Parametrize the contours of integration and write the integrals in terms of the parametrizations. Do not calculate them. ∫ ( ¯ z ) z 3 d zwhere Γis the arc of the circle of radius 2 centered at the origin with initial point at 1+i and terminal point at 1-i that lies in the right half-plane and is transversed once.I asked a question similar to this last night and felt that everyone was very helpful. I'm hoping to see the solution to this problem as I feel completely lost.

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Parametrize the contours of integration and write the integrals in terms of the parametrizations. Do not calculate them.  ∫                              (          ¯                    z      )              z      3        d  zwhere  Γis the arc of the circle of radius      2  centered at the origin with initial point at 1+i and terminal point at 1-i that lies in the right half-plane and is transversed once.I asked a question similar to this last night and felt that everyone was very helpful. I&#039;m hoping to see the solution to this problem as I feel completely lost.

dalennauf5q69 · Accepted Answer

The circle of radius 2 centered at the origin is parametrized by   z  =      2        e          i      θ      ,   0  ≤  θ  ≤  2  π. However, we want just the arc   Γ of the circle. So the range of   θ will be restricted. In polar form,   1  +  i  =      2        e          i      π              /            4       and   1  −  i  =      2        e          −      i      π              /            4      . So   θ ranges from   −  π      /    4 to   π      /    4. Further, since   Γ is traversed from   1  +  i to   1  −  i, it follows that   Γ is oriented clockwise. Thus      ∫          Γ                          z        ¯                    z      3          d  z  =  −      ∫          −      π              /            4              π              /            4                          2                    e                  −          i          θ                                    2                  3                      /                    2                            e                  3          i          θ                      i      2        e          i      θ          d  θ  =  −      i          2            ∫          −      π              /            4              π              /            4            e          −      3      i      θ          d  θ  .

Parametrize the contours of integration and write the integrals in terms of the parametrizations. Do

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