Sentence examples for expansion of the functions from inspiring English sources
After a suitable expansion of the functions, the governing equations are transformed into two coupled ordinary differential equations, and they are solved by a general eigenvalue solution procedure.
In addition, there is no need for additional equations for treating the boundary conditions, because the boundary conditions (15) are imposed at the two collocation points (x^{(alpha,beta)}_{L,N,0}) and (x^{(alpha,beta)}_{L,N,N}) in the expansion of the functions.
In this case, it is sufficient to consider the potential at time n = 0. Thanks to the bounds (53), one can make a series expansion of the functions log and log ( 1 − π ) and rewrite the potential under the form of the expansion: (57).
Proof We start by considering the Hermite expansion of the functions G i and write G i ( X 1 ( t ), X 2 ′ ( t ) ) = lim Q → ∞ ∑ q = 1 Q ∑ k 1 + k 2 = q c k 1 k 2, G i H k 1 ( X 1 ( t ) ) H k 2 ( X 2 ′ ( t ) ) (52).
The primary objectives of this paper are as follows: (A) We introduce a new set of formulas to be calculated at the tangent and cotangent factor nodes, to better approximate the mean and variance of the function values by utilizing the first-order TS expansion of the functions, so that the Gaussianity assumption still holds.
Note that the Taylor expansion of the function in (72) is of a real function in real parameters, even if the likelihood function is for complex quantities.
The following result is a theorem on the recurrence relation of coefficients in the series expansion of the function (ln (1-2x^{2}/15-px^{6})).
where is the transform of the output sequence, is the transform of the input sequence, and denotes the expression, which results from the power series expansion of the function.
On the other hand, recalling the power-series expansion of the function e x, we have another expression for the Estrada index of G as follows: E E ( G ) = ∑ k = 0 ∞ M k ( G ) k !
