For example, the automorphisms of the Riemann sphere are Möbius transformations.

For example, the Riemann hypothesis is a conjecture from number theory that ( amongst other things ) makes predictions about the distribution of prime numbers.

For s an even positive integer, the product sin ( πs / 2 ) Γ ( 1 − s ) is regular and the functional equation relates the values of the Riemann zeta function at odd negative integers and even positive integers.

For a great many functions and practical applications, the Riemann integral can also be readily evaluated by using the fundamental theorem of calculus or ( approximately ) by numerical integration.

For functions on the real line, the Henstock integral is an even more general notion of integral ( based on Riemann's theory rather than Lebesgue's ) that subsumes both Lebesgue integration and improper Riemann integration.

For every Riemann surface, a meromorphic function is the same as a holomorphic function that maps to the Riemann sphere and which is not constant ∞.

For a two-dimensional surface, the Bianchi identities imply that the Riemann tensor can be expressed as

For example, in general relativity gravitation is associated with a tensor field ( in particular, with the Riemann curvature tensor ).

For example, the corresponding complex variety is the Riemann sphere and its initial Betti numbers are 1, 0, 1.

For the purposes of the Riemann – Roch theorem, the surface X is always assumed to be compact.

For discrete-time dynamical systems the orbits are sequences, for real dynamical systems the orbits are curves and for holomorphic dynamical systems the orbits are Riemann surfaces.

For instance, Möbius transformations ( transformations of the complex projective line, or Riemann sphere ) are affine ( transformations of the complex plane ) if and only if they fix the point at infinity.

* For the Riemann integral ( or the Darboux integral, which is equivalent to it ), improper integration is necessary both for unbounded intervals ( since one cannot divide the interval into finitely many subintervals of finite length ) and for unbounded functions with finite integral ( since, supposing it is unbounded above, then the upper integral will be infinite, but the lower integral will be finite ).

* For the Henstock – Kurzweil integral, improper integration is not necessary, and this is seen as a strength of the theory: it encompasses all Lebesgue integrable and improper Riemann integrable functions.

For example, in Riemann surface theory, the deformation theory of complex structures is studied classically by means of quadratic differentials ( namely sections of L ( K < sup > 2 </ sup >)).

For example, consider a CFT on the Riemann sphere.

For x -> ∞, the functions diverge ; the integral without prefactor is given by the Riemann zeta function:

For a compact Riemann surface S of genus greater than 1, its universal covering space is the unit disc D on which its fundamental group Γ acts by Möbius transformations.

For example, the Riemann zeta function has a functional equation relating its value at the complex number s with its value at 1 − s. In every case this relates to some value ζ ( s ) that is only defined by analytic continuation from the infinite series definition.

For example, it is easy to prove that the analytic functions from the Riemann sphere to itself are either

For if such a function f is nonconstant, then since the set of z where f ( z ) is infinity is isolated and the Riemann sphere is compact, there are finitely many z with f ( z ) equal to infinity.

For example, consider G = SL < sub > 2 </ sub >( C ), for which G / B is the Riemann sphere, an integral weight is specified simply by an integer n, and ρ = 1.

For number theorists his main fame is the series for the Riemann zeta function ( the leading function in Riemann's exact prime-counting function ).

For phosphor to fiber and fiber to air surfaces, and assuming Af, we obtain Af percent.

For higher speeds, the flow will compress more significantly as it comes into contact with surfaces and slows.

For images of fractal patterns, this has been expressed by phrases such as " smoothly piling up surfaces " and " swirls upon swirls ".

For closed ( orientable or non-orientable ) surfaces with positive genus, the maximum number p of colors needed depends on the surface's Euler characteristic χ according to the formula

For example, air hoar is a deposit of hoar frost on objects above the surface, such as tree branches, plant stems, wires ; surface hoar is formed by fernlike ice crystals directly deposited on snow, ice or already frozen surfaces ; crevasse hoar consists of crystals that form in glacial crevasses where water vapour can accumulate under calm weather conditions ; depth hoar refers to cup shaped, faceted crystals formed within dry snow, beneath the surface.

For this reason spherical reference surfaces are frequently used in mapping programs.

For example once one gets to 3-dimensional surfaces, one has to deal with solid objects with knot-shaped holes, and then one needs the whole of knot theory just to classify them.

For mirrors with parabolic surfaces, parallel rays incident on the mirror produce reflected rays that converge at a common focus.

For compact space | compact 2-dimensional surfaces without boundary ( topology ) | boundary, if every loop can be continuously tightened to a point, then the surface is topologically Homeomorphism | homeomorphic to a 2-sphere ( usually just called a sphere ).

For instance, the naturally occurring citric and ascorbic acids in lemon or other citrus juice can inhibit the action of the enzyme phenolase which turns surfaces of cut apples and potatoes brown if a small amount of the juice is applied to the freshly cut produce.

For similar reasons, objects intended to avoid detection will not have inside corners or surfaces and edges perpendicular to likely detection directions, which leads to " odd " looking stealth aircraft.

For example if the light source emitted white light and the two diffuse surfaces were blue, then the resulting color of the pixel is blue.

For most points on most surfaces, different sections will have different curvatures ; the maximum and minimum values of these are called the principal curvatures.

#: For most surfaces the focal surface forms two sheets each of which is a surface and which come together at umbilical points.

For channel surfaces one sheet forms a curve and the other sheet is a surface ; For cones, cylinders, toruses and cyclides both sheets form curves.

For most uses, dilution with water is recommended for safety and to avoid damaging the surfaces being cleaned.

For this reason wormholes have been defined geometrically, as opposed to topologically, as regions of spacetime that constrain the incremental deformation of closed surfaces.

For still water, this is the difference in height between the inlet and outlet surfaces.

* For specular surfaces, such as glass or polished metal, reflectivity will be nearly zero at all angles except at the appropriate reflected angle-that is, reflected radiation will follow a different path from incident radiation for all cases other than radiation normal to the surface.

* For diffuse surfaces, such as matte white paint, reflectivity is uniform ; radiation is reflected in all angles equally or near-equally.

For surfaces with b boundary components, the equation reads χ = 2 − 2g − b.

For a long cylinder far from other surfaces, this reduces the effective area by a factor of π / 2

For example, PCC trolley brakes include a flat shoe which is clamped to the rail with an electromagnet ; the Murphy brake pinches a rotating drum, and the Ausco Lambert disc brake uses a hollow disc ( two parallel discs with a structural bridge ) with shoes that sit between the disc surfaces and expand laterally.

For the case of two colliding bodies in two-dimensions, the overall velocity of each body must be split into two perpendicular velocities: one tangent to the common normal surfaces of the colliding bodies at the point of contact, the other along the line of collision.