duplication%20formulas

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1: 5.5 Functional Relations

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§5.5(ii) Reflection

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5.5.3 $$\mathrm{\Gamma }\left(z\right)\mathrm{\Gamma }\left(1-z\right)=\pi /\sin \left(\pi z\right),$$ $z\ne 0,\pm 1,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function and $z$: complex variable

A&S Ref:

6.1.17 (without the condition on $z$.)

Referenced by:

§10.19(i), §15.4(ii), §15.8(ii), (25.9.2), §5.21, (9.10.18), (9.12.15)

Permalink:

http://dlmf.nist.gov/5.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(ii), §5.5 and Ch.5

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§5.5(iii) Multiplication

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Duplication Formula

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Gauss’s Multiplication Formula

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2: 20.7 Identities

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§20.7(ii) Addition Formulas

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§20.7(iii) Duplication Formula

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§20.7(iv) Reduction Formulas for Products

… ►See Lawden (1989, pp. 19–20). … ►

§20.7(ix) Addendum to 20.7(iv) Reduction Formulas for Products

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3: 10.60 Sums

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§10.60(ii) Duplication Formulas

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4: 20.11 Generalizations and Analogs

… ► … ►Such sets of twelve equations include derivatives, differential equations, bisection relations, duplication relations, addition formulas (including new ones for theta functions), and pseudo-addition formulas. …

5: 19.11 Addition Theorems

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§19.11(i) General Formulas

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§19.11(iii) Duplication Formulas

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6: 23.10 Addition Theorems and Other Identities

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§23.10(ii) Duplication Formulas

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§23.10(iii) $n$-Tuple Formulas

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7: 19.36 Methods of Computation

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§19.36(i) Duplication Method

►Numerical differences between the variables of a symmetric integral can be reduced in magnitude by successive factors of 4 by repeated applications of the duplication theorem, as shown by (19.26.18). … ►The duplication method starts with computation of ${\lambda }_0=\sqrt{x}\sqrt{y}+\sqrt{y}\sqrt{z}+\sqrt{z}\sqrt{x}$. …Alternatively, the first duplication is done analytically as in Carlson and FitzSimons (2000), where further information can be found. … ►Here $R_C$ is computed either by the duplication algorithm in Carlson (1995) or via (19.2.19). …

8: 19.26 Addition Theorems

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§19.26(i) General Formulas

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§19.26(iii) Duplication Formulas

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9: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

10: Wolter Groenevelt

… ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …