Reflection formula explained

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. 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 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

$$\backslash Gamma(z)\backslash Gamma(1-z)\; =\; \backslash frac,\; \backslash qquad\; z\; \backslash not\backslash in\; \backslash mathbb\; Z$$

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

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

$$\backslash psi^\; (1-z)+(-1)^\backslash psi^\; (z)\; =\; (-1)^n\; \backslash pi\; \backslash frac\; \backslash cot$$

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

The dilogarithm also satisfies a reflection formula,^{[1] [2]}

$$\backslash operatorname\_2(z)+\backslash operatorname\_2(1-z)=\backslash zeta(2)-\backslash ln\; (z)\backslash ln(1-z)$$

The Riemann zeta function ζ(z) satisfies

$$\backslash frac\; =\; \backslash frac\; \backslash cos\backslash left(\backslash frac\backslash right),$$

and the Riemann Xi function ξ(z) satisfies

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

Notes and References

1. Web site: Weisstein . Eric W. . Dilogarithm . 2024-08-01 . mathworld.wolfram.com . en.
2. Web site: Dilogarithm Reflection Formula - ProofWiki . 2024-08-01 . proofwiki.org.