Cauchy-Hadamard theorem

In mathematics, the Cauchy-Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard.

tatement of the theorem

Consider the formal power series in one complex variable z of the form

:$f(z)\; =\; sum\_\{n\; =\; 0\}^\{infty\}\; c\_\{n\}\; (z-a)^\{n\}.$

where $a,c\_ninmathbb\{C\}.$

Then the radius of convergence of "f" is given by

:

where the limsup is defined as

:$limsup\_\{n\; oinfty\}\; u\_n:=lim\_\{n\; oinfty\}sup\{u\_k:kgeq\; n\}$

where the sup is the least upper bound of a set.

Note:

: $frac\{1\}\{0\}\; :=\; infty$

for this theorem (and vice versa).

External links

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