Reflection formula

<h2 id="known-formulae">Known formulae</h2>
The <a href="/facts/Even_and_odd_functions/tGHWJqAa">even and odd functions</a> 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 <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a> 
 
 
 
 Γ
 (
 z
 )
 
 
 {\textstyle \Gamma (z)}
 
, due to <a href="/facts/Leonhard_Euler/z2fsbjgj">Leonhard Euler</a>.
There is also a reflection formula for the general n-th order <a href="/facts/Polygamma_function/IF6W68aL">polygamma function</a> 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 <a href="/facts/Dilogarithm/lsnCj3Gb">dilogarithm</a> also satisfies a reflection formula,<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a><a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

 
 
 
 
 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 <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a> 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 <a href="/facts/Riemann_Xi_function/Uz0j5Kkr">Riemann Xi function</a> ξ(z) satisfies

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

<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/ReflectionRelation.html">"Reflection Relation"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/PolygammaFunction.html">"Polygamma Function"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Weisstein, Eric W. "Dilogarithm". mathworld.wolfram.com. Retrieved 2024-08-01. <a href="https://mathworld.wolfram.com/" target="_blank">https://mathworld.wolfram.com/</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">"Dilogarithm Reflection Formula - ProofWiki". proofwiki.org. Retrieved 2024-08-01. <a href="https://proofwiki.org/wiki/Dilogarithm_Reflection_Formula" target="_blank">https://proofwiki.org/wiki/Dilogarithm_Reflection_Formula</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Reflection formula open-in-new

Reflection formula