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1: Customize DLMF

…You can customize the appearance and functionality of the DLMF site using these selections. …

… ►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). … ►Nevertheless, it is quite possible that errors remain and we would appreciate knowing about them, so that we can make the necessary corrections. …

3: 27.20 Methods of Computation: Other Number-Theoretic Functions

… ►The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function

p\left(n\right)

for

n<N

. … ►A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function

\tau\left(n\right)

, and the values can be checked by the congruence (27.14.20). …

4: 17.19 Software

… ►Citations in the bulleted list refer to papers for which research software has been made available and can be downloaded via the Web. …

5: Need Help?

… ►Alas, sometimes things go wrong; Perhaps some of these topics can help. … ►

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Links to definitions, keywords, annotations and other interesting information can be found in the Info boxes by clicking or hovering the mouse over the icon next to each formula, table, figure, and section heading.

6: 29.20 Methods of Computation

… ►The normalization of Lamé functions given in §29.3(v) can be carried out by quadrature (§3.5). … ►Initial approximations to the eigenvalues can be found, for example, from the asymptotic expansions supplied in §29.7(i). Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. … ►Zeros of Lamé polynomials can be computed by solving the system of equations (29.12.13) by employing Newton’s method; see §3.8(ii). Alternatively, the zeros can be found by locating the maximum of function

g

in (29.12.11).

7: 21.11 Software

… ►Citations in the bulleted list refer to papers for which research software has been made available and can be downloaded via the Web. …

8: 27.15 Chinese Remainder Theorem

… ►Most of the calculation can be done with five-digit integers as follows. …By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod

m_{1}

), (mod

m_{2}

), (mod

m_{3}

), and (mod

m_{4}

), respectively. Because each residue has no more than five digits, the arithmetic can be performed efficiently on these residues with respect to each of the moduli, yielding answers

a_{1}\pmod{m_{1}}

a_{2}\pmod{m_{2}}

a_{3}\pmod{m_{3}}

, and

a_{4}\pmod{m_{4}}

, where each

a_{j}

has no more than five digits. … ►Details of a machine program describing the method together with typical numerical results can be found in Newman (1967). …

9: 36.15 Methods of Computation

… ►Close to the origin

\mathbf{x}=\boldsymbol{{0}}

of parameter space, the series in §36.8 can be used. … ►Close to the bifurcation set but far from

\mathbf{x}=\boldsymbol{{0}}

, the uniform asymptotic approximations of §36.12 can be used. … ►Direct numerical evaluation can be carried out along a contour that runs along the segment of the real

t

-axis containing all real critical points of

\Phi

and is deformed outside this range so as to reach infinity along the asymptotic valleys of

\exp\left(i\Phi\right)

. … ►This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of

\Phi

, with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …

10: 5.21 Methods of Computation

… ►Or we can use forward recurrence, with an initial value obtained e. …For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3). …