Analytic Continuation - Hadamard's Gap Theorem

Hadamard's Gap Theorem

For a power series

with coefficients mostly zero in the precise sense that they vanish outside a sequence of exponents k(i) with

\lim_{i\to\infty} \frac{k(i+1)}{k(i)} > 1 + \delta \,

for some fixed δ > 0, the circle centre z0 and with radius the radius of convergence is a natural boundary. Such a power series defines a lacunary function.

Read more about this topic:  Analytic Continuation

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