connection formulas

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1: 31.18 Methods of Computation

… ►Subsequently, the coefficients in the necessary connection formulas can be calculated numerically by matching the values of solutions and their derivatives at suitably chosen values of $z$; see Laĭ (1994) and Lay et al. (1998). …

2: 10.4 Connection Formulas

§10.4 Connection Formulas

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3: 25.13 Periodic Zeta Function

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25.13.2 $$F(x,s)=\frac{\mathrm{\Gamma }\left(1-s\right)}{{(2\pi )}^{1-s}}\left({\mathrm{e}}^{\pi \mathrm{i}(1-s)/2}\zeta (1-s,x)+{\mathrm{e}}^{\pi \mathrm{i}(s-1)/2}\zeta (1-s,1-x)\right),$$ , $\mathrm{\Re }s>1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F(x,s)$: periodic zeta function, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

connectionformula

Source:

Apostol (1976, Exercise 2, p. 273)

Permalink:

http://dlmf.nist.gov/25.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

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25.13.3 $$\zeta (1-s,x)=\frac{\mathrm{\Gamma }\left(s\right)}{{(2\pi )}^s}\left({\mathrm{e}}^{-\pi \mathrm{i}s/2}F(x,s)+{\mathrm{e}}^{\pi \mathrm{i}s/2}F(-x,s)\right),$$ $\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $x=1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F(x,s)$: periodic zeta function, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

connectionformula

Source:

Apostol (1976, (10), p. 257)

Notes:

This formula is equivalent to (25.11.9).

Referenced by:

Erratum (V1.1.4) for Equation (25.13.3)

Permalink:

http://dlmf.nist.gov/25.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

4: 14.21 Definitions and Basic Properties

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§14.21(iii) Properties

… ►This includes, for example, the Wronskian relations (14.2.7)–(14.2.11); hypergeometric representations (14.3.6)–(14.3.10) and (14.3.15)–(14.3.20); results for integer orders (14.6.3)–(14.6.5), (14.6.7), (14.6.8), (14.7.6), (14.7.7), and (14.7.11)–(14.7.16); behavior at singularities (14.8.7)–(14.8.16); connection formulas (14.9.11)–(14.9.16); recurrence relations (14.10.3)–(14.10.7). …

5: 28.22 Connection Formulas

§28.22 Connection Formulas

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6: 14.9 Connection Formulas

§14.9 Connection Formulas

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§14.9(i) Connections Between ${𝖯}_{\nu }^{\pm \mu }\left(x\right)$, ${𝖯}_{-\nu -1}^{\pm \mu }\left(x\right)$, ${𝖰}_{\nu }^{\pm \mu }\left(x\right)$, ${𝖰}_{-\nu -1}^{\mu }\left(x\right)$

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§14.9(ii) Connections Between ${𝖯}_{\nu }^{\pm \mu }\left(\pm x\right)$, ${𝖰}_{\nu }^{-\mu }\left(\pm x\right)$, ${𝖰}_{\nu }^{\mu }\left(x\right)$

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§14.9(iii) Connections Between $P_{\nu }^{\pm \mu }\left(x\right)$, $P_{-\nu -1}^{\pm \mu }\left(x\right)$, ${𝑸}_{\nu }^{\pm \mu }\left(x\right)$, ${𝑸}_{-\nu -1}^{\mu }\left(x\right)$

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7: 19.21 Connection Formulas

§19.21 Connection Formulas

… ►The complete cases of $R_F$ and $R_G$ have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). … ►Connection formulas for $R_{-a}(𝐛;𝐳)$ are given in Carlson (1977b, pp. 99, 101, and 123–124). …

8: Bibliography O

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Connection formulas for second-order differential equations with multiple turning points. SIAM J. Math. Anal. 8 (1), pp. 127–154.

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External Links:

Document, MathReview (Anthony Leung), ZentralBlatt

Referenced by:

§2.8(v), (9.13.13), (9.13.17), §9.13(i), §9.13(i).

Encodings:

BibTeX

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Connection formulas for second-order differential equations having an arbitrary number of turning points of arbitrary multiplicities. SIAM J. Math. Anal. 8 (4), pp. 673–700.

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External Links:

Document, MathReview (W. Wasow), ZentralBlatt

Referenced by:

§2.8(v).

Encodings:

BibTeX

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General connection formulae for Liouville-Green approximations in the complex plane. Philos. Trans. Roy. Soc. London Ser. A 289, pp. 501–548.

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External Links:

ISSN 0080-4614, FullText, MathReview (W. Wasow), ZentralBlatt

Referenced by:

§2.8(v), (9.13.18), §9.13(i), §9.13(i).

Encodings:

BibTeX

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9: 10.27 Connection Formulas

§10.27 Connection Formulas

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10: 9.2 Differential Equation

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§9.2(v) Connection Formulas

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