Generalized functions: Dirac delta function, Distribution, Heaviside step function, Green's function, Generalized function - Tapa blanda

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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 40. Chapters: Dirac delta function, Distribution, Heaviside step function, Green's function, Generalized function, Poisson summation formula, Homogeneous distribution, Symmetry of second derivatives, Paley-Wiener theorem, Pseudo-differential operator, Wave front set, FBI transform, Fourier inversion theorem, Cauchy principal value, Weak solution, Dirac comb, Fundamental solution, Hyperfunction, Weak derivative, Rigged Hilbert space, Schwartz kernel theorem, Singularity function, Boehmians, Principal part, Algebraic analysis, Microlocal analysis. Excerpt: The Dirac delta function, or δ function, is (informally) a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical physicist Paul Dirac. In the context of signal processing it is often referred to as the unit impulse function. It is a continuous analog of the Kronecker delta function which is usually defined on a finite 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. While for many purposes the Dirac delta can be manipulated as a function, formally it can 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. The graph of the delta function is usually thought of as following the whole x-axis and the positive y-axis. (This informal picture can sometimes be misleading, for example in ...