Let a&gt;0 and A&gt;0, I am looking for the decay rate of the integral ∫ M ∞ + ∫ − ∞ − M x 2 exp ⁡ ( − ( x − a ) 2 ) 1 + A a − 1 exp ⁡ ( − x 2 / ( 1 + a 2 ) ) d x as a → 0 There is no closed form answer for the integral. I have tried on Matlab that it should converge to zero much faster than power growth. I think the growth should be exponential types. Do we have some literature discussing this kind of issue? Thanks!I have successfully obtained the growth rate of ∫ M ∞ + ∫ − ∞ − M x 2 exp ⁡ ( − ( x − a ) 2 ) 1 + A a − 1 exp ⁡ ( − x 2 / ( 1 + a 2 ) ) d x as a → 0 be expanding the denominator in power series.But I do not know to deal with the integral in [−M,M].

Question

Let a&amp;gt;0 and A&amp;gt;0, I am looking for the decay rate of the integral      ∫          M        ∞    +      ∫          −      ∞              −      M                          x        2            exp      ⁡      (      −      (      x      −      a              )        2            )              1      +      A              a                  −          1                    exp      ⁡      (      −              x        2                    /            (      1      +              a        2            )      )        d  x         as           a      →      0      There is no closed form answer for the integral. I have tried on Matlab that it should converge to zero much faster than power growth. I think the growth should be exponential types. Do we have some literature discussing this kind of issue? Thanks!I have successfully obtained the growth rate of      ∫          M        ∞    +      ∫          −      ∞              −      M                          x        2            exp      ⁡      (      −      (      x      −      a              )        2            )              1      +      A              a                  −          1                    exp      ⁡      (      −              x        2                    /            (      1      +              a        2            )      )        d  x         as           a      →      0      be expanding the denominator in power series.But I do not know to deal with the integral in [−M,M].

Mario Monroe · Accepted Answer

Let us assume   a  ≪  1. Approximating the function (or simply plotting it), we can see that the maximum of the integrand is at       |    x      |    ≥  1 (in fact it is       |    x      |    =  1 for   A  ≪  a and becomes larger when increasing A).Because if this we can approximate   1  +      a    2    ≈  1 and   (  x  −  a      )    2    ≈      x    2   in the exponents of the integrand to obtain the leading order behavior.Denoting the integral by I(a,A), we obtain  I  (  a  ,  A  )  ≃      I    0    (  a  ,  A  )  =      ∫          −      ∞        ∞                                x          2                exp        ⁡        (        −                  x          2                )                    1        +        A                  a                      −            1                          exp        ⁡        (        −                  x          2                )              d  x  =  −                    π            a              2      A            Li          3              /            2        ⁡  (  −  A      /    a  )    .Here, Lis is the polylogarithm function. In fact, I0 is an excellent approximation to I for   a  ≲  0.1The polylogarithm function has a known asymptotic expansions in terms of logx. In particular, we have   x  ≫  1      Li    s    ⁡  (  −  x  )  ∼  −                    log                  s                    ⁡      (      x      )              Γ      (      s      )                  (        1  +  O  (      log          −      2        ⁡  (  x  )  )            )          .As a result, we obtain  I  (  a  ,  A  )  ≃      I    0    (  a  ,  A  )  ∼            2      a              3      A            log          3              /            2        ⁡  (  A      /    a  )    .Note that this is not a rigorous mathematical answer. What remains is to show that   I  −      I    0   is small for   a  →  0. Numerically, it seems to hold that   I  −      I    0    =  o  (      a    2    )

Answered question

Answer & Explanation

New Questions in Algebra I