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21: 3.2 Linear Algebra

… ►By repeatedly subtracting multiples of each row from the subsequent rows we obtain a matrix of the form … ►During this reduction process we store the multipliers ${\mathrm{\ell }}_{jk}$ that are used in each column to eliminate other elements in that column. This yields a lower triangular matrix of the form … ►To avoid instability the rows are interchanged at each elimination step in such a way that the absolute value of the element that is used as a divisor, the pivot element, is not less than that of the other available elements in its column. …

22: 4.45 Methods of Computation

… ►The other trigonometric functions can be found from the definitions (4.14.4)–(4.14.7). … ►The inverses $\mathrm{arcsinh}$, $\mathrm{arccosh}$, and $\mathrm{arctanh}$ can be computed from the logarithmic forms given in §4.37(iv), with real arguments. … ►

Other Methods

… ►The trigonometric functions may be computed from the definitions (4.14.1)–(4.14.7), and their inverses from the logarithmic forms in §4.23(iv), followed by (4.23.7)–(4.23.9). … ►For other methods see Miel (1981). …

23: 10.40 Asymptotic Expansions for Large Argument

… ►Corresponding expansions for $I_{\nu }\left(z\right)$, $K_{\nu }\left(z\right)$, $I_{\nu }^{\prime }\left(z\right)$, and $K_{\nu }^{\prime }\left(z\right)$ for other ranges of $\mathrm{ph}z$ are obtainable by combining (10.34.3), (10.34.4), (10.34.6), and their differentiated forms, with (10.40.2) and (10.40.4). …

24: 12.14 The Function $W(a,x)$

… ►In other cases the general theory of (12.2.2) is available. $W(a,x)$ and $W(a,-x)$ form a numerically satisfactory pair of solutions when . … ►

§12.14(vii) Relations to Other Functions

►

Bessel Functions

… ►

Confluent Hypergeometric Functions

…

25: 32.10 Special Function Solutions

… ►For certain combinations of the parameters, ${\text{P}}_{\text{II}}$–${\text{P}}_{\text{VI}}$ have particular solutions expressible in terms of the solution of a Riccati differential equation, which can be solved in terms of special functions defined in other chapters. All solutions of ${\text{P}}_{\text{II}}$–${\text{P}}_{\text{VI}}$ that are expressible in terms of special functions satisfy a first-order equation of the form … ►Solutions for other values of $\alpha$ are derived from $w(z;\pm \frac{1}{2})$ by application of the Bäcklund transformations (32.7.1) and (32.7.2). … ►The solution (32.10.34) is an essentially transcendental function of both constants of integration since ${\text{P}}_{\text{VI}}$ with $\alpha =\beta =\gamma =0$ and $\delta =\frac{1}{2}$ does not admit an algebraic first integral of the form $P(z,w,w^{\prime },C)=0$, with $C$ a constant. …

26: 28.20 Definitions and Basic Properties

… ►with its algebraic form … ►

§28.20(ii) Solutions ${\mathrm{Ce}}_{\nu }$, ${\mathrm{Se}}_{\nu }$, ${\mathrm{Me}}_{\nu }$, ${\mathrm{Fe}}_n$, ${\mathrm{Ge}}_n$

►

28.20.3 $${\mathrm{Ce}}_{\nu }(z,q)={\mathrm{ce}}_{\nu }(\pm \mathrm{i}z,q),$$ $\nu \ne -1,-2,\mathrm{\dots }$,

ⓘ

Defines:

${\mathrm{Ce}}_{\nu }(z,q)$: modified Mathieu function

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

20.6.1 (in slightly different form) 20.6.2 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.20.E3

Encodings:

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See also:

Annotations for §28.20(ii), §28.20 and Ch.28

… ►

28.20.5 $${\mathrm{Me}}_{\nu }(z,q)={\mathrm{me}}_{\nu }(-\mathrm{i}z,q),$$

ⓘ

Defines:

${\mathrm{Me}}_{\nu }(z,q)$: modified Mathieu function

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

20.6.1 (in slightly different form) 20.6.2 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.20.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(ii), §28.20 and Ch.28

… ►For other values of $z$, $h$, and $\nu$ the functions ${\mathrm{M}}_{\nu }^{(j)}(z,h)$, $j=1,2,3,4$, are determined by analytic continuation. …

27: 13.27 Mathematical Applications

… ►The elements of this group are of the form …The other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions. …

28: 18.9 Recurrence Relations and Derivatives

… ►

First Form

… ►For the other classical OP’s see Table 18.9.1; compare also §18.2(iv). … ►

Second Form

… ►For the other classical OP’s see Table 18.9.2. …

29: 2.6 Distributional Methods

… ►This leads to integrals of the form … ►The distribution method outlined here can be extended readily to functions $f(t)$ having an asymptotic expansion of the form … ►It is easily seen that $K^{+}$ forms a commutative, associative linear algebra. … ►For proofs and other examples, see McClure and Wong (1979) and Wong (1989, Chapter 6). … ►On inserting this identity into (2.6.54), we immediately encounter divergent integrals of the form …

30: 33.23 Methods of Computation

… ►Use of extended-precision arithmetic increases the radial range that yields accurate results, but eventually other methods must be employed, for example, the asymptotic expansions of §§33.11 and 33.21. … ►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). …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). … ►

§33.23(vi) Other Numerical Methods

►Curtis (1964a, §10) describes the use of series, radial integration, and other methods to generate the tables listed in §33.24. …