This article is about reflection in number theory and calculus. For reflection formulas in geometry, see Reflection (mathematics).
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.
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In mathematics, a reflectionformula 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...
reflection, a reflection across a point Reflection (mathematics), a transformation of a space Reflectionformula, a relation in a function Reflective subcategory...
example: the reflectionformula essentially represents the sine function as the product of two gamma functions. Starting from this formula, the exponential...
The Fresnel equations (or Fresnel coefficients) describe the reflection and transmission of light (or electromagnetic radiation in general) when incident...
n+{\frac {1}{8n}}}}\right)\qquad n\geq 1} which both spring off the reflectionformula via 0 = ψ ( 1 − x n ) = ψ ( x n ) + π tan π x n {\displaystyle 0=\psi...
in three dimensions Euler's reflectionformula, reflectionformula for the gamma function Local Euler characteristic formula The Euler function, a modular...
_{j=1}^{n}{\frac {H_{j,c}}{(j+1)^{b}}}} The above MZVs satisfy the Euler reflectionformula: ζ ( a , b ) + ζ ( b , a ) = ζ ( a ) ζ ( b ) − ζ ( a + b ) {\displaystyle...
factorial. In particular, and by means of the reflectionformula, In analogy with the half-integer formula, Γ ( n + 1 3 ) = Γ ( 1 3 ) ( 3 n − 2 ) ! ! !...
formula as stated here is only valid for arguments in the right complex half-plane, it can be extended to the entire complex plane by the reflection formula...
{x^{2}}{n^{2}}}\right)} and is related to the gamma function Γ(x) through Euler's reflectionformula: sin ( π x ) π x = 1 Γ ( 1 + x ) Γ ( 1 − x ) . {\displaystyle {\frac...
Unigrafia, p. 27, ISBN 978-951-51-8695-9, ISSN 2814-4031 "Euler's ReflectionFormula - ProofWiki", proofwiki.org, retrieved 2020-09-02 Paris, R. B. (2010)...
In physics, total internal reflection (TIR) is the phenomenon in which waves arriving at the interface (boundary) from one medium to another (e.g., from...
Specular reflection, or regular reflection, is the mirror-like reflection of waves, such as light, from a surface. The law of reflection states that a...
Euler's reflectionformula Γ ( z ) Γ ( 1 − z ) = π sin π z . {\displaystyle \Gamma (z)\Gamma (1-z)={\frac {\pi }{\sin \pi z}}.} However, this formula cannot...
if n is odd. Given a vector a in the Euclidean space Rn, the formula for the reflection of a across the point p is R e f p ( a ) = 2 p − a . {\displaystyle...
(z\not \in ~]-\infty ;0]),} and applying the reflectionformula to each dilogarithm we find the inversion formula Li 2 ( z ) + Li 2 ( 1 / z ) = − 1 6 π...
{\displaystyle \Gamma (s)} is the Gamma function. The functional equation (or reflectionformula) for Landau's ξ {\displaystyle ~\xi ~} is ξ ( 1 − s ) = ξ ( s...
One of the most common reflection principles for ZFC is a theorem schema that can be described as follows: for any formula ϕ ( x 1 , … , x n ) {\displaystyle...
q+{\frac {n}{k}}\right).} The periodic zeta function occurs in the reflectionformula for the Hurwitz zeta function, which is why the relation that it obeys...
{2}}}{{\sqrt {3}}+1}}\right)} In general, from the Barnes G-function reflectionformula, Cl 2 ( 2 π z ) = 2 π log ( G ( 1 − z ) G ( z ) ) − 2 π log ...
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