Computation of ∫ − π π e i n θ − e i ( n − 1 ) θ ∣ sin ⁡ θ ∣ d θ .

Question

Computation of       ∫          −      π        π                      e                  i          n          θ                    −              e                  i          (          n          −          1          )          θ                            ∣      sin      ⁡              θ            ∣        d  θ  .

Cara Cannon · Accepted Answer

It seems to me that      lim          θ      →              π        −                  (          e              i        n        θ              −          e              i        (        n        −        1        )        θ              )    =      lim          θ      →              π        −                  e          i      n      θ        (  1  −      e          −      i      θ        )  =  2      e          i      n      π        ≠  0So the integral diverges because      lim          θ      →              π        −                                |            sin      ⁡      θ              |                    (      π      −      θ      )        =  1So the integrand looks like            2              e                  i          n          θ                            π      −      θ      As   θ  →      π    −

revistasbibxjm87 · Accepted Answer

A possible integration strategy could be derived from putting                                    e                                  i                        n                        θ                          −                  e                                  i                                      (                              n                −                1                            )                                    θ                          =                  e                                  i                                      (                              n                −                                                      1                    2                                                  +                                                      1                    2                                                              )                                    θ                          −                  e                                  i                                      (                              n                −                                                      1                    2                                                  −                                                      1                    2                                                              )                                    θ                          =                                  =                  e                                  i                                      (                              n                −                                                      1                    2                                                              )                                    θ                                    (                                    e                                              i                                                                      1                    2                                                                  θ                                      −                          e                                              −                i                                                                      1                    2                                                                  θ                                              )                =        2                i                          e                                  i                                      (                              2                n                −                1                            )                                                      θ                2                                                                sin        ⁡                  (                                                    θ                2                                              )                    Thereby reaching to:                          I        =                  ∫                      −            π                    π                                                                                      e                                                          i                                        n                                        θ                                                  −                                  e                                                          i                                                              (                                              n                        −                        1                                            )                                                            θ                                                                                                |                                      sin                    ⁡                    θ                                    |                                                                        d          θ                =                  ∫                      −            π                    π                                                                    2                                i                                                  e                                                          i                                                              (                                              2                        n                        −                        1                                            )                                                                                      θ                        2                                                                                                                sin                ⁡                                  (                                                                                    θ                        2                                                                              )                                                                              |                                      2                    sin                    ⁡                                          (                                                                                                    θ                            2                                                                                              )                                        cos                    ⁡                                          (                                                                                                    θ                            2                                                                                              )                                                        |                                                                        d          θ                =                                  =        2                i                          ∫                      −            π                          /                        2                                π                          /                        2                                                                                                e                                                          i                                                              (                                              2                        n                        −                        1                                            )                                        β                                                                      sin                ⁡                β                                                              |                                      sin                    ⁡                    β                    cos                    ⁡                    β                                    |                                                                        d          β                =        2                i                          ∫                      −            π                          /                        2                                π                          /                        2                                                                                                e                                                          i                                                              (                                              2                        n                        −                        1                                            )                                        β                                                                                        s                  i                  g                  n                                                  (                                      sin                    ⁡                    β                                    )                                                            cos                ⁡                β                                                        d          β                =                                  =        2                i                          (                                    ∫              0                              π                                  /                                2                                                                                                                                e                                                                      i                                                                          (                                                      2                            n                            −                            1                                                    )                                                β                                                                                                                          cos                    ⁡                    β                                                                            d              β                        −                          ∫                              −                π                                  /                                2                            0                                                                                                                  e                                                                      i                                                                          (                                                      2                            n                            −                            1                                                    )                                                β                                                                                                                          cos                    ⁡                    β                                                                            d              β                                )                =                                  =        2                i                          (                                    ∫              0                              π                                  /                                2                                                                                                                                e                                                                      i                                                                          (                                                      2                            n                            −                            1                                                    )                                                β                                                                                                                          cos                    ⁡                    β                                                                            d              β                        +                          ∫              0                              −                π                                  /                                2                                                                                                                                e                                                                      i                                                                          (                                                      2                            n                            −                            1                                                    )                                                β                                                                                                                          cos                    ⁡                    β                                                                            d              β                                )                =                                  =        2                i                          (                                    ∫              0                              π                                  /                                2                                                                                                                                e                                                                      i                                                                          (                                                      2                            n                            −                            1                                                    )                                                β                                                                                                                          cos                    ⁡                    β                                                                            d              β                        −                          ∫              0                              π                                  /                                2                                                                                                                                e                                                                      −                        i                                                                          (                                                      2                            n                            −                            1                                                    )                                                β                                                                                                                          cos                    ⁡                    β                                                                            d              β                                )                =                                  =        −        4                  ∫          0                      π                          /                        2                                                                              sin                ⁡                                  (                                                            (                                              2                        n                        −                        1                                            )                                        β                                    )                                                            cos                ⁡                β                                                        d          β                    From here various methods can be applied, e.g. :                          I        =        −        4                  ∫          0                      π                          /                        2                                                                              sin                ⁡                                  (                                                            (                                              2                        n                        −                        1                                            )                                        β                                    )                                                            cos                ⁡                β                                                        d          β                =                                  =        −        4                  ∫          0                      π                          /                        2                                                                              sin                ⁡                                  (                                      2                    n                    β                                    )                                cos                ⁡                                  (                  β                  )                                −                cos                ⁡                                  (                                      2                    n                    β                                    )                                sin                ⁡                                  (                  β                  )                                                            cos                ⁡                β                                                        d          β                =                                  =        −        4                  ∫          0                      π                          /                        2                                    sin          ⁡                      (                          2              n              β                        )                              d          β                −        4                  ∫                      β            =            0                                π                          /                        2                                                                              cos                ⁡                                  (                                      2                    n                    β                                    )                                                            cos                ⁡                β                                                        d          cos          ⁡          β                    from which you see that you end with expressions that contain the term   ln  ⁡      (          cos      ⁡      β        )   thus confirming the answer above that the integral does not converge.

Computation of <msubsup> ∫ <mrow class="MJX-TeXAtom-ORD"> &#x2

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