Integrating ∫ 0 + ∞ e − s t cos ⁡ ( a t ) d t using the complex exponentialOther similar questions I have found used integration by part of alternative complicated transformations that I do not understand. Using the cos ⁡ ( x ) = 1 2 ∗ ( e i x + e − i x ) identity, is there a way to find the solution, which should be a / ( a 2 + s 2 )?

Name: Integrating ∫ 0 + ∞ e − s t cos ⁡ ( a t ) d t using the complex exponentialOther similar questions I have found used integration by part of alternative complicated transformations that I do not understand. Using the cos ⁡ ( x ) = 1 2 ∗ ( e i x + e − i x ) identity, is there a way to find the solution, which should be a / ( a 2 + s 2 )?
Uploaded: 2022-02-23T17:22:57+00:00
Description: Integrating ∫ 0 + ∞ e − s t cos ⁡ ( a t ) d t using the complex exponentialOther similar questions I have found used integration by part of alternative complicated transformations that I do not understand. Using the cos ⁡ ( x ) = 1 2 ∗ ( e i x + e − i x ) identity, is there a way to find the solution, which should be a / ( a 2 + s 2 )?

Question

Integrating       ∫          0              +      ∞            e          −      s      t        cos  ⁡  (  a  t  )    d  t using the complex exponentialOther similar questions I have found used integration by part of alternative complicated transformations that I do not understand. Using the   cos  ⁡  (  x  )  =      1    2    ∗  (      e          i      x        +      e          −      i      x        ) identity, is there a way to find the solution, which should be   a      /    (      a    2    +      s    2    )?

Milton Hampton · Accepted Answer

For all complex number z,  cos  ⁡  (  z  )  =                    e                  i          z                    +              e                  −          i          z                      2    .It follows that:                                 ∫                      0                                +            ∞                                    e                      −            s            t                          cos        ⁡        (        a        t        )                d        t                            =                                  1          2                          ∫                      0                                +            ∞                                                [                                    e                      −            s            t                                    e                      i            a            t                          +                  e                      −            s            t                                    e                      −            i            a            t                                                ]                                  d        t                                          =                  1          2                          ∫                      0                                +            ∞                                    e                      −            (            s            −            i            a            )            t                                  d        t        +                  1          2                          ∫                      0                                +            ∞                                    e                      −            (            s            +            i            a            )            t                                  d        t        .            Now, if z is a non-zero complex number,                    (1)                              ∫                      0                                +            ∞                                    e                      −            z            t                                  d        t        =                              [                          −                  1          z                          e                      −            z            t                                                              ]                                            t            =            0                                t            =            +            ∞                          =                  1          z                .            Therefore, with   z  =  s  −  i  a:                              ∫                      0                                +            ∞                                    e                      −            s            t                          cos        ⁡        (        a        t        )                d        t                            =                                  1                      2            z                          +                  1                      2                          z              ¯                                                                        =                              z            +                          z              ¯                                            2            z                          z              ¯                                                                        =                              ℜ            (            z            )                                |            z                          |              2                                                                        =                  s                                    s              2                        +                          a              2                                      .            Note that (1) is true if s&amp;gt;0 because       |          e              −        z        t              |    =          e              −        ℜ        (        z        )        t              =          e              −        s        t                  →        0   as   t  →  +  ∞

Nathalie Case · Accepted Answer

∫          0                      +            ∞                                    e                      −            s            t                          cos        ⁡        (        a        t        )                          d                t                            =                              [                          e                              −                s                t                                                                    sin                ⁡                (                a                t                )                            a                        ]                                t            =            0                                t            =            +            ∞                          +                  s          a                          ∫          0                      +            ∞                                    e                      −            s            t                          sin        ⁡        (        a        t        )                          d                t                                          =                              [            −                          s                              a                2                                                    e                              −                s                t                                      cos            ⁡            (            s            t            )            ]                                t            =            0                                t            =            +            ∞                          −                              s            2                                a            2                                    ∫          0                      +            ∞                                    e                      −            s            t                          cos        ⁡        (        a        t        )                          d                t                                          =                  s                      a            2                          −                              s            2                                a            2                                    ∫          0                      +            ∞                                    e                      −            s            t                          cos        ⁡        (        a        t        )                          d                t            and therefore      ∫    0          +      ∞            e          −      s      t        cos  ⁡  (  a  t  )        d    t  =            s              a        2                    1      +                        s          2                          a          2                      =      s                  s        2            +              a        2              .

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