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mathematical physics > special function > Heine's identity

mathematical analysis > functional analysis > special function > Heine's identity

algebra > elementary algebra > identity > Heine's identity

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Heine's identity

In mathematical analysis, Heine's identity, named after Heinrich Eduard Heine is a Fourier expansion of a reciprocal square root which Heine presented as
${\\displaystyle {\ rac {1}{\\sqrt {z-\\cos \\psi }}}={\ rac {\\sqrt {2}}{\\pi }}\\sum _{m=-\\infty }^{\\infty }Q_{m-{\ rac {1}{2}}}(z)e^{im\\psi }}$
where ${\\displaystyle Q_{m-{\ rac {1}{2}}}}$ is a Legendre function of the second kind, which has degree, m − 1⁄2, a half-integer, and argument, z, real and greater than one. This expression can be generalized for arbitrary half-integer powers as follows
${\\displaystyle (z-\\cos \\psi )^{n-{\ rac {1}{2}}}={\\sqrt {\ rac {2}{\\pi }}}{\ rac {(z^{2}-1)^{\ rac {n}{2}}}{\\Gamma ({\ rac {1}{2}}-n)}}\\sum _{m=-\\infty }^{\\infty }{\ rac {\\Gamma (m-n+{\ rac {1}{2}})}{\\Gamma (m+n+{\ rac {1}{2}})}}Q_{m-{\ rac {1}{2}}}^{n}(z)e^{im\\psi },}$
where ${\\displaystyle \\scriptstyle \\,\\Gamma }$ is the Gamma function.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Heine%27s_identity)

identité de Heine

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RDF/XML TURTLE JSON-LD Date de création 13/07/2023, dernière modification le 13/07/2023