Calculus of residue

Question

i try to do the following

Assuming[k\[Element] Integers,Residue[Cos[x]/x^(k+1),{x,0}]]

but any respond how to calculate for any k the k derivate of Cos function thanks

Answer 1

You could use SeriesCoefficient:

res[k_] = Assuming[
    k ∈ Integers && k > 0,
    FullSimplify @ SeriesCoefficient[x^(-1-k) Cos[x], {x, 0, -1}]
]

Cos[(k π)/2]/k!

Check:

res[Range[10]]

Table[Residue[x^(-1-k) Cos[x], {x, 0}], {k, 10}]

{0, -(1/2), 0, 1/24, 0, -(1/720), 0, 1/40320, 0, -(1/3628800)}

{0, -(1/2), 0, 1/24, 0, -(1/720), 0, 1/40320, 0, -(1/3628800)}