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1: Bibliography D

… ►

Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung. Birkhäuser Verlag, Basel und Stuttgart (German).

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External Links:

MathReview (I. I. Hirschman), ZentralBlatt

Referenced by:

§2.5(i).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions for oblate spheroidal functions I: Positive separation parameter $\lambda$ . Proc. Roy. Soc. Edinburgh Sect. A 121 (3-4), pp. 303–320.

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External Links:

ISSN 0308-2105, MathReview (Archibald D. MacGillivray), ZentralBlatt

Referenced by:

§30.9(ii).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions for oblate spheroidal functions II: Negative separation parameter $\lambda$ . Proc. Roy. Soc. Edinburgh Sect. A 125 (4), pp. 719–737.

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External Links:

ISSN 0308-2105, MathReview (Archibald D. MacGillivray), ZentralBlatt

Referenced by:

§30.9(ii).

Encodings:

BibTeX

…

2: Bibliography H

… ►

Coulomb Wave Functions. In Handbuch der Physik, Bd. 41/1, S. Flügge (Ed.), pp. 408–465.

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External Links:

MathReview (G. Källén)

Referenced by:

§33.2(iii), §33.23(vii), §33.23(vii), §33.24, §33.5(ii), §33.7, Ch.33.

Encodings:

BibTeX

…

3: Bibliography G

… ►

Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est a points critiques fixes. Acta Math. 33 (1), pp. 1–55.

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External Links:

ISSN 0001-5962, Document, MathReview Entry, ZentralBlatt

Referenced by:

§32.10(ii), §32.10(iv), §32.7(ii).

Encodings:

BibTeX

…

4: 31.8 Solutions via Quadratures

… ►Denote $𝐦=(m_0,m_1,m_2,m_3)$ and $\lambda =-4q$. Then …Here ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$ is a polynomial of degree $g$ in $\lambda$ and of degree $N=m_0+m_1+m_2+m_3$ in $z$, that is a solution of the third-order differential equation satisfied by a product of any two solutions of Heun’s equation. …The variables $\lambda$ and $\nu$ are two coordinates of the associated hyperelliptic (spectral) curve $\mathrm{\Gamma }:{\nu }^2={\prod }_{j=1}^{2g+1}(\lambda -{\lambda }_j)$. …Lastly, ${\lambda }_j$, $j=1,2,\mathrm{\dots },2g+1$, are the zeros of the Wronskian of $w_{+}(𝐦;\lambda ;z)$ and $w_{-}(𝐦;\lambda ;z)$. …

5: 28.29 Definitions and Basic Properties

… ►A generalization of Mathieu’s equation (28.2.1) is Hill’s equation … ►The basic solutions $w_{\text{I}}(z,\lambda )$, $w_{\text{II}}(z,\lambda )$ are defined in the same way as in §28.2(ii) (compare (28.2.5), (28.2.6)). … ►This is the characteristic equation of (28.29.1), and $\cos \left(\pi \nu \right)$ is an entire function of $\lambda$. Given $\lambda$ together with the condition (28.29.6), the solutions $\pm \nu$ of (28.29.9) are the characteristic exponents of (28.29.1). … ►In the symmetric case $Q(z)=Q(-z)$, $w_{\text{I}}(z,\lambda )$ is an even solution and $w_{\text{II}}(z,\lambda )$ is an odd solution; compare §28.2(ii). …

6: 32.4 Isomonodromy Problems

… ►

${\displaystyle \frac{\partial 𝚿}{\partial \lambda }}=𝐀(z,\lambda )𝚿,$

►

${\displaystyle \frac{\partial 𝚿}{\partial z}}=𝐁(z,\lambda )𝚿,$

►is a linear system in which $𝐀$ and $𝐁$ are matrices and $\lambda$ is independent of $z$. … ►

32.4.2 $$\frac{{\partial }^2𝚿}{\partial z\partial \lambda }=\frac{{\partial }^2𝚿}{\partial \lambda \partial z},$$

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Symbols:

$\partial x$: partial differential of $x$ and $z$: real

Permalink:

http://dlmf.nist.gov/32.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §32.4(i), §32.4 and Ch.32

… ►

32.4.3 $$\frac{\partial 𝐀}{\partial z}-\frac{\partial 𝐁}{\partial \lambda }+𝐀𝐁-𝐁𝐀=0.$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $z$: real

Referenced by:

§32.4(i)

Permalink:

http://dlmf.nist.gov/32.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §32.4(i), §32.4 and Ch.32

…

7: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

… ►For $x\ge 0$ and $p>1$ let $x=\lambda p$ and define $A_0(\lambda )=1$, ►

8.20.4 $$A_{k+1}(\lambda )=(1-2k\lambda )A_k(\lambda )+\lambda (\lambda +1)\frac{d{A}_k(\lambda )}{d\lambda },$$ $k=0,1,2,\mathrm{\dots }$,

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $k$: nonnegative integer and $A_k(\lambda )$

Permalink:

http://dlmf.nist.gov/8.20.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.20(ii), §8.20 and Ch.8

►so that $A_k(\lambda )$ is a polynomial in $\lambda$ of degree $k-1$ when $k\ge 1$. … ►

$A_1(\lambda )=1,$

… ►uniformly for $\lambda \in [0,\mathrm{\infty })$. …

8: 18.35 Pollaczek Polynomials

… ►Thus type 3 with $c=0$ reduces to type 2, and type 3 with $c=0$ and $\lambda =\frac{1}{2}$ reduces to type 1, also in subsequent formulas. … ►The Pollaczek polynomials of type 3 are defined by the recurrence relation (in first form (18.2.8)) … ►For type 1 take $\lambda =\frac{1}{2}$ and for Gauss’ hypergeometric function $F$ see (15.2.1). … ►Hence, only in the case $a=b=0$ does $\ln \left(w^{(\lambda )}(x;a,b)\right)$ satisfy the condition (18.2.39) for the Szegő class $𝒢$. … ►More generally, the $P_n^{(\lambda )}(x;a,b)$ are OP’s if and only if one of the following three conditions holds (in case (iii) work with the monic polynomials (18.35.2_2)). …

9: 19.26 Addition Theorems

… ►In this subsection, and also §§19.26(ii) and 19.26(iii), we assume that $\lambda ,x,y,z$ are positive, except that at most one of $x,y,z$ can be 0. ►

19.26.1 $$R_F(x+\lambda ,y+\lambda ,z+\lambda )+R_F(x+\mu ,y+\mu ,z+\mu )=R_F(x,y,z),$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\lambda$: positive, $x$: positive, $y$: positive, $z$: positive and $\mu >0$: parameter

Referenced by:

§19.26(i)

Permalink:

http://dlmf.nist.gov/19.26.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.26(i), §19.26 and Ch.19

… ►where $\lambda >0$, $y>0$, $x\ge 0$, and … ►If $x=0$, then $\lambda \mu =yz$. … ►The equations inverse to $z+\lambda =(\sqrt{z}+\sqrt{x})(\sqrt{z}+\sqrt{y})$ and the two other equations obtained by permuting $x,y,z$ (see (19.26.19)) are …

10: 31.11 Expansions in Series of Hypergeometric Functions

… ►

31.11.2

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Defines:

$P_j$: General solution of generalized hypergeometric differential equation (locally)

Symbols:

: Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $j$: nonnegative integer, $\lambda$ and $\mu$

Referenced by:

§31.11(iii)

Permalink:

http://dlmf.nist.gov/31.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

… ►

31.11.3 $$\lambda +\mu =\gamma +\delta -1=\alpha +\beta -ϵ.$$

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Defines:

$\lambda$ (locally) and $\mu$ (locally)

Symbols:

$\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

… ►

31.11.3_1 $$P_j^5=\frac{{\left(\lambda \right)}_j{\left(1-\gamma +\lambda \right)}_j}{{\left(1+\lambda -\mu \right)}_{2j}}{z}^{-\lambda -j}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\lambda +j,1-\gamma +\lambda +j}{1+\lambda -\mu +2j};\frac{1}{z}),$$

ⓘ

Defines:

$P_j^5$: first Kummer solution at $\mathrm{\infty }$ of generalized hypergeometric differential equation (locally)

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $\lambda$ and $\mu$

Referenced by:

§31.11(ii), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/31.11.E3_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

Suggested 2022-09-14 by Hans Volkmer

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

… ► $\lambda$, $\mu$ must also satisfy the condition … ►Here …