initial values

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21: 4.45 Methods of Computation

… ►Initial approximations are obtainable, for example, from the power series (4.13.6) (with $t\ge 0$) when $x$ is close to $-1/\mathrm{e}$, from the asymptotic expansion (4.13.10) when $x$ is large, and by numerical integration of the differential equation (4.13.4) (§3.7) for other values of $x$. …

22: 31.16 Mathematical Applications

… ►It describes the monodromy group of Heun’s equation for specific values of the accessory parameter. … ►

31.16.3 $$A_0=\frac{n!}{{\left(\gamma +\delta \right)}_n}{\mathrm{𝐻𝑝}}_{n,m}\left(1\right),Q_0{A}_0+R_0{A}_1=0,$$

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Symbols:

${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\beta$: real or complex parameter, $A_j$: coefficients, $Q_j$ and $R_j$

Referenced by:

§31.16(ii), Erratum (V1.0.21) for Equations (31.16.2) and (31.16.3), Erratum (V1.0.21) for Equations (31.16.2) and (31.16.3)

Permalink:

http://dlmf.nist.gov/31.16.E3

Encodings:

pMML, png, TeX

Correction (effective with 1.0.21):

The initial data $A_0$, previously missing, has now been included.

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

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23: 32.2 Differential Equations

… ►In general the singularities of the solutions are movable in the sense that their location depends on the constants of integration associated with the initial or boundary conditions. … ►For arbitrary values of the parameters $\alpha$, $\beta$, $\gamma$, and $\delta$, the general solutions of ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ are transcendental, that is, they cannot be expressed in closed-form elementary functions. However, for special values of the parameters, equations ${\text{P}}_{\text{II}}$–${\text{P}}_{\text{VI}}$ have special solutions in terms of elementary functions, or special functions defined elsewhere in the DLMF. …

24: Mathematical Introduction

… ►Also, valuable initial advice on all aspects of the project was provided by ten external associate editors. … ►These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). … ►With two real variables, special functions are depicted as 3D surfaces, with vertical height corresponding to the value of the function, and coloring added to emphasize the 3D nature. … ►Special functions with a complex variable are depicted as colored 3D surfaces in a similar way to functions of two real variables, but with the vertical height corresponding to the modulus (absolute value) of the function. … ►05, and the corresponding function values are tabulated to 8 decimal places or 8 significant figures. …

25: Errata

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The constraints in (14.5.3), (14.5.4) on $\nu +\mu$ have been corrected to exclude all negative integers since the Ferrers function of the second kind is not defined for these values.

Reported by Hans Volkmer on 2021-06-02

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The asymptotic results were originally for $\nu$ real valued and $\nu \to +\mathrm{\infty }$. However, these results are also valid for complex values of $\nu$. The maximum sectors of validity are now specified.

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In the line just below (18.15.4), it was previously stated “is less than twice the first neglected term in absolute value.” It now states “is less than twice the first neglected term in absolute value, in which one has to take $\cos {\theta }_{n,m,\mathrm{\ell }}=1$.”

Reported by Gergő Nemes on 2019-02-08

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31.16.2

$xy=a{\sin }^2\theta {\cos }^2\varphi ,$

$(x-1)(y-1)=(1-a){\sin }^2\theta {\sin }^2\varphi ,$

$(x-a)(y-a)=a(a-1){\cos }^2\theta$

31.16.3 $$A_0=\frac{n!}{{\left(\gamma +\delta \right)}_n}{\mathrm{𝐻𝑝}}_{n,m}\left(1\right),Q_0{A}_0+R_0{A}_1=0$$

Originally $x,y$ were incorrectly defined by the set of equations (31.16.2), given previously as “$x={\sin }^2\theta {\cos }^2\varphi ,$ $y={\sin }^2\theta {\sin }^2\varphi$”. In fact, $x,y$ are implicitly defined by the corrected set of equations. In (31.16.3), the initial data $A_0$, previously missing, has now been included.

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Two corrections have been made in this paragraph. First, the correct range of the initial displacement $a$ is . Previously it was $\sqrt{1/\beta }\le |a|\le \sqrt{2/\beta }$. Second, the correct period of the oscillations is $2K\left(k\right)/\sqrt{\eta }$. Previously it was given incorrectly as $4K\left(k\right)/\sqrt{\eta }$.

Reported 2014-05-02 by Svante Janson.

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26: DLMF Project News

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