Calculate ∫ − ∞ ∞ d x 1 + x 4 x sin ⁡ [ n π ϕ ( x ) ] ; ϕ ( x ) ∝ ∫ 0 x d y 1 + y 4

Question

Calculate       ∫          −      ∞              ∞                               d        x                    1        +                  x                      4                                     x         sin    ⁡    [    n    π    ϕ    (    x    )    ]       ;   ϕ  (  x  )  ∝      ∫          0              x                  d      y              1      +              y                  4

kissmystarzlfq70 · Accepted Answer

The asymptotics of   I  (  n  )  =      ∫          −      1        1    x  (  ϕ  )  sin  ⁡  n  π  ϕ    d  ϕ depends on the behavior of   x  (  ϕ  ) at   ϕ  →  ±  1 (simple poles, considering   x  (  ϕ  ) analytically continued). Observe that   ϕ  (  x  )  +  ϕ  (  1      /    x  )  =  1, hence   x  (  ϕ  )  x  (  1  −  ϕ  )  =  1. Thus the residues of   x  (  ϕ  ) at   ϕ  ∈  {  ±  1  } are equal to   −  1      /    α, as well as of  y  (  ϕ  )  =      π          2      α        tan  ⁡            π      ϕ        2    ,so that   x  (  ϕ  )  −  y  (  ϕ  ) is regular at   ϕ  ∈  {  ±  1  }. This gives, as   n  →  ∞  I  (  n  )  ≍      ∫          −      1        1    y  (  ϕ  )  sin  ⁡  n  π  ϕ    d  ϕ  =      π    α    (  −  1      )          n      −      1        .The behavior of   B  (  m  ,  n  ) may be analysed similarly, using      π    2        ∫          −      1        1    tan  ⁡            π      ϕ        2      cos  ⁡      (    m    +          1      2        )    π  ϕ    sin  ⁡  n  π  ϕ    d  ϕ  =  (  −  1      )          m      +      n      −      1        (      S          m      ,      n        +      S          m      ,      n      −      1        )  ,        S          m      ,      n        =      ∑          k      =      −      n        n        1          2      m      +      2      k      +      1        .

Calculate <msubsup> ∫ <mrow class="MJX-TeXAtom-ORD"> −<!--

Answered question

Answer & Explanation

New Questions in Integral Calculus