* By the convolution theorem, Fourier transforms turn the complicated convolution operation into simple multiplication, which means that they provide an efficient way to compute convolution-based operations such as polynomial multiplication and multiplying large numbers.

By similar arguments, it can be shown that the discrete convolution of sequences and is given by:

By the convolution theorem, the FT of an arbitrary transparency function-multiplied ( or truncated ) by an aperture function-is equal to the FT of the non-truncated transparency function convolved against the FT of the aperture function, which in this case becomes a type of " Greens function " or " impulse response function " in the spectral domain.

Consider two waveforms f and g. By calculating the convolution, we determine how much a reversed function g must be shifted along the x-axis to become identical to function f. The convolution function essentially reverses and slides function g along the axis, and calculates the integral of their ( f and the reversed and shifted g ) product for each possible amount of sliding.

has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.

By a theorem of Gelfand and Naimark, given a B * algebra A there exists a Hilbert space H and an isometric *- homomorphism from A into the algebra B ( H ) of all bounded linear operators on H. Thus every B * algebra is isometrically *- isomorphic to a C *- algebra.

By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.

By setting K = ker ( f ) we immediately get the first isomorphism theorem.

By " satisfactory " one would mean at least the equivalent of Plancherel theorem.

By the fundamental theorem of arithmetic, every positive integer has a unique prime factorization.

By the Glivenko – Cantelli theorem, if the sample comes from distribution F ( x ), then D < sub > n </ sub > converges to 0 almost surely.

By this theorem, once a star's chemical composition and its position on the main sequence is known, so too is the star's mass and radius.

By Gödel's incompleteness theorem, Peano arithmetic is incomplete and its consistency is not internally provable.

By Rice's theorem, the 1-halting problem is undecidable.

By the corollary to the recursion theorem, there is an index such that returns.

By Liouville's theorem, Hamiltonian flows preserve the volume form on the phase space.

By the fundamental theorem, we may replace the new set by the old set subject to a unitary transformation.

By the first isomorphism theorem, the image of A under ƒ is a substructure of B isomorphic to the quotient of A by this congruence.

By the central limit theorem, this distribution approaches the normal distribution as n increases.

By analogy to the prior and posterior probability terms in Bayes ' theorem, Bayes ' rule can be seen as Bayes ' theorem in odds form.

By the extreme value theorem, a continuous function on a closed interval must attain its minimum and maximum values at least once.

By the rank-nullity theorem, a system of n vectors in k dimensions ( where all dimensions are necessary ) satisfies a ( p = n − k )- dimensional space of relations.

By Tychonoff's theorem we have that is compact since is, so the closure of in is a compactification of.

By Stone's representation theorem every Boolean ring is isomorphic to a field of sets ( treated as a ring with these operations ).

By induction, Hilbert's basis theorem establishes that, the ring of all polynomials in n variables with coefficients in, is a Noetherian ring.

By Euler's rotation theorem, we may replace the vector with where is a 3x3 rotation matrix and is the position of the particle at some fixed point in time, say t = 0.

By the divergence theorem Gauss's law can alternatively be written in the differential form:

By connecting outward light to inward light, via an interaction point, this equation stands for the whole ' light transport ' — all the movement of light — in a scene.

By direct substitution, the solution to this equation can be readily shown to be the scalar Green's function, which in the spherical coordinate system ( and using the physics time convention ) is:

By a proper choice of coordinate system, the ellipse can be described by the canonical implicit equation

By eliminating ρ and J, using Maxwell's equations, and manipulating using the theorems of vector calculus, this form of the equation can be used to derive the Maxwell stress tensor σ, in turn this can be combined with the Poynting vector S to obtain the electromagnetic stress-energy tensor T used in general relativity.

By interchanging the roles of x and y one obtains the corresponding equation of a parabola with a vertical axis as

By assuming that light actually consisted of discrete energy packets, Einstein wrote an equation for the photoelectric effect that agreed with experimental results.

By solving this equation, one arrives at the " equations of motion " of the field.

By using the London equation, one can obtain the dependence of the magnetic field inside the superconductor on the distance to the surface.

By the radiation pressure equation σT < sup > 4 </ sup >/ c ; the sun-facing photon pressure is 3. 61 µPa ( 3. 6 N / km², 2. 08 lbf / mi² ).

By finding the derivative of the equation while assuming that y is a constant, the slope of ƒ at the point is found to be:

By using the definition of an average, this equation states that when the acceleration is constant average velocity times time equals displacement.

By analogy, the magnetic equation is an inductive current involving spin.

By putting the measured period T, and the measured distance between the pivot blades L, into the period equation ( 1 ), g could be calculated very accurately.

By implicit differentiation, one can show that all branches of W satisfy the differential equation

By using the equation of hydrostatic equilibrium, combined with conservation of angular momentum and assuming that the disc is thin, the equations of disk structure may be solved in terms of the parameter.

By Late Antiquity, some Gnostic and Neoplatonist thinkers had expanded this syncretic equation to include Aion, Adonis, Attis, Mithras and other gods of the mystery religions.

By 1928 Dirac deduced his equation from the first successful unified combination of special relativity and quantum mechanics to the electron-the Dirac equation.

By contrast the inverse heat equation, deducing a previous distribution of temperature from final data is not well-posed in that the solution is highly sensitive to changes in the final data.

By the Kelvin – Stokes theorem, this equation can also be written in a " differential form ".

By examining Laplace's equation in spherical coordinates, Thomson and Tait recovered Laplace's spherical harmonics.

By separation of variables, two differential equations result by imposing Laplace's equation:

By combining the beamwidth equation with the gain equation, the relation is:

By quantizing this, we get the non-relativistic Schrödinger equation for a free particle,