# representations by Euler–Maclaurin formula

(0.003 seconds)

## 9 matching pages

##### 3: Errata
• Expansion

§4.13 has been enlarged. The Lambert $W$-function is multi-valued and we use the notation $W_{k}\left(x\right)$, $k\in\mathbb{Z}$, for the branches. The original two solutions are identified via $\operatorname{Wp}\left(x\right)=W_{0}\left(x\right)$ and $\operatorname{Wm}\left(x\right)=W_{\pm 1}\left(x\mp 0\mathrm{i}\right)$.

Other changes are the introduction of the Wright $\omega$-function and tree $T$-function in (4.13.1_2) and (4.13.1_3), simplification formulas (4.13.3_1) and (4.13.3_2), explicit representation (4.13.4_1) for $\frac{{\mathrm{d}}^{n}W}{{\mathrm{d}z}^{n}}$, additional Maclaurin series (4.13.5_1) and (4.13.5_2), an explicit expansion about the branch point at $z=-{\mathrm{e}}^{-1}$ in (4.13.9_1), extending the number of terms in asymptotic expansions (4.13.10) and (4.13.11), and including several integrals and integral representations for Lambert $W$-functions in the end of the section.

• Equation (27.14.2)
27.14.2 $\mathit{f}\left(x\right)=\prod_{m=1}^{\infty}(1-x^{m})=\left(x;x\right)_{% \infty},$ $|x|<1$

The representation in terms of $\left(x;x\right)_{\infty}$ was added to this equation.

• Paragraph Inversion Formula (in §35.2)

The wording was changed to make the integration variable more apparent.

• Equations (15.6.1)–(15.6.9)

The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$, previously omitted from the left-hand sides to make the formulas more concise, has been added. In Equations (15.6.1)–(15.6.5), (15.6.7)–(15.6.9), the constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ has been added. In (15.6.6), the constraint $|\operatorname{ph}\left(-z\right)|<\pi$ has been added. In Section 15.6 Integral Representations, the sentence immediately following (15.6.9), “These representations are valid when $|\operatorname{ph}\left(1-z\right)|<\pi$, except (15.6.6) which holds for $|\operatorname{ph}\left(-z\right)|<\pi$.”, has been removed.

• Equation (5.11.8)

It was reported by Nico Temme on 2015-02-28 that the asymptotic formula for $\operatorname{Ln}\Gamma\left(z+h\right)$ is valid for $h$ $(\in\mathbb{C})$; originally it was unnecessarily restricted to $[0,1]$.

• ##### 4: Bibliography E
• G. P. Egorychev (1984) Integral Representation and the Computation of Combinatorial Sums. Translations of Mathematical Monographs, Vol. 59, American Mathematical Society, Providence, RI.
• D. Elliott (1998) The Euler-Maclaurin formula revisited. J. Austral. Math. Soc. Ser. B 40 (E), pp. E27–E76 (electronic).
• T. Estermann (1959) On the representations of a number as a sum of three squares. Proc. London Math. Soc. (3) 9, pp. 575–594.
• L. Euler (1768) Institutiones Calculi Integralis. Opera Omnia (1), Vol. 11, pp. 110–113.
• J. A. Ewell (1990) A new series representation for $\zeta(3)$ . Amer. Math. Monthly 97 (3), pp. 219–220.
• ##### 5: Bibliography B
• Á. Baricz and T. K. Pogány (2013) Integral representations and summations of the modified Struve function. Acta Math. Hungar. 141 (3), pp. 254–281.
• W. Barrett (1981) Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.
• B. C. Berndt (1975a) Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (4), pp. 413–445.
• B. C. Berndt (1975b) Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.
• P. L. Butzer and M. Hauss (1992) Riemann zeta function: Rapidly converging series and integral representations. Appl. Math. Lett. 5 (2), pp. 83–88.
where …
##### 7: Bibliography F
• P. Flajolet and B. Salvy (1998) Euler sums and contour integral representations. Experiment. Math. 7 (1), pp. 15–35.
• J. P. M. Flude (1998) The Edmonds asymptotic formulas for the $3j$ and $6j$ symbols. J. Math. Phys. 39 (7), pp. 3906–3915.
• W. B. Ford (1960) Studies on Divergent Series and Summability & The Asymptotic Developments of Functions Defined by Maclaurin Series. Chelsea Publishing Co., New York.
• G. Freud (1976) On the coefficients in the recursion formulae of orthogonal polynomials. Proc. Roy. Irish Acad. Sect. A 76 (1), pp. 1–6.
• ##### 8: 13.29 Methods of Computation
Although the Maclaurin series expansion (13.2.2) converges for all finite values of $z$, it is cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. … For $U\left(a,b,z\right)$ and $W_{\kappa,\mu}\left(z\right)$ we may integrate along outward rays from the origin in the sectors $\tfrac{1}{2}\pi<|\operatorname{ph}z|<\tfrac{3}{2}\pi$, with initial values obtained from connection formulas in §13.2(vii), §13.14(vii). …
###### §13.29(iii) Integral Representations
The integral representations (13.4.1) and (13.4.4) can be used to compute the Kummer functions, and (13.16.1) and (13.16.5) for the Whittaker functions. …
13.29.8 $w(n)\sim\frac{\sqrt{\pi}e^{\frac{1}{2}z}z^{\frac{1}{4}(4a-2b+1)}}{\Gamma\left(% a\right)\Gamma\left(a+1-b\right)}n^{\frac{1}{4}(4a-2b-3)}e^{-2\sqrt{nz}},$
##### 9: 18.17 Integrals
###### Legendre
For formulas for Jacobi and Laguerre polynomials analogous to (18.17.5) and (18.17.6), see Koornwinder (1974, 1977). … and three formulas similar to (18.17.9)–(18.17.11) by symmetry; compare the second row in Table 18.6.1. …