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Suppose you have a function with double poles somewhere on the complex plane. Are there complex analysis techniques that can be used to split the double pole into two single isolated poles?

Some example functions might be

f(z) = 1/(z - 2)^2

or

g(z) = csc(pi*z)*csc(2*pi*z) = 1/sin(pi*z)*1/sin(2*pi*z)

of the complex variable z = a + bi.

The 2nd function g(z) has doubles at all the integers, and single poles at all points 1/2 + n and 1/2 - n where n is an integer 1, 2, 3...etc.

Are there any techniques that can allow us to split the double poles at the integers of the 2nd function into isolated singularities off the real axis, while leaving the single poles on the real axis?

Inquisitively,

Edwin G. Schasteen

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# Complex Analysis Question

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