Charlemagne expanded the function of these meetings to make them an instrument for cementing the king's personal ties with counts, bishops, abbots, and powerful magnates.
Her work -- which included setting up a cooperative breeding program for tamarins among zoos worldwide and making minute observations of decades of pandas' social, sexual and gastronomic lives -- helped expand the function of the modern zoo from mere exhibition to concerted, scientifically informed conservation.
Thus, we can expand the function f ^ λ with the aid of Whittaker-Shannon-Kotel'nikov theorem.
Compared with the traditional oscilloscope, the designed oscilloscope not only expanded the function, simplified the structure, but also improved economical efficiency.
That is, elements of a cellular network and a WSN are mixed in order to easily expand the function of each network.
They expanded the function in q-analogues of Lidstone polynomials which are in fact q-Bernoulli polynomials as in the classical case (see Section 2).
In order to show that the one-periodic wave (3.7) degenerates to the one-soliton solution (2.12) under the limit (rhorightarrow0), we expand the function f in the form begin{aligned} f=1+rho^{2}bigl(e^{2pixi}+e^{-2pipixi}bigrgr)+ rho^{8}bigl(e^{4pixi}+e^{-4pipixi}bigrgr)+cdots.
In order to verify that the one-periodic wave (3.6) tends to the one-soliton solution (2.3) when (rhorightarrow0), we expand the function f in the form begin{aligned} f=1+rho^{2}bigl(ie^{2pi ixi}-ixi}-ie^{-2pi bigr -rho^{8}bigl(e^{4pi ixi }+e^{-4pi ixi}bigr -rho^{8}bigl
Assuming that the assumption (i) is fulfilled we expand the function f by the double Fourier series f varphi,z =sum_{n=0}^{infty}sum _{m=0}^{infty} alpha_{mn} cos mvarphi+ beta_{mn}sin mvarphi cosgamma_{n}z,quad varphi,z in overline{S}, (54) where begin{aligned}& left.
Therefore the roots x ℓ ( b ) are simple and | Im x ℓ ( b ) | → ∞ as b → ∞ for all ℓ = 1, … K. Let us expand the function x k ( Q ( x ) - i b ) - 1 in a linear combination of the functions ( x - x ℓ ) - 1 and of the constant term 1 (for k = K ).
