Rectangular functionThe rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as rect(tT)=Π(tT)={0,if |t|>T212,if |t|=T21,if |t|<T2.{\displaystyle \operatorname {rect} \left({\frac {t}{T}} ight)=\Pi \left({\frac {t}{T}} ight)=\left\{{\begin{array}{rl}0,&{\text{if }}|t|>{\frac {T}{2}}\\{\frac {1}{2}},&{\text{if }}|t|={\frac {T}{2}}\\1,&{\text{if }}|t|<{\frac {T}{2}}.\end{array}} ight.}Alternative definitions of the function define rect(t=±T2){\textstyle \operatorname {rect} \left(t=\pm {\frac {T}{2}} ight)} to be 0, 1, or undefined.
Fourier transformMathematical transform that expresses a function of time as a function of frequencyThe Fourier transform applied to the waveform of a C majorpianochord (with logarithmic horizontal (frequency) axis). The first three peaks on the left correspond to the fundamental frequencies of the chord (C, E, G).
Whittaker–Shannon interpolation formulaThe Whittaker–Shannon interpolation formula or sinc interpolation is a method to construct a continuous-timebandlimited function from a sequence of real numbers. The formula dates back to the works of E. Borel in 1898, and E. T. Whittaker in 1915, and was cited from works of J. M. Whittaker in 1935, and in the formulation of the Nyquist–Shannon sampling theorem by Claude Shannon in 1949.
Nyquist–Shannon sampling theoremExample of magnitude of the Fourier transform of a bandlimited functionThe Nyquist–Shannon sampling theorem, or the sampling theorem, is a theorem in the field of signal processing which serves as a fundamental bridge between continuous-time signals and discrete-time signals.
Analytic functionIn mathematical analysis, an analytic function is a function that is locally represented by a convergent power series. More precisely, a real or complex function is analytic at a point if, in some neighborhood of that point, it is equal to a power series centered there. The analytic function is therefore locally determined by coefficients of the series, or equivalently by derivatives of the function evaluated at the center of the series expansion.Entire functionIn complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functionssine and cosine and their hyperbolic counterpartssinh and cosh, as well as derivatives and integrals of entire functions such as the error function.
Removable singularityIn complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourhood of that point. For instance, the (unnormalized) sinc function, as defined by sinc ( z ) = sin z z {\displaystyle {\text{sinc}}(z)={\frac {\sin z}{z}}} has a singularity at z = 0 {\displaystyle z=0} .
Cartesian coordinate systemIn geometry, a Cartesian coordinate system (UK: , US: ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called coordinates, which are the signed distances to the point from two fixed perpendicularoriented lines, called coordinate lines, coordinate axes or just axes (plural of axis) of the system. The point where the axes meet is called the origin and has (0, 0) as coordinates. The axes directions represent an orthogonal basis.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.PhysicsPhysics is the scientific study of matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. It is one of the most fundamental scientific disciplines. A scientist who specializes in the field of physics is called a physicist. Physics is one of the oldest academic disciplines.
EngineeringEngineering is the practice of systematically applying natural science and mathematics to design and improve systems, devices, or processes that solve problems under constraints. It is typically motivated by satisfying human needs, resulting in creations such as bridges, engines, smartphones, pacemakers, the internet, spacecraft, and washing machines.
Dependent and independent variablesA variable is considered dependent if it depends on (or is hypothesized to depend on) an independent variable. Dependent variables are the outcome of the test they depend on, by some law or rule (e.g., by a mathematical function). Independent variables, on the other hand, are not seen as depending on any other variable in the scope of the experiment in question. Rather, they are controlled by the experimenter.*As an Amazon Associate I earn from qualifying purchases.