2}e^{xcos^2theta}dtheta=frac{1}{2pi}int_{-pi/2}^{pi/2}e^{x(1-sin^2theta)}dtheta$$
            $$theta=pi/2-phi$$
            $$dtheta=-dphi$$
            $$I(x)=frac{1}{2pi}int_0^pi e^{xsin^2phi}dphi$$



            EDIT:



            taking the other approach, I have:
            $$I'(x)=frac{1}{2pi}int_{-pi/2}^{pi/2}cos^2theta e^{xcos^2theta}dtheta$$
            then using $u=costheta$ you can obtain:
            $$I'(x)=frac1piint_0^1frac{u^2}{sqrt{1-u^2}}e^{xu^2}du$$
            now using $v=sqrt{1-u^2}$ we can get:
            $$I'(x)=frac1piint_0^1sqrt{1-v^2}e^{x(1-v^2)}dv$$
            which can be re-written as:
            $$I'(x)=frac1piint_0^1sqrt{1-v^2}e^{-(sqrt{x}v)^2}dv$$
            if you continue and differentiate once again you get:
            $$I''(x)=frac{1}{2pi x^2}int_0^xbeta e^beta dbeta$$
            so:
            $$I(x)=frac1{2pi}iintfrac{(x-1)e^x+1}{x^2}dxdx$$







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Nov 18 at 17:08

























            answered Nov 17 at 21:57









            Henry Lee

            1,666118




            1,666118








            • 1




              How does this help find the asymptotic?
              – Yuriy S
              Nov 17 at 21:58










            • You could try to get a term for $I^2(x)$ then use polar coordinates
              – Henry Lee
              Nov 17 at 21:59






            • 1




              I think you are onto something here. I've seen large and small x expansions for the erf function and this could be the trick to get it into that form. Will take a look!
              – niagarajohn
              Nov 17 at 23:26










            • ^That didn't work...
              – niagarajohn
              Nov 18 at 14:56










            • Ok try and find $I’(x)$ then make a substitution, find a term for $I’(x)$ in terms of x then integrate this
              – Henry Lee
              Nov 18 at 15:35














            • 1




              How does this help find the asymptotic?
              – Yuriy S
              Nov 17 at 21:58










            • You could try to get a term for $I^2(x)$ then use polar coordinates
              – Henry Lee
              Nov 17 at 21:59






            • 1




              I think you are onto something here. I've seen large and small x expansions for the erf function and this could be the trick to get it into that form. Will take a look!
              – niagarajohn
              Nov 17 at 23:26










            • ^That didn't work...
              – niagarajohn
              Nov 18 at 14:56










            • Ok try and find $I’(x)$ then make a substitution, find a term for $I’(x)$ in terms of x then integrate this
              – Henry Lee
              Nov 18 at 15:35








            1




            1




            How does this help find the asymptotic?
            – Yuriy S
            Nov 17 at 21:58




            How does this help find the asymptotic?
            – Yuriy S
            Nov 17 at 21:58












            You could try to get a term for $I^2(x)$ then use polar coordinates
            – Henry Lee
            Nov 17 at 21:59




            You could try to get a term for $I^2(x)$ then use polar coordinates
            – Henry Lee
            Nov 17 at 21:59




            1




            1




            I think you are onto something here. I've seen large and small x expansions for the erf function and this could be the trick to get it into that form. Will take a look!
            – niagarajohn
            Nov 17 at 23:26




            I think you are onto something here. I've seen large and small x expansions for the erf function and this could be the trick to get it into that form. Will take a look!
            – niagarajohn
            Nov 17 at 23:26












            ^That didn't work...
            – niagarajohn
            Nov 18 at 14:56




            ^That didn't work...
            – niagarajohn
            Nov 18 at 14:56












            Ok try and find $I’(x)$ then make a substitution, find a term for $I’(x)$ in terms of x then integrate this
            – Henry Lee
            Nov 18 at 15:35




            Ok try and find $I’(x)$ then make a substitution, find a term for $I’(x)$ in terms of x then integrate this
            – Henry Lee
            Nov 18 at 15:35


















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