asymptotic expansion in a sentence
1) There are always many different functions with any given asymptotic expansion .
2) High ambiguity of functions given by asymptotic expansions .
3) Bernoulli numbers are also frequently used in other kinds of asymptotic expansions .
4) Observations in 2000 indicate infinite expansion, but theoretical elegance argues for asymptotic expansion .
asymptotic expansion example sentences5) Boundary-layer theory is amenable to the method of matched asymptotic expansions for deriving approximate solutions.
6) Detailed understanding of non-linear thread behavior requires the use of asymptotic expansions to generate the appropriate scaling behavior.
7) This is a typical example of the fact that Borel summation will sometimes "correctly" sum divergent asymptotic expansions .
8) This function has an asymptotic expansion as "z" tends to 0 that is given by the original divergent series.
9) They introduced the small parameter method, fixed points, integral invariants, variational equations, the convergence of the asymptotic expansions .
10) We used calculus of variations to find the optimal velocity field and solved the resulting nonlinear Euler-Lagrange equations both numerically and analytically (using matched asymptotic expansion ).
11) The process is iterative in that the result obtained by performing the method once can be used as input when the method is repeated, to obtain as many terms in the asymptotic expansion as desired.
12) Fate of the universe . In the first few decades after 2000, humans will learn the fate of the universe: collapse, infinite expansion, or asymptotic expansion .
13) These have an asymptotic expansion for small positive "t", which can be used to evaluate the limit as "t" tends to 0, giving a proof of the Atiyah–Singer index theorem.
14) Singular perturbation theory can, however, be applied here and this amounts to 'zooming in' at the boundaries (using the method of matched asymptotic expansions ).
example sentences with asymptotic15) In mathematics, it is commonly used to describe how closely a finite series approximates a given function, especially in the case of a truncated Taylor series or asymptotic expansion .
16) asymptotic series, otherwise asymptotic expansions, are infinite series whose partial sums become good approximations in the limit of some point of the domain.
17) A more accurate approximation can be obtained by using more terms from the asymptotic expansions of ln(Γ("z")) and Γ("z"), which are based on Stirling's approximation.
18) The following example is the classical Poincaré-type asymptotic expansion of the
19) has been studied by Srinivasa Ramanujan and has asymptotic expansion :
20) asymptotic expansion : Approximation of functions generalizing Taylor's formula
21) For the modified Bessel functions, Hankel developed asymptotic expansions as well:
22) Hardy and Ramanujan obtained an asymptotic expansion with this approximation as the first term:
23) Substituting the asymptotic expansion for each of the cosines into the spherical relation for a right triangle yields
24) By expressing the Maclaurin series for the cosine function as an asymptotic expansion with the remainder term in big O notation,
25) The constant term of the asymptotic expansion does not depend on "f": it is necessarily the same value given by analytic continuation, −1/12.
26) Finally, using an integral estimate such as Euler–MacLaurin summation, or the asymptotic expansion of the "n"th harmonic number, we obtain
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