which will not be solvable with regular power series methods if either p(z)/z or q(z)/z2 are not analytic at z = 0.
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Given that the function is analytic within each of these quarters, a nonzero winding number N (always an integer) identifies N zeros of the function inside the quarter in question by Rouché's theorem, each zero counted as many times as its multiplicity.
In the 1990s, Alex Wilkie showed that one has the same result if instead of adding every analytic function, one just adds the exponential function to R to get the ordered real field with exponentiation, Rexp, a result known as Wilkie's theorem.
1930s: British mathematician Mary Cartwright proved her theorem, now known as Cartwright's theorem, which gives an estimate for the maximum modulus of an analytic function that takes the same value no more than p times in the unit disc.
From a mathematical view the Z-transform can also be viewed as a Laurent series where one views the sequence of numbers under consideration as the (Laurent) expansion of an analytic function.