Reflection formula

Template:Short description In mathematics, a reflection formula or reflection relation for a function Template:Mvar is a relationship between Template:Math and Template:Math. It is a special case of a functional equation. It is common in mathematical literature to use the term "functional equation" for what are specifically reflection formulae.

Reflection formulae 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 by definition simple reflection relations around Template:Math. 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

$$\mathrm{\Gamma }(z)\mathrm{\Gamma }(1-z)=\frac{\pi }{\sin (\pi z)},z\in ̸\mathbb{\mathbb{Z}}$$

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

There is also a reflection formula for the general Template:Mvar-th order polygamma function Template:Math,

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

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

The dilogarithm also satisfies a reflection formula,

$${\mathrm{Li}}_2(z)+{\mathrm{Li}}_2(1-z)=\zeta (2)-\ln (z)\ln (1-z)$$

The Riemann zeta function Template:Math satisfies

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

and the Riemann Xi function Template:Math satisfies

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