general values

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31: 12.15 Generalized Parabolic Cylinder Functions

§12.15 Generalized Parabolic Cylinder Functions

… ►can be viewed as a generalization of (12.2.4). This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function. …

32: 32.8 Rational Solutions

… ► ${\text{P}}_{\text{II}}$–${\text{P}}_{\text{VI}}$ possess hierarchies of rational solutions for special values of the parameters which are generated from “seed solutions” using the Bäcklund transformations and often can be expressed in the form of determinants. …

33: 16.3 Derivatives and Contiguous Functions

… ►Two generalized hypergeometric functions ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ are (generalized) contiguous if they have the same pair of values of $p$ and $q$, and corresponding parameters differ by integers. …

34: 28.12 Definitions and Basic Properties

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§28.12(i) Eigenvalues ${\lambda }_{\nu +2n}\left(q\right)$

… ►Without loss of generality, from now on we replace $\nu +2n$ by $\nu$. … ►As in §28.7 values of $q$ for which (28.2.16) has simple roots $\lambda$ are called normal values with respect to $\nu$. … ►If $q$ is a normal value of the corresponding equation (28.2.16), then these functions are uniquely determined as analytic functions of $z$ and $q$ by the normalization …

35: 28.13 Graphics

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§28.13(i) Eigenvalues ${\lambda }_{\nu }\left(q\right)$ for General $\nu$

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36: 32.2 Differential Equations

… ►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. …

37: 10.15 Derivatives with Respect to Order

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Noninteger Values of $\nu$

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10.15.1 $$\frac{\partial J_{\pm \nu }\left(z\right)}{\partial \nu }=\pm J_{\pm \nu }\left(z\right)\ln \left(\frac{1}{2}z\right)\mp {(\frac{1}{2}z)}^{\pm \nu }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{\psi \left(k+1\pm \nu \right)}{\mathrm{\Gamma }\left(k+1\pm \nu \right)}\frac{{(\frac{1}{4}{z}^2)}^k}{k!},$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.64

Referenced by:

§10.15, §10.38, Erratum (V1.0.17) for Equations (10.15.1), (10.38.1)

Permalink:

http://dlmf.nist.gov/10.15.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation has been generalized to include the additional case of $\partial J_{-\nu }\left(z\right)/\partial \nu$ because it will help the user who wants to combine it with (10.15.2).

See also:

Annotations for §10.15, §10.15 and Ch.10

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Integer Values of $\nu$

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Half-Integer Values of $\nu$

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38: 3.10 Continued Fractions

… ►For special functions see §5.10 (gamma function), §7.9 (error function), §8.9 (incomplete gamma functions), §8.17(v) (incomplete beta function), §8.19(vii) (generalized exponential integral), §§10.10 and 10.33 (quotients of Bessel functions), §13.6 (quotients of confluent hypergeometric functions), §13.19 (quotients of Whittaker functions), and §15.7 (quotients of hypergeometric functions). … ►To achieve a prescribed accuracy, either a priori knowledge is needed of the value of $n$, or $n$ is determined by trial and error. In general this algorithm is more stable than the forward algorithm; see Jones and Thron (1974). …

39: 18.2 General Orthogonal Polynomials

§18.2 General Orthogonal Polynomials

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Orthogonality on General Sets

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Generalizations of the Szegő Class

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40: 10.74 Methods of Computation

… ►The power-series expansions given in §§10.2 and 10.8, together with the connection formulas of §10.4, can be used to compute the Bessel and Hankel functions when the argument $x$ or $z$ is sufficiently small in absolute value. … ►For large positive real values of $\nu$ the uniform asymptotic expansions of §§10.20(i) and 10.20(ii) can be used. … ►If values of the Bessel functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, or the other functions treated in this chapter, are needed for integer-spaced ranges of values of the order $\nu$, then a simple and powerful procedure is provided by recurrence relations typified by the first of (10.6.1). … ►

§10.74(vi) Zeros and Associated Values

… ►Necessary values of the first derivatives of the functions are obtained by the use of (10.6.2), for example. …