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In mathematics, certain types of entire functions can be expressed as a certain polynomial expansion known as the Lidstone series.## References

Let f(z) be an entire function of exponential type less than (N + 1)π, as defined below. Then f(z) can be expanded in terms of polynomials A_{n} as follows:

- $f(z)=sum\_\{n=0\}^infty\; left[A\_n(1-z)\; f^\{(2n)\}(0)\; +\; A\_n(z)\; f^\{(2n)\}(1)\; right]\; +\; sum\_\{k=1\}^N\; C\_k\; sin\; (kpi\; z)$.

Here A_{n}(z) is a polynomial in z of degree n, C_{k} a constant, and f^{(n)}(a) the derivative of f at a.

A function is said to be of exponential type of less than t if the function

- $h(theta;\; f)\; =\; lim\; sup\; frac\{1\}\{r\}\; log\; |f(r\; e^\{itheta\})|,$

is bounded above by t. Thus,the constant N used in the summation above is given by

- $t=\; lim\; sup\; h(theta;\; f),$

with

- $Npi\; leq\; t\; <\; (N+1)pi.,$

- Ralph P. Boas, Jr. and C. Creighton Buck, Polynomial Expansions of Analytic Functions, (1964) Academic Press, NY. ISBN 3-540-03123-5

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Last updated on Monday April 23, 2007 at 23:07:19 PDT (GMT -0700)

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