Solve this integral using the Residue Theorem
$\oint_{C} \frac{e^{z}}{(z + 2) (z - 2)} d z$

Question

Solve this integral using the Residue Theorem      ∮    C              e      z              (      z      +      2      )      (      z      −      2      )        d  z

Marlee Norman · Accepted Answer

Notice that            e      z              (      z      +      2      )      (      z      −      2      )      has simple poles at -2 and 2 The contour C encloses those poles, since       |    −  2      |    =      |    2      |    &amp;lt;      |    3      |    . Therefore, you can use the residue theorem and conclude that      ∮    C              e      z              (      z      +      2      )      (      z      −      2      )              d    z  =  2  π      i        [    Res    ⁡          (      −      2      ,                        e          z                          (          z          +          2          )          (          z          −          2          )                    )        +    Res    ⁡          (      2      ,                        e          z                          (          z          +          2          )          (          z          −          2          )                    )        ]    =  2  π      i        [          lim              z        →        −        2              (    z    +    2    )                  e        z                    (        z        +        2        )        (        z        −        2        )              +          lim              z        →        2              (    z    −    2    )                  e        z                    (        z        +        2        )        (        z        −        2        )              ]    .You should be able to do the rest.

Solve this integral using the Residue Theorem ∮ C </msub>

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