Compute using residuals the integral of the following function over the positively oriented circle | z | = 3 f ( z ) = e − z z 2 My solution: The only singular point of f in | z | ≤ 3 is z = 0 (double pole) and its remainder is therefore Res z = 0 ⁡ f ( z ) = lim z → 0 1 ( 2 − 1 ) ! ( e − z z 2 z 2 ) ′ = lim z → 0 − e − z = − 1 Consequently, ∫ | z = 3 | f ( z ) = 2 π i Res z = 0 ⁡ f ( z ) = − 2 π i . this right?

Question

Compute using residuals the integral of the following function over the positively oriented circle   |  z  |  =  3  f     (  z  )  =                    e                  −          z                            z        2            My solution: The only singular point of   f in       |      z    |    ≤  3 is   z  =  0 (double pole) and its remainder is therefore      Res          z      =      0        ⁡  f  (  z  )  =            lim              z        →        0                            1                  (          2          −          1          )          !                                      (                                                                      e                                      −                    z                                                                    z                  2                                                            z                2                                              )                ′            =                        lim                      z            →            0                          −                  e                      −            z                          =        −        1            Consequently,             ∫                    |        z        =        3              |              f    (    z    )    =    2    π    i          Res              z        =        0              ⁡    f    (    z    )    =    −    2    π    i    .  this right?

RamPatWeese2w · Accepted Answer

That is correct, although it is simpler, in order to compute the residue, to note that            e              −        z                    z      2        =            1      −      z      +              1                  2          !                            z        2            −              1                  3          !                            z        3            +      ⋯              z      2        =      z          −      2        −      z          −      1        +      1          2      !        −      1          3      !        z  +  ⋯and that therefore, by definition,      res          z      =      0        ⁡      (                  e                  −          z                            z        2              )    =  −  1.

Compute using residuals the integral of the following function over the positively oriented circle |z|=3, f(z)=(e^-z)/(z^2)

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