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Euler-Maclaurin expansions for integrals with endpoint singularities: A new perspective. (English) Zbl 1081.65003

This paper provides a new and important asymptotic expansion for the discretization error associated with applying an $m$-panel offset trapezoidal rule

${\stackrel{˜}{T}}_{m}\left[f;\theta \right]=\frac{1}{m}\sum _{i=0}^{m-1}f\left(\frac{i+\theta }{m}\right)$

to approximate the integral $I\left[f\right]={\int }_{0}^{1}f\left(x\right)\phantom{\rule{0.166667em}{0ex}}dx$. A minor reformulation of the classical Euler-Maclaurin summation formula leads to the familiar result that when $f\in {C}^{\infty }\left[0,1\right]$,

$\stackrel{˜}{T}\left[f;\theta \right]-I\left[f\right]\sim \sum _{s=1}^{\infty }{b}_{s}/{m}^{s},$

where ${b}_{s}$ depends on $f$ and on $\theta$ but not on $m$. The reader, identifying the step length as $h=1/m$, will identify this as a basis for extrapolation quadrature, such as Romberg integration and its many variants.

During the last half century, this formula has been extended to many functions $f\left(x\right)$ having end-point singularities by various authors, principally Navot, Lyness, Ninham, Monegato, and Sidi. At present, expansions of this general nature have been shown to exist for integrands of the form

$f\left(x\right)={x}^{\gamma }\left(1-{x}^{\delta }\right)P\left(logx\right)Q\left(log\left(1-x\right)\right)g\left(x\right)$

with $g\left(x\right)\in {C}^{\infty }\left[0,1\right]$ and $P$ and $Q$ polynomials and general $\gamma$, $\delta$.

In this more general case, the discretization error expansion includes, in addition to the terms of form ${b}_{s}/{m}^{s}$, terms of the form ${a}_{s+\gamma }{log}^{m}/{m}^{s+\gamma }$ for all positive integer $s$ and for all nonnegative integers $p$ up to the degree of $P$ together with corresponding terms involving $\delta$ and $Q$. In cases where the integral diverges, and $\gamma$ and $\delta$ are not negative integers, the theory replaces the indeterminate quantity in a natural way with the Hadamard finite part integral.

In the present paper the author has generalized almost all these results to a function with more sophisticated singularities at the end point. To illustrate this, suppose $P$ is a constant so no logarithmic terms appear. The singularity of $f\left(x\right)={x}^{\gamma }g\left(x\right)$ with $g\left(x\right)\in {C}^{\infty }\left[0,1\right]$ at $x=0$ is of the form $f\left(x\right)\sim {\sum }_{s=0}^{\infty }{C}_{s}{x}^{\gamma +s}$ as $x\to 0+$ and one set of terms in the resulting expansion is ${a}_{\gamma +s}/{m}^{\gamma +s}$. The generalization established in this paper allows functions

$f\left(x\right)\sim \sum _{s=0}^{\infty }{C}_{s}{x}^{{\gamma }_{s}}\phantom{\rule{1.em}{0ex}}\text{as}\phantom{\rule{1.em}{0ex}}x\to 0+,$

where the sequence of distinct complex number ${\gamma }_{s}$ satisfies

$\text{Re}\phantom{\rule{0.166667em}{0ex}}{\gamma }_{s+1}\ge \text{Re}\phantom{\rule{0.166667em}{0ex}}{\gamma }_{s};\phantom{\rule{1.em}{0ex}}{\gamma }_{s}\notin \text{negative}\phantom{\rule{4.pt}{0ex}}\text{integer};\phantom{\rule{1.em}{0ex}}\underset{s\to \infty }{lim}\phantom{\rule{0.166667em}{0ex}}\text{Re}\phantom{\rule{0.166667em}{0ex}}{\gamma }_{s+1}=\infty ·$

Terms in the expansion may now include all of ${a}_{{\gamma }_{s}}/{m}^{{\gamma }_{s}}$

With the possible exception of integrands involving negative integer exponents $\gamma$ or $\delta$, this new result is an ‘umbrella’ result and almost all previous results known to this reviewer are special cases of this.

A casual reader might imagine that integrand functions such as these occur only rarely. However, an immediate application is to the analysis of Atkinson’s transformation for integration over the surface of a sphere, and other smooth surfaces in ${ℝ}^{3}$.

Two points stand out when reviewing the totality of the results covered here. The first is that, while the proofs are long and cumbersome, the resulting expansions are very straightforward in character. They can easily be committed to memory. The second is that all the mathematics required to establish these results is over one hundred years old. Only the most elementary properties of asymptotic expansions are required. A reader from the first decade of the twentieth century would appreciate the judicious use of neutralizer functions, would admire the author’s dexterity, and would understand the proof and the results with little difficulty. No modern theory seems to be required at all.

This in no way detracts from the intrinsic value of this work. This referee believes that this is a major contribution to extrapolation quadrature and has a useful role to play in practical procedures in scientific computation.

##### MSC:
 65B15 Euler-Maclaurin formula (numerical analysis) 30E15 Asymptotic representations in the complex domain 40A25 Approximation to limiting values (summation of series, etc.) 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 65D32 Quadrature and cubature formulas (numerical methods) 41A55 Approximate quadratures
##### References:
 [1] Apostol, T.M.: Mathematical Analysis. Addison–Wesley, London, 1957 [2] Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration. Academic Press, New York, second edition, 1984 [3] Lyness, J.N.: Finite-part integrals and the Euler–Maclaurin expansion. In R.V.M. Zahar, editor, Approximation and Computation, number 119 in ISNM, pages 397–407, Boston–Basel–Berlin, 1994. Birkhäuser Verlag [4] Lyness, J.N., Ninham, B.W.: Numerical quadrature and asymptotic expansions. Math. Comp. 21, 162–178 (1967) · doi:10.1090/S0025-5718-1967-0225488-X [5] Monegato, G., Lyness, J.N.: The Euler–Maclaurin expansion and finite-part integrals. Numer. Math. 81, 273–291 (1998) · Zbl 0932.41027 · doi:10.1007/s002110050392 [6] Navot, I.: An extension of the Euler–Maclaurin summation formula to functions with a branch singularity. J. Math. and Phys. 40, 271–276 (1961) [7] Navot, I.: A further extension of the Euler–Maclaurin summation formula. J. Math. and Phys. 41, 155–163 (1962) [8] Ninham, B.W.: Generalised functions and divergent integrals. Numer. Math. 8, 444–457 (1966) · Zbl 0143.38701 · doi:10.1007/BF02166670 [9] Sidi, A.: Practical Extrapolation Methods: Theory and Applications. Number 10 in Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, 2003 [10] Steffensen, J.F.: Interpolation. Chelsea, New York, 1950 [11] Verlinden, P.: Cubature formulas and asymptotic expansions. PhD thesis, Katholieke Universiteit Leuven, 1993. Supervised by A. Haegemans