Sentence examples for expansion formula for from inspiring English sources
These vectors are important to get the spectral expansion formula for this BVP.
An asymptotic Euler Maclaurin formula, by which we mean an asymptotic expansion formula for Riemann sums over lattice polytopes, was first obtained by Guillemin and Sternberg (2007) [11].
According to the partial fraction expansion formula for the cotangent function ([14], Chapter 5, Section 2) we know that varphi(x)=2+frac{x}{x+1}+sum_{n=2}^{infty}biggl(frac {x}{x-n}+frac {x}{x+n} biggr).
This follows from an asymptotic expansion formula for (rho ) obtained by Lee and Melrose [10]: begin{aligned} rho (z)=r z)Big (a_0 z)+sum _{j=1}^infty a_j (r^{n+1} log (-r))^jBig ), end{aligned} (1.9 where (rin C^infty (overline{D})) is any defining function for D and (a_jin C^infty (overline{D})) and (a_0 z)>0) on (partial D).
We are now ready to seek an expansion formula for (u^{epsilon } t, x,y)) with respect to ϵ of the form begin{aligned}& u^{epsilon } t,x,y)=u_{0} t,x,y)+epsilon u_{1} t,x,y)+r^{epsilon } t,x,y), end{aligned} (4.1) where (u_{0}) and (u_{1}) are smooth functions, which will be constructed further, and (r^{epsilon }) is the remainder term.
In fact, we obtain a more general polynomial expansion formula for translation invariant Minkowski valuations when the arguments are Minkowski sums of zonoids.
In this paper, we present three new expansion formulas for the generalized Hurwitz-Lerch zeta function.
In particular, it greatly prompted establishments of many expansion formulas for calculating the ranks of matrices and their operations, and these rank formulas, as demonstrated below, now are widely used in matrix theory and applications.
Alternate derivations of the expansion formulae for wave structure interaction problems are obtained in case of water of infinite depth and utilized to analyze the hydroelastic behavior of large floating structures.
By the Neumann expansion formula, can be expressed by (3.5).
