# Thread: Contour intergration sinh z/ { z^100 dz }

1. ## 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 ?? )

2. Originally Posted by Chris0724
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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