Rectified tesseract

29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church. In mathematics, a reflection formula or reflection relation for a 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.

Reflection formulas are useful for numerical computation of special functions. 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.

Known formulae

The even and odd functions satisfy simple reflection relations around a = 0. For all even functions,

$f(-x)=f(x),\,\!$

and for all odd functions,

$f(-x)=-f(x).\,\!$

A famous relationship is Euler's reflection formula

$\Gamma (z)\Gamma (1-z)={\frac {\pi }{\sin {(\pi z)}}}\!$

for the Gamma function Γ(z), due to Leonhard Euler.

There is also a reflection formula for the general n-th order polygamma function ψ⁽ⁿ⁾(z),

$\psi ^{(n)}(1-z)+(-1)^{n+1}\psi ^{(n)}(z)=(-1)^{n}\pi {\frac {d^{n}}{dz^{n}}}\cot {(\pi z)}\,$

which springs trivally from the fact that the polygamma functions are defined as the derivations of the $\ln \Gamma$ and thus its reflection formula is inherited to them.

The Riemann zeta function ζ(z) satisfies

${\frac {\zeta (1-z)}{\zeta (z)}}={\frac {2\,\Gamma (z)}{(2\pi )^{z}}}\cos \left({\frac {\pi z}{2}}\right),\,\!$

and the Riemann Xi function ξ(z) satisfies

$\xi (z)=\xi (1-z).\,\!$

References

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