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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.
In telecommunications and transmission line theory, the reflection coefficient is the ratio of the complex amplitude of the reflected wave to that of the incident wave. The voltage and current at any point along a transmission line can always be resolved into forward and reflected traveling waves given a specified reference impedance Z 0.
A reflection through an axis. In mathematics, a reflection (also spelled reflexion) [1] is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as the set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection.
In that unusual situation we have θ t = θ i (that is, the transmitted ray is undeviated), so that the cosines in equations , , , , and to cancel out, and all the reflection and transmission ratios become independent of the angle of incidence; in other words, the ratios for normal incidence become applicable to all angles of incidence. [35]
To find the amplitudes for reflection and transmission for incidence from the left, we set in the above equations A → = 1 (incoming particle), A ← = √ R (reflection), B ← = 0 (no incoming particle from the right) and B → = √ Tk 1 /k 2 (transmission [1]). We then solve for T and R. The result is:
Reflectivity is the square of the magnitude of the Fresnel reflection coefficient, [4] which is the ratio of the reflected to incident electric field; [5] as such the reflection coefficient can be expressed as a complex number as determined by the Fresnel equations for a single layer, whereas the reflectance is always a positive real number.
These equations can be proved through straightforward matrix multiplication and application of trigonometric identities, specifically the sum and difference identities. The set of all reflections in lines through the origin and rotations about the origin, together with the operation of composition of reflections and rotations, forms a group .
The reflection angle is equal to the incidence angle, and the amount of light that is reflected is determined by the reflectivity of the surface. The reflectivity can be calculated from the refractive index and the incidence angle with the Fresnel equations, which for normal incidence reduces to [42]: 44