List all the harmonic conjugates of the following rational function. The integration is almost impossible, Symbolab and Microsoft Math fails to arrive at the answer: μ ( x , y ) = x 2 + x + y 2 x 2 + y 2

Question

List all the harmonic conjugates of the following rational function. The integration is almost impossible, Symbolab and Microsoft Math fails to arrive at the answer:  μ  (  x  ,  y  )  =                    x        2            +      x      +              y        2                            x        2            +              y        2

Philip Deleon · Accepted Answer

Look at                    (1)                    μ        (        x        ,        y        )        =                                                            x                2                            +              x              +                              y                2                                                                    x                2                            +                              y                2                                                        in polar coordinates:                    (2)                    μ        (        x        ,        y        )        =                                                            x                2                            +              x              +                              y                2                                                                    x                2                            +                              y                2                                                    =                                                            r                2                            +              r              cos              ⁡              θ                                      r              2                                      =        1        +                                            cos              ⁡              θ                        r                          ;            with                    (3)                    z        =        x        +        i        y        =        r        cos        ⁡        θ        +        i        r        sin        ⁡        θ        =        r                  e                      i            θ                              we have                    (4)                              z                      −            1                          =                  r                      −            1                                    e                      −            i            θ                          =                                            cos              ⁡              θ              −              i              sin              ⁡              θ                        r                          =                                            cos              ⁡              θ                        r                          −        i                                            sin              ⁡              θ                        r                              is holomorphic on       C    ∖  {  0  }; it follows that, as long as we stay away from   r  =  0, where   μ is not in any event defined, that the harmonic conjugate of   cos  ⁡  θ      /    r is   −  sin  ⁡  θ      /    r, up to an addidive real constant. In   x-  y coordinates, we have                    (5)                    −                                            sin              ⁡              θ                        r                          =        −                                            r              sin              ⁡              θ                                      r              2                                      =        −                              y                                          x                2                            +                              y                2                                                    .            Thus, every harmonic conjugate of   μ  (  x  ,  y  ) is of the form                    (6)                    α        −                              y                                          x                2                            +                              y                2                                                    ,              α  ∈      R   a constant, in agreement with the comments of Thomas Andrews.

List all the harmonic conjugates of the following rational function. The integration is almost impos

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