I ran into some trouble understanding how my professor went from this step in his solution: ∫ − π π | 1 2 π e i m x − 1 2 π e i n x | 2 d x ∫ − π π | 1 2 π e i m x − 1 2 π e i n x | 2 d xto this step: ∫ − π π 1 2 π ( 1 − e i ( m − n ) x − e i ( n − m ) x + 1 ) d xI cannot see how this was done, please help!

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I ran into some trouble understanding how my professor went from this step in his solution:      ∫          −      π        π              |                        1                      2            π                                    e                      i            m            x                          −                  1                      2            π                                    e                      i            n            x                              |        2        d    x      ∫          −      π        π              |                        1                      2            π                                    e                      i            m            x                          −                  1                      2            π                                    e                      i            n            x                              |        2        d    xto this step:      ∫          −      π              π            1          2      π            (    1    −          e              i        (        m        −        n        )        x              −          e              i        (        n        −        m        )        x              +    1    )        d    xI cannot see how this was done, please help!

kamphundg4 · Accepted Answer

I would recommend using the complex conjugate to calculate the squared absolute value.      ∫          −      π        π              |                        1                      2            π                                    e                      i            m            x                          −                  1                      2            π                                    e                      i            n            x                              |        2        d    x      ∫          −      π        π        (                  1                  2          π                            e                  i          m          x                    −              1                  2          π                            e                  i          n          x                      )        (                  1                  2          π                            e                  −          i          m          x                    −              1                  2          π                            e                  −          i          n          x                      )        d    x      ∫          −      π        π        (                  1                  2          π                    −              1                  2          π                            e                  i          (          n          −          m          )          x                      −                  1                  2          π                            e                  i          (          m          −          n          )          x                    +              1                  2          π                      )        d    x      ∫          −      π              π            1          2      π            (    1    −          e              i        (        m        −        n        )        x              −          e              i        (        n        −        m        )        x              +    1    )        d    x

I am in a Fourier Transforms course, and I ran into some trouble understanding how my professor went from this step in his solution: int_(-pi)^pi |(1)/(sqrt(2pi))e^(imx)-(1)/(sqrt(2pi))e^(i nx)|^2 dx

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