central differences in imaginary direction

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21: 33.23 Methods of Computation

… ►When numerical values of the Coulomb functions are available for some radii, their values for other radii may be obtained by direct numerical integration of equations (33.2.1) or (33.14.1), provided that the integration is carried out in a stable direction (§3.7). Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii. On the other hand, the irregular solutions of §§33.2(iii) and 33.14(iii) need to be integrated in the direction of decreasing radii beginning, for example, with values obtained from asymptotic expansions (§§33.11 and 33.21). … ►In a similar manner to §33.23(iii) the recurrence relations of §§33.4 or 33.17 can be used for a range of values of the integer $\mathrm{\ell }$, provided that the recurrence is carried out in a stable direction (§3.6). … ►Bardin et al. (1972) describes ten different methods for the calculation of $F_{\mathrm{\ell }}$ and $G_{\mathrm{\ell }}$, valid in different regions of the ($\eta ,\rho$)-plane. …

22: 3.6 Linear Difference Equations

§3.6 Linear Difference Equations

… ►where $\mathrm{\Delta }{w}_{n-1}=w_n-w_{n-1}$, ${\mathrm{\Delta }}^2{w}_{n-1}=\mathrm{\Delta }{w}_n-\mathrm{\Delta }{w}_{n-1}$, and $n\in \mathbb{Z}$. … ► … ►The difference equation … …

23: 3.3 Interpolation

… ►

§3.3(iii) Divided Differences

… ►Explicitly, the divided difference of order $n$ is given by …If $f$ is analytic in a simply-connected domain $D$, then for $z\in D$, … ►This represents the Lagrange interpolation polynomial in terms of divided differences: … ►Then by using $x_3$ in Newton’s interpolation formula, evaluating $\left[x_0,x_1,x_2,x_3\right]f=-\mathrm{0.26608\hspace{0.33em}28233}$ and recomputing $f^{\prime }(x)$, another application of Newton’s rule with starting value $x_3$ gives the approximation $x=\mathrm{2.33810\hspace{0.33em}7373}$, with 8 correct digits. …

24: 2.11 Remainder Terms; Stokes Phenomenon

… ►Two different asymptotic expansions in terms of elementary functions, (2.11.6) and (2.11.7), are available for the generalized exponential integral in the sector . … ►On the other hand, when $\pi +\delta \le \theta \le 3\pi -\delta$, $c(\theta )$ is in the left half-plane and $\mathrm{erfc}\left(\sqrt{\frac{1}{2}\rho }c(\theta )\right)$ differs from 2 by an exponentially-small quantity. … ►We now compute the forward differences ${\mathrm{\Delta }}^j$, $j=0,1,2,\mathrm{\dots }$, of the moduli of the rounded values of the first 6 neglected terms: …Multiplying these differences by ${(-1)}^j{2}^{-j-1}$ and summing, we obtain … ►However, direct numerical transformations need to be used with care. …

25: Simon Ruijsenaars

… ► 1948 in Hillegom, The Netherlands) is a Professor of Mathematical Physics at the University of Leeds. His main research interests cover integrable systems, special functions, analytic difference equations, classical and quantum mechanics, and the relations between these areas. ►Ruijsenaars served as a Validator for the original release and publication in May 2010 of the NIST Digital Library of Mathematical Functions and the NIST Handbook of Mathematical Functions. ►In November 2015, Ruijsenaars was named Senior Associate Editor of the DLMF and Associate Editor of the following DLMF Chapters …

26: 29.20 Methods of Computation

… ►The eigenvalues $a_{\nu }^m\left(k^2\right)$, $b_{\nu }^m\left(k^2\right)$, and the Lamé functions ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$, can be calculated by direct numerical methods applied to the differential equation (29.2.1); see §3.7. The normalization of Lamé functions given in §29.3(v) can be carried out by quadrature (§3.5). … ►These matrices are the same as those provided in §29.15(i) for the computation of Lamé polynomials with the difference that $n$ has to be chosen sufficiently large. …The numerical computations described in Jansen (1977) are based in part upon this method. … ►The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. …

27: Qiming Wang

… ► 1943 in Shanghai, China) received a degree in Computational Mathematics from Tsinghua University, Beijing, in 1966. She started to work for NIST in 1990 and was on the staff of the Visualization and Usability Group in the Information Access Division of the Information Technology Laboratory in the National Institute of Standards and Technology when she retired in March, 2008. … ►She has applied VRML and X3D techniques to several different fields including interactive mathematical function visualization, 3D human body modeling, and manufacturing-related modeling. …

28: Daniel W. Lozier

… ► 1941 in Portland, Oregon) was the Group Leader of the Mathematical Software Group in the Applied and Computational Mathematics Division of NIST until his retirement in 2013. … ►Lozier received a degree in mathematics from Oregon State University in 1962 and his Ph. …in applied mathematics from the University of Maryland, College Park, in 1979. …Army Engineer Research and Development Laboratory in Virginia on finite-difference solutions of differential equations associated with nuclear weapons effects. Then he transferred to NIST (then known as the National Bureau of Standards), where he collaborated for several years with the Building and Fire Research Laboratory developing and applying finite-difference and spectral methods to differential equation models of fire growth. …

29: Possible Errors in DLMF

Possible Errors in DLMF

►We have strived for the utmost in correctness, clarity and accuracy, having validated each chapter with both internal and external experts. ►One source of confusion, rather than actual errors, are some new functions which differ from those in Abramowitz and Stegun (1964) by scaling, shifts or constraints on the domain; see the Info box (click or hover over the icon) for links to defining formula. There are also cases where browser bugs or poor fonts can be misleading; you can verify MathML display by comparing the to the images or TeXfound under Encodings in the Info boxes (see About MathML). Errors in the printed Handbook may already have been corrected in the online version; please consult Errata. …

30: 2.9 Difference Equations

§2.9 Difference Equations

… ►or equivalently the second-order homogeneous linear difference equation …in which $\mathrm{\Delta }$ is the forward difference operator (§3.6(i)). … ►For analogous results for difference equations of the form … ►