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The '''[[Dirac delta function]]''' is a function introduced in 1930 by Paul Adrien Maurice Dirac in his seminal book on quantum mechanics. A physical model that visualizes a delta function is a mass distribution of finite total mass ''M''—the integral over the mass distribution.  When the distribution becomes smaller and smaller,  while ''M'' is constant, the mass distribution shrinks to a ''point mass'', which by definition has zero extent and yet has a finite-valued integral equal to total mass ''M''. In the limit of a point mass the distribution becomes a Dirac delta function.
{{:{{FeaturedArticleTitle}}}}
 
<small>
Heuristically, the Dirac delta function can be seen as an extension of the Kronecker delta from integral indices (elements of <font style="vertical-align: 13%"> <math>\mathbb{Z}</math></font>) to real indices (elements of <font style="vertical-align: 13%"><math>\mathbb{R}</math></font>). Note that the Kronecker delta acts as a "filter" in a summation:
==Footnotes==
:<math>
{{reflist|2}}
\sum_{i=m}^n \; f_i\; \delta_{ia} =
</small>
\begin{cases}
f_a & \quad\hbox{if}\quad  a\in[m,n] \sub\mathbb{Z} \\
0  & \quad \hbox{if}\quad a \notin [m,n].
\end{cases}
</math>
 
In analogy, the Dirac delta function &delta;(''x''&minus;''a'')  is defined by (replace ''i'' by ''x'' and the summation over ''i'' by an integration over ''x''),
:<math>
\int_{a_0}^{a_1} f(x)  \delta(x-a) \mathrm{d}x =
\begin{cases}
f(a) & \quad\hbox{if}\quad  a\in[a_0,a_1] \sub\mathbb{R},  \\
0  & \quad \hbox{if}\quad a \notin [a_0,a_1].
\end{cases}
</math>
 
The Dirac delta function is ''not'' an ordinary well-behaved map  <font style="vertical-align: 12%"><math>\mathbb{R} \rightarrow \mathbb{R}</math></font>, but a distribution, also known as an ''improper'' or ''generalized function''. Physicists express its special character by stating that the Dirac delta function makes only sense as a factor in an integrand ("under the integral"). Mathematicians say that the delta function is a linear functional on a space of test functions.
 
====Properties====
Most commonly one takes the lower and the upper bound in the definition of the delta function equal to <math>-\infty</math> and <math> \infty</math>, respectively. From here on this will be done.
:<math>
\begin{align}
\int_{-\infty}^{\infty} \delta(x)\mathrm{d}x &= 1, \\
\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ikx} \mathrm{d}k &= \delta(x) \\
\delta(x-a) &= \delta(a-x), \\
(x-a)\delta(x-a) &= 0, \\
\delta(ax) &= |a|^{-1} \delta(x) \quad (a \ne 0), \\
f(x) \delta(x-a) &= f(a) \delta(x-a), \\
\int_{-\infty}^{\infty} \delta(x-y)\delta(y-a)\mathrm{d}y &= \delta(x-a)
\end{align}
</math>
The physicist's proof of these properties proceeds by making proper substitutions into the integral and using the ordinary rules of integral calculus. The delta function as a Fourier transform of the unit function ''f''(''x'') = 1 (the second property) will be proved below.
The last property is the analogy of the multiplication of two identity matrices,
:<math>
\sum_{j=1}^n \;\delta_{ij}\;\delta_{jk} = \delta_{ik}, \quad i,k=1,\ldots, n.
</math>
''[[Dirac delta function|.... (read more)]]''

Latest revision as of 10:19, 11 September 2020

1901 photograph of a stentor (announcer) at the Budapest Telefon Hirmondó.

Telephone newspaper is a general term for the telephone-based news and entertainment services which were introduced beginning in the 1890s, and primarily located in large European cities. These systems were the first example of electronic broadcasting, and offered a wide variety of programming, however, only a relative few were ever established. Although these systems predated the invention of radio, they were supplanted by radio broadcasting stations beginning in the 1920s, primarily because radio signals were able to cover much wider areas with higher quality audio.

History

After the electric telephone was introduced in the mid-1870s, it was mainly used for personal communication. But the idea of distributing entertainment and news appeared soon thereafter, and many early demonstrations included the transmission of musical concerts. In one particularly advanced example, Clément Ader, at the 1881 Paris Electrical Exhibition, prepared a listening room where participants could hear, in stereo, performances from the Paris Grand Opera. Also, in 1888, Edward Bellamy's influential novel Looking Backward: 2000-1887 foresaw the establishment of entertainment transmitted by telephone lines to individual homes.

The scattered demonstrations were eventually followed by the establishment of more organized services, which were generally called Telephone Newspapers, although all of these systems also included entertainment programming. However, the technical capabilities of the time meant that there were limited means for amplifying and transmitting telephone signals over long distances, so listeners had to wear headphones to receive the programs, and service areas were generally limited to a single city. While some of the systems, including the Telefon Hirmondó, built their own one-way transmission lines, others, including the Electrophone, used standard commercial telephone lines, which allowed subscribers to talk to operators in order to select programming. The Telephone Newspapers drew upon a mixture of outside sources for their programs, including local live theaters and church services, whose programs were picked up by special telephone lines, and then retransmitted to the subscribers. Other programs were transmitted directly from the system's own studios. In later years, retransmitted radio programs were added.

During this era telephones were expensive luxury items, so the subscribers tended to be the wealthy elite of society. Financing was normally done by charging fees, including monthly subscriptions for home users, and, in locations such as hotel lobbies, through the use of coin-operated receivers, which provided short periods of listening for a set payment. Some systems also accepted paid advertising.

Footnotes

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