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Math Help - Contour intergration sinh z/ { z^100 dz }

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    Contour intergration sinh z/ { z^100 dz }

    contour integral of sinh z/ z^100 dz where C1 : |z-1| =5. (this part i understand using Laurent) i achieve 2.pi.j [1 / 99! ]

    hence , or otherwise, show 98th derivative of function sinh z / z w.r.t z evaluated at z = 0 is equal to 1 / 99.

    (but this part i'm not sure how to show the factorial disappear ?? )
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    Quote Originally Posted by Chris0724 View Post
    contour integral of sinh z/ z^100 dz where C1 : |z-1| =5. (this part i understand using Laurent) i achieve 2.pi.j [1 / 99! ]

    hence , or otherwise, show 98th derivative of function sinh z / z w.r.t z evaluated at z = 0 is equal to 1 / 99.

    (but this part i'm not sure how to show the factorial disappear ?? )
    Use the following theorem:

    If f(z) has a pole of order n at z = z_1 then Res(f(z), z = z_1) = \frac{1}{(n-1)!} \lim_{z \to z_1} \frac{d^{n-1}}{dz^{n-1}} \left[(z - z_1)^n f(z)\right].
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