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Many approximate practical calculations, particularly on the Earth-Moon-Sun system, were done using series expansions involving thousands of algebraic terms. … In 1912 Karl Sundman did however find an infinite series that could in principle be summed to give the solution—but which converges exceptionally slowly.
How the Discoveries in This Chapter Were Made
This chapter —and the last —have described a series of surprising discoveries that I have made about what simple programs typically do.
Known methods for high-precision evaluation of special functions—usually based in the end on series representations—typically require of order n 1/s m[n] operations, where s is often 2 or 3. (Examples of more difficult cases include HypergeometricPFQ[a, b, 1] and StieltjesGamma[k] , where logarithmic series can require an exponential number of terms.
Embryo development
Starting from a single egg cell, embryos first exhibit a series of geometrically quite precise cell divisions corresponding, I suspect, to a simple neighbor-independent substitution system.
Given the functional equation one can find a power series for g[x] for any a . The series has an accumulation of poles on the circle Abs[a] 2 1 ; the coefficient of x m turns out to have denominator
2^(m - DigitCount[m, 2, 1]) Apply[Times, Table[Cyclotomic[s, a]^Floor[(m - 1)/s], {s, m - 1}]]
For other iterated maps general formulas also seem rare.
, yielding a series that formally diverges. … (The high-order terms often seem to be associated with asymptotic series for things like Exp[-1/ α ] .)
I am not sure whether in my approach one should expect an infinite series of progressively more massive particles.
As a increases, the repetition period goes through a series of doublings.
The same basic setup also applies to spectra associated with linear filters and ARMA time series processes (see page 1083 ), in which elements in a sequence are generated from external random noise by forming linear combinations of the noise with definite configurations of elements in the sequence.
When problems are originally stated as differential equations, results in terms of integrals ("quadrature") are sometimes considered exact solutions—as occasionally are convergent series.