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| {{about|reflection in number theory and calculus|reflection formulas in geometry|Reflection (mathematics)}}
| | Nice to satisfy you, my name is Numbers Held although I don't really like becoming known as like that. Hiring is his occupation. Years ago we moved to North Dakota. What I adore doing is playing baseball but I haven't made a dime with it.<br><br>my page: [http://faculty.jonahmancini.com/groups/solid-advice-for-dealing-with-a-candida-albicans/members/ std home test] |
| In [[mathematics]], a '''reflection formula''' or '''reflection relation''' for a [[function (mathematics)|function]] ''f'' is a relationship between ''f''(''a'' − ''x'') and ''f''(''x''). It is a special case of a [[functional equation]], and it is very common in the literature to use the term "functional equation" when "reflection formula" is meant.
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| Reflection formulas are useful for [[numerical analysis|numerical computation]] of [[special function]]s. In effect, an approximation that has greater accuracy or only converges on one side of a reflection point (typically in the positive half of the [[complex plane]]) can be employed for all arguments.
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| == Known formulae ==
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| The [[even and odd functions]] satisfy simple reflection relations around ''a'' = 0. For all even functions,
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| :<math>f(-x) = f(x),\,\!</math>
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| and for all odd functions,
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| :<math>f(-x) = -f(x).\,\!</math> | |
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| A famous relationship is '''Euler's reflection formula'''
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| :<math>\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin{(\pi z)}}\!</math>
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| for the [[Gamma function]] Γ(''z''), due to [[Leonhard Euler]].
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| There is also a reflection formula for the general ''n''-th order [[polygamma function]] ψ<sup>(n)</sup>(''z''),
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| :<math>\psi^{(n)} (1-z)+(-1)^{n+1}\psi^{(n)} (z) = (-1)^n \pi \frac{d^n}{d z^n} \cot{(\pi z)} \,</math>
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| which springs trivally from the fact that the polygamma functions are defined as the derivations of the <math>\ln \Gamma</math> and thus its reflection formula is inherited to them.
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| The [[Riemann zeta function]] ζ(''z'') satisfies
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| :<math>\frac{\zeta(1-z)}{\zeta(z)} = \frac{2\, \Gamma(z)}{(2\pi)^{z}} \cos\left(\frac{\pi z}{2}\right),\,\!</math>
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| and the [[Riemann Xi function]] ξ(''z'') satisfies
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| :<math>\xi(z) = \xi(1-z). \,\!</math>
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| ==References==
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| * {{MathWorld|urlname=ReflectionRelation|title=Reflection Relation}}
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| * {{MathWorld|urlname=PolygammaFunction|title=Polygamma Function}}
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| [[Category:Calculus]]
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Nice to satisfy you, my name is Numbers Held although I don't really like becoming known as like that. Hiring is his occupation. Years ago we moved to North Dakota. What I adore doing is playing baseball but I haven't made a dime with it.
my page: std home test