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Known formulae

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

f ( − x ) = f ( x ) , {\displaystyle f(-x)=f(x),}

and for all odd functions,

f ( − x ) = − f ( x ) . {\displaystyle f(-x)=-f(x).}

A famous relationship is Euler's reflection formula

Γ ( z ) Γ ( 1 − z ) = π sin ⁡ ( π z ) , z ∉ Z {\displaystyle \Gamma (z)\Gamma (1-z)={\frac {\pi }{\sin {(\pi z)}}},\qquad z\not \in \mathbb {Z} }

for the gamma function Γ ( z ) {\textstyle \Gamma (z)} , due to Leonhard Euler.

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

ψ ( n ) ( 1 − z ) + ( − 1 ) n + 1 ψ ( n ) ( z ) = ( − 1 ) n π d n d z n cot ⁡ ( π z ) {\displaystyle \psi ^{(n)}(1-z)+(-1)^{n+1}\psi ^{(n)}(z)=(-1)^{n}\pi {\frac {d^{n}}{dz^{n}}}\cot {(\pi z)}}

which springs trivially from the fact that the polygamma functions are defined as the derivatives of ln ⁡ Γ {\textstyle \ln \Gamma } and thus inherit the reflection formula.

The dilogarithm also satisfies a reflection formula,¹²

Li 2 ⁡ ( z ) + Li 2 ⁡ ( 1 − z ) = ζ ( 2 ) − ln ⁡ ( z ) ln ⁡ ( 1 − z ) {\displaystyle \operatorname {Li} _{2}(z)+\operatorname {Li} _{2}(1-z)=\zeta (2)-\ln(z)\ln(1-z)}

The Riemann zeta function ζ(z) satisfies

ζ ( 1 − z ) ζ ( z ) = 2 Γ ( z ) ( 2 π ) z cos ⁡ ( π z 2 ) , {\displaystyle {\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

ξ ( z ) = ξ ( 1 − z ) . {\displaystyle \xi (z)=\xi (1-z).}

• Weisstein, Eric W. "Reflection Relation". MathWorld.
• Weisstein, Eric W. "Polygamma Function". MathWorld.

References

1. Weisstein, Eric W. "Dilogarithm". mathworld.wolfram.com. Retrieved 2024-08-01. https://mathworld.wolfram.com/ ↩

2. "Dilogarithm Reflection Formula - ProofWiki". proofwiki.org. Retrieved 2024-08-01. https://proofwiki.org/wiki/Dilogarithm_Reflection_Formula ↩