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In mathematics, the Dirac delta function, or function, is a generalized function, or distribution, on the real number line that is zero everywhere except at zero, with an integral of one over the entire real line.〔, p. 58〕 The delta function is sometimes thought of as an infinitely high, infinitely thin spike at the origin, with total area one under the spike, and physically represents the density of an idealized point mass or point charge. It was introduced by theoretical physicist Paul Dirac. In the context of signal processing it is often referred to as the unit impulse symbol (or function). Its discrete analog is the Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1. From a purely mathematical viewpoint, the Dirac delta is not strictly a function, because any extended-real function that is equal to zero everywhere but a single point must have total integral zero. The delta function only makes sense as a mathematical object when it appears inside an integral. While from this perspective the Dirac delta can usually be manipulated as though it were a function, formally it must be defined as a distribution that is also a measure. In many applications, the Dirac delta is regarded as a kind of limit (a weak limit) of a sequence of functions having a tall spike at the origin. The approximating functions of the sequence are thus "approximate" or "nascent" delta functions. ==Overview== The graph of the delta function is usually thought of as following the whole ''x''-axis and the positive ''y''-axis. Despite its name, the delta function is not truly a function, at least not a usual one with range in real numbers. For example, the objects and are equal everywhere except at yet have integrals that are different. According to Lebesgue integration theory, if ''f'' and ''g'' are functions such that almost everywhere, then ''f'' is integrable if and only if ''g'' is integrable and the integrals of ''f'' and ''g'' are identical. Rigorous treatment of the Dirac delta requires measure theory or the theory of distributions. The Dirac delta is used to model a tall narrow spike function (an ''impulse''), and other similar abstractions such as a point charge, point mass or electron point. For example, to calculate the dynamics of a baseball being hit by a bat, one can approximate the force of the bat hitting the baseball by a delta function. In doing so, one not only simplifies the equations, but one also is able to calculate the motion of the baseball by only considering the total impulse of the bat against the ball rather than requiring knowledge of the details of how the bat transferred energy to the ball. In applied mathematics, the delta function is often manipulated as a kind of limit (a weak limit) of a sequence of functions, each member of which has a tall spike at the origin: for example, a sequence of Gaussian distributions centered at the origin with variance tending to zero. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dirac delta function」の詳細全文を読む スポンサード リンク
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