Function of exponential type

Lidstone series

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

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 An 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 An(z) is a polynomial in z of degree n, Ck 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),


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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