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Thursday, September 14, 2023

Renaissance humanism in Northern Europe

Renaissance humanism came much later to Germany and Northern Europe in general than to Italy, and when it did, it encountered some resistance from the scholastic theology which reigned at the universities. Humanism may be dated from the invention of the printing press about 1450. Its flourishing period began at the close of the 15th century and lasted only until about 1520, when it was absorbed by the more popular and powerful religious movement, the Reformation, as Italian humanism was superseded by the papal counter-Reformation.

However, the Netherlands was influenced by humanism and the Renaissance until arguably roughly 1550.

Marked features distinguished the new culture north of the Alps from the culture of the Italians. The university and school played a much more important part than in the South according to Catholic historians. The representatives of the new scholarship were teachers; even Erasmus taught in Cambridge and was on intimate terms with the professors at Basel. During the progress of the movement new universities sprang up, from Basel to Rostock. Again, in Germany, there were no princely patrons of arts and learning to be compared in intelligence and munificence to the Renaissance popes and the Medici.[citation needed] Nor was the new culture here exclusive and aristocratic. It sought the general spread of intelligence, and was active in the development of primary and grammar schools. In fact, when the currents of the Italian Renaissance began to set toward the North, a strong, independent, intellectual current was pushing down from the flourishing schools conducted by the Brethren of the Common Life. In the humanistic movement, the German people was far from being a slavish imitator. It received an impulse from the South, but made its own path.

Overview

Albrecht Dürer, self-portrait, 1500

In the North, humanism entered into the service of religious progress. German scholars were less brilliant and elegant, but more serious in their purpose and more exact in their scholarship than their Italian predecessors and contemporaries. In the South, the ancient classics absorbed the attention of the literati. It was not so in the North. There was no consuming passion to render the classics into German as there had been in Italy. Nor did Italian literature, with its often relaxed moral attitude, find imitators in the North. Giovanni Boccaccio’s Decameron was first translated into German by the physician, Henry Stainhowel, who died in 1482. North of the Alps, attention was chiefly centred on the Old and New Testaments. Greek and Hebrew were studied, not with the purpose of ministering to a cult of antiquity, but to reach the fountains of the Christian system more adequately. In this way, preparation was made for the work of the Protestant Reformation. This focus on translation was a feature of the Christian humanists who helped to launch the new, post-scholastic era, among them Erasmus and Luther. In so doing, they also placed biblical texts above any human or institutional authority, an approach that emphasised the role of the reader in understanding a text for him or herself. Closely allied to the late medieval shift of scholarship from the monastery to the university, Christian humanism engendered a new freedom of expression, even though some of its proponents opposed that freedom of expression elsewhere, such as in their censure of the Anabaptists.

What was true of the scholarship of Germany was also true of its art. The painters, Albrecht Dürer (1471–1528), who was born and died at Nuremberg, Lucas Cranach the Elder (1472–1553), and for the most part Hans Holbein the Younger (1497–1543), took little interest in mythology, apart from Cranach's nudes, and were persuaded by the Reformation, though most continued to take commissions for traditional Catholic subjects. Dürer and Holbein had close contacts with leading humanists. Cranach lived in Wittenberg after 1504 and painted portraits of Martin Luther, Philip Melanchthon and other leaders of the German Reformation. Holbein made frontispieces and illustrations for Protestant books and painted portraits of Erasmus and Melanchthon.

The Italian roots of humanism in Germany

Pope Pius II

If any one individual more than another may be designated as the connecting link between the learning of Italy and Germany, it is Aeneas Sylvius. By his residence at the court of Frederick III and at Basel, as one of the secretaries of the council, he became a well-known character north of the Alps long before he was chosen pope. The mediation, however, was not effected by any single individual. The fame of the Renaissance was carried over the pathways of trade which led from Northern Italy to Augsburg, Nuremberg, Konstanz and other German cities. The visits of Frederick III and the campaigns of Charles VIII and the ascent of the throne of Naples by the princes of Aragon carried Germans, Frenchmen and Spaniards to the greater centres of the peninsula. A constant stream of pilgrims travelled to Rome and the Spanish popes drew to the city throngs of Spaniards. As the fame of Italian culture spread, scholars and artists began to travel to Venice, Florence and Rome, and caught the inspiration of the new era.

To the Italians, Germany was a land of barbarians. They despised the German people for their rudeness and intemperance in eating and drinking. Aeneas was impressed by the beauty of Vienna, though it was quite small when compared to the greatest Italian cities. However, he found that the German princes and nobles cared more for horses and dogs than for poets and scholars and loved their wine-cellars better than the muses. Campanus, a witty poet of the papal court, who was sent as legate to the Diet of Regensburg (1471) by Pope Paul II, and afterwards was made a bishop by Pope Pius II, abused Germany for its dirt, cold climate, poverty, sour wine and miserable fare. He lamented his unfortunate nose, which had to smell everything, and praised his ears, which understood nothing. Johannes Santritter, himself being a German living in Italy, admitted that Italy was slightly ahead of Germany in the humanities. However, he also contended that many Italians were jealous of German science and technology, which he considered superior taking the examples of the printing press and the work of the astronomer Regiomontanus.

Such impressions were soon offset by the sound scholarship which arose in Germany and the Netherlands. And, if Italy contributed to Germany an intellectual impulse, Germany sent out to the world the printing press, the most important agent in the history of intellectual culture since the invention of the alphabet.

Universities

Before the first swell of the new movement was felt, the older German universities were already established: University of Vienna in 1365, University of Heidelberg in 1386, University of Cologne in 1388, University of Erfurt in 1392, University of Würzburg in 1402, University of Leipzig in 1409 and University of Rostock in 1419. During the last half of the 15th century, there were quickly added to this list universities at Greifswald and Freiburg 1457, Trier 1457, Basel 1459, Ingolstadt 1472, Tübingen and Mainz 1477, and Wittenberg 1502. Ingolstadt lost its distinct existence by incorporation in the University of Munich, 1826, and Wittenberg by removal to Halle.

Most of these universities had the four faculties, although the popes were slow to give their assent to the sanction of the theological department, as in the case of Vienna and Rostock, where the charter of the secular prince authorized their establishment. Strong as the religious influences of the age were, the social and moral habits of the students were by no means such as to call for praise. Parents, Luther said, in sending their sons to the universities, were sending them to destruction, and an act of the Leipzig university, dating from the close of the 15th century, stated that students came forth from their homes obedient and pious, but "how they returned, God alone knew", to university archives and library.

In the Netherlands, universities or "Latin schools" spurred on by Renaissance humanists helped the majority of people in the region become more literate than in most other European kingdoms.

Education

The theological teaching was ruled by the Schoolmen, and the dialectic method prevailed in all departments. In clashing with the scholastic method and curricula, the new teaching met with many a repulse, and in no case was it thoroughly triumphant till the era of the Reformation opened. Erfurt may be regarded as having been the first to give the new culture a welcome. In 1466, it received Peter Luder of Kislau, who had visited Greece and Asia Minor, and had been previously appointed to a chair in Heidelberg, 1456. He read on Virgil, Jerome, Ovid and other Latin writers. There Agricola studied and there Greek was taught by Nicolas Marschalck, under whose supervision the first Greek book printed in Germany issued from the press, 1501. There John of Wesel taught. It was Luther's alma mater and, among his professors, he singled out Trutvetter for special mention as the one who directed him to the study of the Scriptures.

Portrait of Huldrych Zwingli by Hans Asper

Heidelberg, chartered by the elector Ruprecht I and Pope Urban VI, showed scant sympathy with the new movement. However, the elector-palatine, Philip, 1476–1508, gathered at his court some of its representatives, among them Reuchlin. Ingolstadt for a time had Reuchlin as professor and, in 1492, Conrad Celtes was appointed professor of poetry and eloquence.

In 1474, a chair of poetry was established at Basel. Founded by Pius II, it had among its early teachers two Italians, Finariensis and Publicius. Sebastian Brant taught there at the close of the century and among its notable students were Reuchlin and the Reformers, Leo Jud and Zwingli. In 1481, Tübingen had a stipend of oratoria. Here Gabriel Biel taught till very near the close of the century. The year after Biel's death, Heinrich Bebel was called to lecture on poetry. One of Bebel's distinguished pupils was Philip Melanchthon, who studied and taught in the university, 1512–1518. Reuchlin was called from Ingolstadt to Tübingen, 1521, to teach Hebrew and Greek, but died a few months later.

Leipzig and Cologne remained inaccessible strongholds of scholasticism, till Luther appeared, when Leipzig changed front. The last German university of the Middle Ages, Wittenberg, founded by Frederick the Wise and placed under the patronage of the Virgin Mary and St. Augustine, acquired a worldwide influence through its professors, Luther and Melanchthon. Not till 1518, did it have instruction in Greek, when Melanchthon, soon to be the chief Greek scholar in Germany, was called to one of its chairs at the age of 21. According to Luther, his lecture-room was at once filled brimful, theologians high and low resorting to it.

As seats of the new culture, Nuremberg and Strasbourg occupied, perhaps, even a more prominent place than any of the university towns. These two cities, with Basel and Augsburg, had the most prosperous German printing establishments. At the close of the 15th century, Nuremberg, the fountain of inventions, had four Latin schools and was the home of Albrecht Dürer the painter and his friend Willibald Pirkheimer, a patron of learning.

Popular education, during the century before the Reformation, was far more advanced in Germany than in other nations. Apart from the traditional monastic and civic schools, the Brothers of the Common Life had schools at Zwolle, Deventer, 's-Hertogenbosch and Liège in the Low Countries. All the leading towns had schools. The town of Sélestat in Alsace was noted as a classical centre. Here, Thomas Platter found Hans Sapidus teaching, and he regarded it as the best school he had found. In 1494, there were five pedagogues in Wesel, teaching reading, writing, arithmetic and singing. One Christmas the clergy of the place entertained the pupils, giving them each cloth for a new coat and a piece of money as begun with the 4th class.

Among the noted schoolmasters was Alexander Hegius, who taught at Deventer for nearly a quarter of a century, till his death in 1498. At the age of 40 he was not ashamed to sit at the feet of Agricola. He made the classics central in education and banished the old text-books. Trebonius, who taught Luther at Eisenach, belonged to a class of worthy men. The penitential books of the day called upon parents to be diligent in keeping their children off the streets and sending them to school.

Leaders of Northern humanism

Portrait of Rudolph Agricola by Lucas Cranach the Elder, ca. 1532

The leading Northern humanists included Rudolph Agricola, Reuchlin and Erasmus. Agricola, whose original name was Roelef Huisman, was born near Groningen in 1443 and died 1485. He enjoyed the highest reputation in his day as a scholar and received unstinted praise from Erasmus and Melanchthon. He has been regarded as doing for Humanism in Germany what was done in Italy by Petrarch, the first biography of whom, in German, Agricola prepared. After studying in Erfurt, Louvain and Cologne, Agricola went to Italy, spending some time at the universities in Pavia and Ferrara. He declined a professor's chair in favor of an appointment at the court of Philip of the Palatinate in Heidelberg. He made Cicero and Quintilian his models. In his last years, he turned his attention to theology and studied Hebrew. Like Pico della Mirandola, he was a monk. The inscription on his tomb in Heidelberg stated that he had studied what is taught about God and the true faith of the Saviour in the books of Scripture.

Johannes Trithemius by Tilman Riemenschneider

Another Humanist was Jacob Wimpheling, 1450–1528, of Schlettstadt, who taught in Heidelberg. He was inclined to be severe on clerical abuses but, at the close of his career, wanted to substitute for the study of Virgil and Horace, Sedulius and Prudentius. The poetic Sebastian Brant, 1457–1521, the author of the Ship of Fools, began his career as a teacher of law in Basel. Mutianus Rufus, in his correspondence, went so far as to declare that Christianity is as old as the world and that Jupiter, Apollo, Ceres and Christ are only different names of the one hidden God.

A name which deserves a high place in the German literature of the last years of the Middle Ages is John Trithemius, 1462–1516, abbot of a Benedictine convent at Sponheim, which, under his guidance, gained the reputation of a learned academy. He gathered a library of 2,000 volumes and wrote a patrology, or encyclopaedia of the Fathers, and a catalogue of the renowned men of Germany. Increasing differences with the convent led to his resignation in 1506, when he decided to take up the offer of the Lord Bishop of Würzburg, Lorenz von Bibra (bishop from 1495 to 1519), to become abbot of the Schottenkloster in Würzburg. He remained there until the end of his life. Prelates and nobles visited him to consult and read the Latin and Greek authors he had collected. These men and others contributed their part to that movement of which Reuchlin and Erasmus were the chief lights and which led to the Protestant Reformation.

German Renaissance

From Wikipedia, the free encyclopedia
Portrait of Emperor Maximilian I (reigned: 1493–1519), the first Renaissance monarch of the Holy Roman Empire, by Albrecht Dürer, 1519
First page of a copy of the Gutenberg Bible, now in Texas

The German Renaissance, part of the Northern Renaissance, was a cultural and artistic movement that spread among German thinkers in the 15th and 16th centuries, which developed from the Italian Renaissance. Many areas of the arts and sciences were influenced, notably by the spread of Renaissance humanism to the various German states and principalities. There were many advances made in the fields of architecture, the arts, and the sciences. Germany produced two developments that were to dominate the 16th century all over Europe: printing and the Protestant Reformation.

One of the most important German humanists was Konrad Celtis (1459–1508). Celtis studied at Cologne and Heidelberg, and later travelled throughout Italy collecting Latin and Greek manuscripts. Heavily influenced by Tacitus, he used the Germania to introduce German history and geography. Eventually he devoted his time to poetry, in which he praised Germany in Latin. Another important figure was Johann Reuchlin (1455–1522) who studied in various places in Italy and later taught Greek. He studied the Hebrew language, aiming to purify Christianity, but encountered resistance from the church.

The most significant German Renaissance artist is Albrecht Dürer especially known for his printmaking in woodcut and engraving, which spread all over Europe, drawings, and painted portraits. Important architecture of this period includes the Landshut Residence, Heidelberg Castle, the Augsburg Town Hall as well as the Antiquarium of the Munich Residenz in Munich, the largest Renaissance hall north of the Alps.

Background

The Renaissance was largely driven by the renewed interest in classical learning, and was also the result of rapid economic development. At the beginning of the 16th century, Germany (referring to the lands contained within the Holy Roman Empire) was one of the most prosperous areas in Europe despite a relatively low level of urbanization compared to Italy or the Netherlands. It benefited from the wealth of certain sectors such as metallurgy, mining, banking and textiles. More importantly, book-printing developed in Germany, and German printers dominated the new book-trade in most other countries until well into the 16th century.

Art

The Heller altar by Albrecht Dürer

The concept of the Northern Renaissance or German Renaissance is somewhat confused by the continuation of the use of elaborate Gothic ornament until well into the 16th century, even in works that are undoubtedly Renaissance in their treatment of the human figure and other respects. Classical ornament had little historical resonance in much of Germany, but in other respects Germany was very quick to follow developments, especially in adopting printing with movable type, a German invention that remained almost a German monopoly for some decades, and was first brought to most of Europe, including France and Italy, by Germans.

Printmaking by woodcut and engraving was already more developed in Germany and the Low Countries than elsewhere in Europe, and the Germans took the lead in developing book illustrations, typically of a relatively low artistic standard, but seen all over Europe, with the woodblocks often being lent to printers of editions in other cities or languages. The greatest artist of the German Renaissance, Albrecht Dürer, began his career as an apprentice to a leading workshop in Nuremberg, that of Michael Wolgemut, who had largely abandoned his painting to exploit the new medium. Dürer worked on the most extravagantly illustrated book of the period, the Nuremberg Chronicle, published by his godfather Anton Koberger, Europe's largest printer-publisher at the time.

After completing his apprenticeship in 1490, Dürer travelled in Germany for four years, and Italy for a few months, before establishing his own workshop in Nuremberg. He rapidly became famous all over Europe for his energetic and balanced woodcuts and engravings, while also painting. Though retaining a distinctively German style, his work shows strong Italian influence, and is often taken to represent the start of the German Renaissance in visual art, which for the next forty years replaced the Netherlands and France as the area producing the greatest innovation in Northern European art. Dürer supported Martin Luther but continued to create Madonnas and other Catholic imagery, and paint portraits of leaders on both sides of the emerging split of the Protestant Reformation.

"The Crucifixion", central panel of the Isenheim Altarpiece by Matthias Grünewald.

Dürer died in 1528, before it was clear that the split of the Reformation had become permanent, but his pupils of the following generation were unable to avoid taking sides. Most leading German artists became Protestants, but this deprived them of painting most religious works, previously the mainstay of artists' revenue. Martin Luther had objected to much Catholic imagery, but not to imagery itself, and Lucas Cranach the Elder, a close friend of Luther, had painted a number of "Lutheran altarpieces", mostly showing the Last Supper, some with portraits of the leading Protestant divines as the Twelve Apostles. This phase of Lutheran art was over before 1550, probably under the more fiercely aniconic influence of Calvinism, and religious works for public display virtually ceased to be produced in Protestant areas. Presumably largely because of this, the development of German art had virtually ceased by about 1550, but in the preceding decades German artists had been very fertile in developing alternative subjects to replace the gap in their order books. Cranach, apart from portraits, developed a format of thin vertical portraits of provocative nudes, given classical or Biblical titles.

Lying somewhat outside these developments is Matthias Grünewald, who left very few works, but whose masterpiece, his Isenheim Altarpiece (completed 1515), has been widely regarded as the greatest German Renaissance painting since it was restored to critical attention in the 19th century. It is an intensely emotional work that continues the German Gothic tradition of unrestrained gesture and expression, using Renaissance compositional principles, but all in that most Gothic of forms, the multi-winged triptych.

Albrecht Altdorfer (c. 1480–1538), Danube landscape near Regensburg c. 1528, one of the earliest Western pure landscapes, from the Danube School in southern Germany.

The Danube School is the name of a circle of artists of the first third of the 16th century in Bavaria and Austria, including Albrecht Altdorfer, Wolf Huber and Augustin Hirschvogel. With Altdorfer in the lead, the school produced the first examples of independent landscape art in the West (nearly 1,000 years after China), in both paintings and prints. Their religious paintings had an expressionist style somewhat similar to Grünewald's. Dürer's pupils Hans Burgkmair and Hans Baldung Grien worked largely in prints, with Baldung developing the topical subject matter of witches in a number of enigmatic prints.

Hans Holbein the Elder and his brother Sigismund Holbein painted religious works in the late Gothic style. Hans the Elder was a pioneer and leader in the transformation of German art from the Gothic to the Renaissance style. His son, Hans Holbein the Younger was an important painter of portraits and a few religious works, working mainly in England and Switzerland. Holbein's well known series of small woodcuts on the Dance of Death relate to the works of the Little Masters, a group of printmakers who specialized in very small and highly detailed engravings for bourgeois collectors, including many erotic subjects.

The outstanding achievements of the first half of the 16th century were followed by several decades with a remarkable absence of noteworthy German art, other than accomplished portraits that never rival the achievement of Holbein or Dürer. The next significant German artists worked in the rather artificial style of Northern Mannerism, which they had to learn in Italy or Flanders. Hans von Aachen and the Netherlandish Bartholomeus Spranger were the leading painters at the Imperial courts in Vienna and Prague, and the productive Netherlandish Sadeler family of engravers spread out across Germany, among other counties.

In Catholic parts of South Germany the Gothic tradition of wood carving continued to flourish until the end of the 18th century, adapting to changes in style through the centuries. Veit Stoss (d. 1533), Tilman Riemenschneider (d.1531) and Peter Vischer the Elder (d. 1529) were Dürer's contemporaries, and their long careers covered the transition between the Gothic and Renaissance periods, although their ornament often remained Gothic even after their compositions began to reflect Renaissance principles.

Architecture

Heidelberg Castle
Juleum in Helmstedt (built 1592), an example of Weser Renaissance architecture

Renaissance architecture in Germany was inspired first by German philosophers and artists such as Albrecht Dürer and Johannes Reuchlin who visited Italy. Important early examples of this period are especially the Landshut Residence, the Castle in Heidelberg, Johannisburg Palace in Aschaffenburg, Schloss Weilburg, the City Hall and Fugger Houses in Augsburg and St. Michael in Munich, the largest Renaissance church north of the Alps.

A particular form of Renaissance architecture in Germany is the Weser Renaissance, with prominent examples such as the City Hall of Bremen and the Juleum in Helmstedt.

In July 1567 the city council of Cologne approved a design in the Renaissance style by Wilhelm Vernukken for a two storied loggia for Cologne City Hall. St Michael in Munich is the largest Renaissance church north of the Alps. It was built by Duke William V of Bavaria between 1583 and 1597 as a spiritual center for the Counter Reformation and was inspired by the Church of il Gesù in Rome. The architect is unknown. Many examples of Brick Renaissance buildings can be found in Hanseatic old towns, such as Stralsund, Wismar, Lübeck, Lüneburg, Friedrichstadt and Stade. Notable German Renaissance architects include Friedrich Sustris, Benedikt Rejt, Abraham van den Blocke, Elias Holl and Hans Krumpper.

Influential people

Johannes Gutenberg (c. 1398–1468)

Born Johannes Gensfleisch zur Laden, Johannes Gutenberg is widely considered the most influential person within the German Renaissance. As a free thinker, humanist, and inventor, Gutenberg also grew up within the Renaissance, but influenced it greatly as well. His best-known invention is the printing press in 1440. Gutenberg's press allowed the humanists, reformists, and others to circulate their ideas. He is also known as the creator of the Gutenberg Bible, a crucial work that marked the start of the Gutenberg Revolution and the age of the printed book in the Western world.

Johann Reuchlin (1455–1522)

Johann Reuchlin was the most important aspect of world culture teaching within Germany at this time. He was a scholar of both Greek and Hebrew. Graduating, then going on to teach at Basel, he was considered extremely intelligent. Yet after leaving Basel, he had to start copying manuscripts and apprenticing within areas of law. However, he is most known for his work within Hebrew studies. Unlike some other "thinkers" of this time, Reuchlin submerged himself into this, even creating a guide to preaching within the Hebrew faith. The book, titled De Arte Predicandi (1503), is possibly one of his best-known works from this period.

Albrecht Dürer (1471–1528)

Albrecht Dürer was at the time, and remains, the most famous artist of the German Renaissance. He was famous across Europe, and greatly admired in Italy, where his work was mainly known through his prints. He successfully integrated an elaborate Northern style with Renaissance harmony and monumentality. Among his best known works are Melencolia I, the Four Horsemen from his woodcut Apocalypse series, and Knight, Death, and the Devil. Other significant artists were Lucas Cranach the Elder, the Danube School and the Little Masters.

Martin Luther (1483–1546)

Martin Luther was a Protestant Reformer who criticized church practices such as selling indulgences, against which he published in his Ninety-Five Theses of 1517. Luther also translated the Bible into German, making the Christian scriptures more accessible to the general population and inspiring the standardization of the German language.

Dirac delta function

From Wikipedia, the free encyclopedia
Schematic representation of the Dirac delta function by a line surmounted by an arrow. The height of the arrow is usually meant to specify the value of any multiplicative constant, which will give the area under the function. The other convention is to write the area next to the arrowhead.
The Dirac delta as the limit as (in the sense of distributions) of the sequence of zero-centered normal distributions

In mathematical physics, the Dirac delta distribution (δ distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one.

The current understanding of the unit impulse is as a linear functional that maps every continuous function (e.g., ) to its value at zero of its domain (), or as the weak limit of a sequence of bump functions (e.g., ), which are zero over most of the real line, with a tall spike at the origin. Bump functions are thus sometimes called "approximate" or "nascent" delta distributions.

The delta function was introduced by physicist Paul Dirac as a tool for the normalization of state vectors. It also has uses in probability theory and signal processing. Its validity was disputed until Laurent Schwartz developed the theory of distributions where it is defined as a linear form acting on functions.

The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is the discrete analog of the Dirac delta function.

Motivation and overview

The graph of the Dirac delta is usually thought of as following the whole x-axis and the positive y-axis. 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 billiard ball being struck, one can approximate the force of the impact by a Dirac delta. In doing so, one not only simplifies the equations, but one also is able to calculate the motion of the ball by only considering the total impulse of the collision without a detailed model of all of the elastic energy transfer at subatomic levels (for instance).

To be specific, suppose that a billiard ball is at rest. At time it is struck by another ball, imparting it with a momentum P, with units kg⋅m⋅s−1. The exchange of momentum is not actually instantaneous, being mediated by elastic processes at the molecular and subatomic level, but for practical purposes it is convenient to consider that energy transfer as effectively instantaneous. The force therefore is P δ(t); the units of δ(t) are s−1.

To model this situation more rigorously, suppose that the force instead is uniformly distributed over a small time interval . That is,

Then the momentum at any time t is found by integration:

Now, the model situation of an instantaneous transfer of momentum requires taking the limit as Δt → 0, giving a result everywhere except at 0:

Here the functions are thought of as useful approximations to the idea of instantaneous transfer of momentum.

The delta function allows us to construct an idealized limit of these approximations. Unfortunately, the actual limit of the functions (in the sense of pointwise convergence) is zero everywhere but a single point, where it is infinite. To make proper sense of the Dirac delta, we should instead insist that the property

which holds for all , should continue to hold in the limit. So, in the equation , it is understood that the limit is always taken outside the integral.

In applied mathematics, as we have done here, 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.

The Dirac delta is not truly a function, at least not a usual one with domain and range in real numbers. For example, the objects f(x) = δ(x) and g(x) = 0 are equal everywhere except at x = 0 yet have integrals that are different. According to Lebesgue integration theory, if f and g are functions such that f = g almost everywhere, then f is integrable if and only if g is integrable and the integrals of f and g are identical. A rigorous approach to regarding the Dirac delta function as a mathematical object in its own right requires measure theory or the theory of distributions.

History

Joseph Fourier presented what is now called the Fourier integral theorem in his treatise Théorie analytique de la chaleur in the form:

which is tantamount to the introduction of the δ-function in the form:

Later, Augustin Cauchy expressed the theorem using exponentials:

Cauchy pointed out that in some circumstances the order of integration is significant in this result (contrast Fubini's theorem).

As justified using the theory of distributions, the Cauchy equation can be rearranged to resemble Fourier's original formulation and expose the δ-function as

where the δ-function is expressed as

A rigorous interpretation of the exponential form and the various limitations upon the function f necessary for its application extended over several centuries. The problems with a classical interpretation are explained as follows:

The greatest drawback of the classical Fourier transformation is a rather narrow class of functions (originals) for which it can be effectively computed. Namely, it is necessary that these functions decrease sufficiently rapidly to zero (in the neighborhood of infinity) to ensure the existence of the Fourier integral. For example, the Fourier transform of such simple functions as polynomials does not exist in the classical sense. The extension of the classical Fourier transformation to distributions considerably enlarged the class of functions that could be transformed and this removed many obstacles.

Further developments included generalization of the Fourier integral, "beginning with Plancherel's pathbreaking L2-theory (1910), continuing with Wiener's and Bochner's works (around 1930) and culminating with the amalgamation into L. Schwartz's theory of distributions (1945) ...", and leading to the formal development of the Dirac delta function.

An infinitesimal formula for an infinitely tall, unit impulse delta function (infinitesimal version of Cauchy distribution) explicitly appears in an 1827 text of Augustin Louis CauchySiméon Denis Poisson considered the issue in connection with the study of wave propagation as did Gustav Kirchhoff somewhat later. Kirchhoff and Hermann von Helmholtz also introduced the unit impulse as a limit of Gaussians, which also corresponded to Lord Kelvin's notion of a point heat source. At the end of the 19th century, Oliver Heaviside used formal Fourier series to manipulate the unit impulse. The Dirac delta function as such was introduced by Paul Dirac in his 1927 paper The Physical Interpretation of the Quantum Dynamics and used in his textbook The Principles of Quantum Mechanics. He called it the "delta function" since he used it as a continuous analogue of the discrete Kronecker delta.

Definitions

The Dirac delta function can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite,

and which is also constrained to satisfy the identity

This is merely a heuristic characterization. The Dirac delta is not a function in the traditional sense as no function defined on the real numbers has these properties.

Another equivalent definition of the Dirac delta function: is a function (in a loose sense) that satisfies

where g(x) is a well-behaved function. The second condition in this definition can be derived by the first definition above:
The Dirac delta function can be rigorously defined either as a distribution or as a measure as described below.

As a measure

One way to rigorously capture the notion of the Dirac delta function is to define a measure, called Dirac measure, which accepts a subset A of the real line R as an argument, and returns δ(A) = 1 if 0 ∈ A, and δ(A) = 0 otherwise. If the delta function is conceptualized as modeling an idealized point mass at 0, then δ(A) represents the mass contained in the set A. One may then define the integral against δ as the integral of a function against this mass distribution. Formally, the Lebesgue integral provides the necessary analytic device. The Lebesgue integral with respect to the measure δ satisfies

for all continuous compactly supported functions f. The measure δ is not absolutely continuous with respect to the Lebesgue measure—in fact, it is a singular measure. Consequently, the delta measure has no Radon–Nikodym derivative (with respect to Lebesgue measure)—no true function for which the property

holds. As a result, the latter notation is a convenient abuse of notation, and not a standard (Riemann or Lebesgue) integral.

As a probability measure on R, the delta measure is characterized by its cumulative distribution function, which is the unit step function.

This means that H(x) is the integral of the cumulative indicator function 1(−∞, x] with respect to the measure δ; to wit,

the latter being the measure of this interval; more formally, δ((−∞, x]). Thus in particular the integration of the delta function against a continuous function can be properly understood as a Riemann–Stieltjes integral:

All higher moments of δ are zero. In particular, characteristic function and moment generating function are both equal to one.

As a distribution

In the theory of distributions, a generalized function is considered not a function in itself but only about how it affects other functions when "integrated" against them. In keeping with this philosophy, to define the delta function properly, it is enough to say what the "integral" of the delta function is against a sufficiently "good" test function φ. Test functions are also known as bump functions. If the delta function is already understood as a measure, then the Lebesgue integral of a test function against that measure supplies the necessary integral.

A typical space of test functions consists of all smooth functions on R with compact support that have as many derivatives as required. As a distribution, the Dirac delta is a linear functional on the space of test functions and is defined by

 

 

 

 

(1)

for every test function φ.

For δ to be properly a distribution, it must be continuous in a suitable topology on the space of test functions. In general, for a linear functional S on the space of test functions to define a distribution, it is necessary and sufficient that, for every positive integer N there is an integer MN and a constant CN such that for every test function φ, one has the inequality

where sup represents the supremum. With the δ distribution, one has such an inequality (with CN = 1) with MN = 0 for all N. Thus δ is a distribution of order zero. It is, furthermore, a distribution with compact support (the support being {0}).

The delta distribution can also be defined in several equivalent ways. For instance, it is the distributional derivative of the Heaviside step function. This means that for every test function φ, one has

Intuitively, if integration by parts were permitted, then the latter integral should simplify to

and indeed, a form of integration by parts is permitted for the Stieltjes integral, and in that case, one does have

In the context of measure theory, the Dirac measure gives rise to distribution by integration. Conversely, equation (1) defines a Daniell integral on the space of all compactly supported continuous functions φ which, by the Riesz representation theorem, can be represented as the Lebesgue integral of φ concerning some Radon measure.

Generally, when the term Dirac delta function is used, it is in the sense of distributions rather than measures, the Dirac measure being among several terms for the corresponding notion in measure theory. Some sources may also use the term Dirac delta distribution.

Generalizations

The delta function can be defined in n-dimensional Euclidean space Rn as the measure such that

for every compactly supported continuous function f. As a measure, the n-dimensional delta function is the product measure of the 1-dimensional delta functions in each variable separately. Thus, formally, with x = (x1, x2, ..., xn), one has

 

 

 

 

(2)

The delta function can also be defined in the sense of distributions exactly as above in the one-dimensional case. However, despite widespread use in engineering contexts, (2) should be manipulated with care, since the product of distributions can only be defined under quite narrow circumstances.

The notion of a Dirac measure makes sense on any set. Thus if X is a set, x0X is a marked point, and Σ is any sigma algebra of subsets of X, then the measure defined on sets A ∈ Σ by

is the delta measure or unit mass concentrated at x0.

Another common generalization of the delta function is to a differentiable manifold where most of its properties as a distribution can also be exploited because of the differentiable structure. The delta function on a manifold M centered at the point x0M is defined as the following distribution:

 

 

 

 

(3)

for all compactly supported smooth real-valued functions φ on M. A common special case of this construction is a case in which M is an open set in the Euclidean space Rn.

On a locally compact Hausdorff space X, the Dirac delta measure concentrated at a point x is the Radon measure associated with the Daniell integral (3) on compactly supported continuous functions φ. At this level of generality, calculus as such is no longer possible, however a variety of techniques from abstract analysis are available. For instance, the mapping is a continuous embedding of X into the space of finite Radon measures on X, equipped with its vague topology. Moreover, the convex hull of the image of X under this embedding is dense in the space of probability measures on X.

Properties

Scaling and symmetry

The delta function satisfies the following scaling property for a non-zero scalar α:

and so

 

 

 

 

(4)

Scaling property proof:

where a change of variable x′ = ax is used. If a is negative, i.e., a = −|a|, then
Thus, .

In particular, the delta function is an even distribution (symmetry), in the sense that

which is homogeneous of degree −1.

Algebraic properties

The distributional product of δ with x is equal to zero:

More generally, for all positive integers .

Conversely, if xf(x) = xg(x), where f and g are distributions, then

for some constant c.

Translation

The integral of the time-delayed Dirac delta is

This is sometimes referred to as the sifting property or the sampling property. The delta function is said to "sift out" the value of f(t) at t = T.

It follows that the effect of convolving a function f(t) with the time-delayed Dirac delta is to time-delay f(t) by the same amount:

The sifting property holds under the precise condition that f be a tempered distribution (see the discussion of the Fourier transform below). As a special case, for instance, we have the identity (understood in the distribution sense)

Composition with a function

More generally, the delta distribution may be composed with a smooth function g(x) in such a way that the familiar change of variables formula holds, that

provided that g is a continuously differentiable function with g′ nowhere zero. That is, there is a unique way to assign meaning to the distribution so that this identity holds for all compactly supported test functions f. Therefore, the domain must be broken up to exclude the g′ = 0 point. This distribution satisfies δ(g(x)) = 0 if g is nowhere zero, and otherwise if g has a real root at x0, then

It is natural therefore to define the composition δ(g(x)) for continuously differentiable functions g by

where the sum extends over all roots (i.e., all the different ones) of g(x), which are assumed to be simple. Thus, for example

In the integral form, the generalized scaling property may be written as

Indefinite integral

For a constant and a "well-behaved" arbitrary real-valued function y(x),

where H(x) is the Heaviside step function and c is an integration constant.

Properties in n dimensions

The delta distribution in an n-dimensional space satisfies the following scaling property instead,

so that δ is a homogeneous distribution of degree n.

Under any reflection or rotation ρ, the delta function is invariant,

As in the one-variable case, it is possible to define the composition of δ with a bi-Lipschitz function g: RnRn uniquely so that the identity

for all compactly supported functions f.

Using the coarea formula from geometric measure theory, one can also define the composition of the delta function with a submersion from one Euclidean space to another one of different dimension; the result is a type of current. In the special case of a continuously differentiable function g : RnR such that the gradient of g is nowhere zero, the following identity holds

where the integral on the right is over g−1(0), the (n − 1)-dimensional surface defined by g(x) = 0 with respect to the Minkowski content measure. This is known as a simple layer integral.

|More generally, if S is a smooth hypersurface of Rn, then we can associate to S the distribution that integrates any compactly supported smooth function g over S:

where σ is the hypersurface measure associated to S. This generalization is associated with the potential theory of simple layer potentials on S. If D is a domain in Rn with smooth boundary S, then δS is equal to the normal derivative of the indicator function of D in the distribution sense,

where n is the outward normal. For a proof, see e.g. the article on the surface delta function.

In three dimensions, the delta function is represented in spherical coordinates by:

Fourier transform

The delta function is a tempered distribution, and therefore it has a well-defined Fourier transform. Formally, one finds

Properly speaking, the Fourier transform of a distribution is defined by imposing self-adjointness of the Fourier transform under the duality pairing of tempered distributions with Schwartz functions. Thus is defined as the unique tempered distribution satisfying

for all Schwartz functions φ. And indeed it follows from this that

As a result of this identity, the convolution of the delta function with any other tempered distribution S is simply S:

That is to say that δ is an identity element for the convolution on tempered distributions, and in fact, the space of compactly supported distributions under convolution is an associative algebra with identity the delta function. This property is fundamental in signal processing, as convolution with a tempered distribution is a linear time-invariant system, and applying the linear time-invariant system measures its impulse response. The impulse response can be computed to any desired degree of accuracy by choosing a suitable approximation for δ, and once it is known, it characterizes the system completely. See LTI system theory § Impulse response and convolution.

The inverse Fourier transform of the tempered distribution f(ξ) = 1 is the delta function. Formally, this is expressed as

and more rigorously, it follows since
for all Schwartz functions f.

In these terms, the delta function provides a suggestive statement of the orthogonality property of the Fourier kernel on R. Formally, one has

This is, of course, shorthand for the assertion that the Fourier transform of the tempered distribution

is
which again follows by imposing self-adjointness of the Fourier transform.

By analytic continuation of the Fourier transform, the Laplace transform of the delta function is found to be[49]

Derivatives of the Dirac delta function

The derivative of the Dirac delta distribution, denoted δ′ and also called the Dirac delta prime or Dirac delta derivative as described in Laplacian of the indicator, is defined on compactly supported smooth test functions φ by

The first equality here is a kind of integration by parts, for if δ were a true function then

By mathematical induction, the k-th derivative of δ is defined similarly as the distribution given on test functions by

In particular, δ is an infinitely differentiable distribution.

The first derivative of the delta function is the distributional limit of the difference quotients:

More properly, one has

where τh is the translation operator, defined on functions by τhφ(x) = φ(x + h), and on a distribution S by

In the theory of electromagnetism, the first derivative of the delta function represents a point magnetic dipole situated at the origin. Accordingly, it is referred to as a dipole or the doublet function.

The derivative of the delta function satisfies a number of basic properties, including:

which can be shown by applying a test function and integrating by parts.

The latter of these properties can also be demonstrated by applying distributional derivative definition, Liebnitz's theorem and linearity of inner product:

Furthermore, the convolution of δ′ with a compactly-supported, smooth function f is

which follows from the properties of the distributional derivative of a convolution.

Higher dimensions

More generally, on an open set U in the n-dimensional Euclidean space , the Dirac delta distribution centered at a point aU is defined by

for all , the space of all smooth functions with compact support on U. If is any multi-index with and denotes the associated mixed partial derivative operator, then the α-th derivative αδa of δa is given by

That is, the α-th derivative of δa is the distribution whose value on any test function φ is the α-th derivative of φ at a (with the appropriate positive or negative sign).

The first partial derivatives of the delta function are thought of as double layers along the coordinate planes. More generally, the normal derivative of a simple layer supported on a surface is a double layer supported on that surface and represents a laminar magnetic monopole. Higher derivatives of the delta function are known in physics as multipoles.

Higher derivatives enter into mathematics naturally as the building blocks for the complete structure of distributions with point support. If S is any distribution on U supported on the set {a} consisting of a single point, then there is an integer m and coefficients cα such that

Representations of the delta function

The delta function can be viewed as the limit of a sequence of functions

where ηε(x) is sometimes called a nascent delta function. This limit is meant in a weak sense: either that

 

 

 

 

(5)

for all continuous functions f having compact support, or that this limit holds for all smooth functions f with compact support. The difference between these two slightly different modes of weak convergence is often subtle: the former is convergence in the vague topology of measures, and the latter is convergence in the sense of distributions.

Approximations to the identity

Typically a nascent delta function ηε can be constructed in the following manner. Let η be an absolutely integrable function on R of total integral 1, and define

In n dimensions, one uses instead the scaling

Then a simple change of variables shows that ηε also has integral 1. One may show that (5) holds for all continuous compactly supported functions f, and so ηε converges weakly to δ in the sense of measures.

The ηε constructed in this way are known as an approximation to the identity. This terminology is because the space L1(R) of absolutely integrable functions is closed under the operation of convolution of functions: fgL1(R) whenever f and g are in L1(R). However, there is no identity in L1(R) for the convolution product: no element h such that fh = f for all f. Nevertheless, the sequence ηε does approximate such an identity in the sense that

This limit holds in the sense of mean convergence (convergence in L1). Further conditions on the ηε, for instance that it be a mollifier associated to a compactly supported function, are needed to ensure pointwise convergence almost everywhere.

If the initial η = η1 is itself smooth and compactly supported then the sequence is called a mollifier. The standard mollifier is obtained by choosing η to be a suitably normalized bump function, for instance

In some situations such as numerical analysis, a piecewise linear approximation to the identity is desirable. This can be obtained by taking η1 to be a hat function. With this choice of η1, one has

which are all continuous and compactly supported, although not smooth and so not a mollifier.

Probabilistic considerations

In the context of probability theory, it is natural to impose the additional condition that the initial η1 in an approximation to the identity should be positive, as such a function then represents a probability distribution. Convolution with a probability distribution is sometimes favorable because it does not result in overshoot or undershoot, as the output is a convex combination of the input values, and thus falls between the maximum and minimum of the input function. Taking η1 to be any probability distribution at all, and letting ηε(x) = η1(x/ε)/ε as above will give rise to an approximation to the identity. In general this converges more rapidly to a delta function if, in addition, η has mean 0 and has small higher moments. For instance, if η1 is the uniform distribution on , also known as the rectangular function, then:

Another example is with the Wigner semicircle distribution

This is continuous and compactly supported, but not a mollifier because it is not smooth.

Semigroups

Nascent delta functions often arise as convolution semigroups. This amounts to the further constraint that the convolution of ηε with ηδ must satisfy

for all ε, δ > 0. Convolution semigroups in L1 that form a nascent delta function are always an approximation to the identity in the above sense, however the semigroup condition is quite a strong restriction.

In practice, semigroups approximating the delta function arise as fundamental solutions or Green's functions to physically motivated elliptic or parabolic partial differential equations. In the context of applied mathematics, semigroups arise as the output of a linear time-invariant system. Abstractly, if A is a linear operator acting on functions of x, then a convolution semigroup arises by solving the initial value problem

in which the limit is as usual understood in the weak sense. Setting ηε(x) = η(ε, x) gives the associated nascent delta function.

Some examples of physically important convolution semigroups arising from such a fundamental solution include the following.

The heat kernel

The heat kernel, defined by

represents the temperature in an infinite wire at time t > 0, if a unit of heat energy is stored at the origin of the wire at time t = 0. This semigroup evolves according to the one-dimensional heat equation:

In probability theory, ηε(x) is a normal distribution of variance ε and mean 0. It represents the probability density at time t = ε of the position of a particle starting at the origin following a standard Brownian motion. In this context, the semigroup condition is then an expression of the Markov property of Brownian motion.

In higher-dimensional Euclidean space Rn, the heat kernel is

and has the same physical interpretation, mutatis mutandis. It also represents a nascent delta function in the sense that ηεδ in the distribution sense as ε → 0.

The Poisson kernel

The Poisson kernel

is the fundamental solution of the Laplace equation in the upper half-plane. It represents the electrostatic potential in a semi-infinite plate whose potential along the edge is held at fixed at the delta function. The Poisson kernel is also closely related to the Cauchy distribution and Epanechnikov and Gaussian kernel functions. This semigroup evolves according to the equation

where the operator is rigorously defined as the Fourier multiplier

Oscillatory integrals

In areas of physics such as wave propagation and wave mechanics, the equations involved are hyperbolic and so may have more singular solutions. As a result, the nascent delta functions that arise as fundamental solutions of the associated Cauchy problems are generally oscillatory integrals. An example, which comes from a solution of the Euler–Tricomi equation of transonic gas dynamics, is the rescaled Airy function

Although using the Fourier transform, it is easy to see that this generates a semigroup in some sense—it is not absolutely integrable and so cannot define a semigroup in the above strong sense. Many nascent delta functions constructed as oscillatory integrals only converge in the sense of distributions (an example is the Dirichlet kernel below), rather than in the sense of measures.

Another example is the Cauchy problem for the wave equation in R1+1:

The solution u represents the displacement from equilibrium of an infinite elastic string, with an initial disturbance at the origin.

Other approximations to the identity of this kind include the sinc function (used widely in electronics and telecommunications)

and the Bessel function

Plane wave decomposition

One approach to the study of a linear partial differential equation

where L is a differential operator on Rn, is to seek first a fundamental solution, which is a solution of the equation

When L is particularly simple, this problem can often be resolved using the Fourier transform directly (as in the case of the Poisson kernel and heat kernel already mentioned). For more complicated operators, it is sometimes easier first to consider an equation of the form

where h is a plane wave function, meaning that it has the form

for some vector ξ. Such an equation can be resolved (if the coefficients of L are analytic functions) by the Cauchy–Kovalevskaya theorem or (if the coefficients of L are constant) by quadrature. So, if the delta function can be decomposed into plane waves, then one can in principle solve linear partial differential equations.

Such a decomposition of the delta function into plane waves was part of a general technique first introduced essentially by Johann Radon, and then developed in this form by Fritz John (1955). Choose k so that n + k is an even integer, and for a real number s, put

Then δ is obtained by applying a power of the Laplacian to the integral with respect to the unit sphere measure of g(x · ξ) for ξ in the unit sphere Sn−1:

The Laplacian here is interpreted as a weak derivative, so that this equation is taken to mean that, for any test function φ,

The result follows from the formula for the Newtonian potential (the fundamental solution of Poisson's equation). This is essentially a form of the inversion formula for the Radon transform because it recovers the value of φ(x) from its integrals over hyperplanes. For instance, if n is odd and k = 1, then the integral on the right hand side is

where (ξ, p) is the Radon transform of φ:

An alternative equivalent expression of the plane wave decomposition is:

Fourier kernels

In the study of Fourier series, a major question consists of determining whether and in what sense the Fourier series associated with a periodic function converges to the function. The n-th partial sum of the Fourier series of a function f of period is defined by convolution (on the interval [−π,π]) with the Dirichlet kernel:

Thus,
where
A fundamental result of elementary Fourier series states that the Dirichlet kernel restricted to the interval [−π,π] tends to a multiple of the delta function as N → ∞. This is interpreted in the distribution sense, that
for every compactly supported smooth function f. Thus, formally one has
on the interval [−π,π].

Despite this, the result does not hold for all compactly supported continuous functions: that is DN does not converge weakly in the sense of measures. The lack of convergence of the Fourier series has led to the introduction of a variety of summability methods to produce convergence. The method of Cesàro summation leads to the Fejér kernel

The Fejér kernels tend to the delta function in a stronger sense that

for every compactly supported continuous function f. The implication is that the Fourier series of any continuous function is Cesàro summable to the value of the function at every point.

Hilbert space theory

The Dirac delta distribution is a densely defined unbounded linear functional on the Hilbert space L2 of square-integrable functions. Indeed, smooth compactly supported functions are dense in L2, and the action of the delta distribution on such functions is well-defined. In many applications, it is possible to identify subspaces of L2 and to give a stronger topology on which the delta function defines a bounded linear functional.

Sobolev spaces

The Sobolev embedding theorem for Sobolev spaces on the real line R implies that any square-integrable function f such that

is automatically continuous, and satisfies in particular

Thus δ is a bounded linear functional on the Sobolev space H1. Equivalently δ is an element of the continuous dual space H−1 of H1. More generally, in n dimensions, one has δHs(Rn) provided s > n/2.

Spaces of holomorphic functions

In complex analysis, the delta function enters via Cauchy's integral formula, which asserts that if D is a domain in the complex plane with smooth boundary, then

for all holomorphic functions f in D that are continuous on the closure of D. As a result, the delta function δz is represented in this class of holomorphic functions by the Cauchy integral:

Moreover, let H2(∂D) be the Hardy space consisting of the closure in L2(∂D) of all holomorphic functions in D continuous up to the boundary of D. Then functions in H2(∂D) uniquely extend to holomorphic functions in D, and the Cauchy integral formula continues to hold. In particular for zD, the delta function δz is a continuous linear functional on H2(∂D). This is a special case of the situation in several complex variables in which, for smooth domains D, the Szegő kernel plays the role of the Cauchy integral.

Resolutions of the identity

Given a complete orthonormal basis set of functions {φn} in a separable Hilbert space, for example, the normalized eigenvectors of a compact self-adjoint operator, any vector f can be expressed as

The coefficients {αn} are found as
which may be represented by the notation:
a form of the bra–ket notation of Dirac. Adopting this notation, the expansion of f takes the dyadic form:

Letting I denote the identity operator on the Hilbert space, the expression

is called a resolution of the identity. When the Hilbert space is the space L2(D) of square-integrable functions on a domain D, the quantity:

is an integral operator, and the expression for f can be rewritten

The right-hand side converges to f in the L2 sense. It need not hold in a pointwise sense, even when f is a continuous function. Nevertheless, it is common to abuse notation and write

resulting in the representation of the delta function:

With a suitable rigged Hilbert space (Φ, L2(D), Φ*) where Φ ⊂ L2(D) contains all compactly supported smooth functions, this summation may converge in Φ*, depending on the properties of the basis φn. In most cases of practical interest, the orthonormal basis comes from an integral or differential operator, in which case the series converges in the distribution sense.

Infinitesimal delta functions

Cauchy used an infinitesimal α to write down a unit impulse, infinitely tall and narrow Dirac-type delta function δα satisfying in a number of articles in 1827. Cauchy defined an infinitesimal in Cours d'Analyse (1827) in terms of a sequence tending to zero. Namely, such a null sequence becomes an infinitesimal in Cauchy's and Lazare Carnot's terminology.

Non-standard analysis allows one to rigorously treat infinitesimals. The article by Yamashita (2007) contains a bibliography on modern Dirac delta functions in the context of an infinitesimal-enriched continuum provided by the hyperreals. Here the Dirac delta can be given by an actual function, having the property that for every real function F one has as anticipated by Fourier and Cauchy.

Dirac comb

A Dirac comb is an infinite series of Dirac delta functions spaced at intervals of T

A so-called uniform "pulse train" of Dirac delta measures, which is known as a Dirac comb, or as the Sha distribution, creates a sampling function, often used in digital signal processing (DSP) and discrete time signal analysis. The Dirac comb is given as the infinite sum, whose limit is understood in the distribution sense,

which is a sequence of point masses at each of the integers.

Up to an overall normalizing constant, the Dirac comb is equal to its own Fourier transform. This is significant because if f is any Schwartz function, then the periodization of f is given by the convolution

In particular,
is precisely the Poisson summation formula. More generally, this formula remains to be true if f is a tempered distribution of rapid descent or, equivalently, if is a slowly growing, ordinary function within the space of tempered distributions.

Sokhotski–Plemelj theorem

The Sokhotski–Plemelj theorem, important in quantum mechanics, relates the delta function to the distribution p.v. 1/x, the Cauchy principal value of the function 1/x, defined by

Sokhotsky's formula states that

Here the limit is understood in the distribution sense, that for all compactly supported smooth functions f,

Relationship to the Kronecker delta

The Kronecker delta δij is the quantity defined by

for all integers i, j. This function then satisfies the following analog of the sifting property: if ai (for i in the set of all integers) is any doubly infinite sequence, then

Similarly, for any real or complex valued continuous function f on R, the Dirac delta satisfies the sifting property

This exhibits the Kronecker delta function as a discrete analog of the Dirac delta function.

Applications

Probability theory

In probability theory and statistics, the Dirac delta function is often used to represent a discrete distribution, or a partially discrete, partially continuous distribution, using a probability density function (which is normally used to represent absolutely continuous distributions). For example, the probability density function f(x) of a discrete distribution consisting of points x = {x1, ..., xn}, with corresponding probabilities p1, ..., pn, can be written as

As another example, consider a distribution in which 6/10 of the time returns a standard normal distribution, and 4/10 of the time returns exactly the value 3.5 (i.e. a partly continuous, partly discrete mixture distribution). The density function of this distribution can be written as

The delta function is also used to represent the resulting probability density function of a random variable that is transformed by continuously differentiable function. If Y = g(X) is a continuous differentiable function, then the density of Y can be written as

The delta function is also used in a completely different way to represent the local time of a diffusion process (like Brownian motion). The local time of a stochastic process B(t) is given by

and represents the amount of time that the process spends at the point x in the range of the process. More precisely, in one dimension this integral can be written
where is the indicator function of the interval

Quantum mechanics

The delta function is expedient in quantum mechanics. The wave function of a particle gives the probability amplitude of finding a particle within a given region of space. Wave functions are assumed to be elements of the Hilbert space L2 of square-integrable functions, and the total probability of finding a particle within a given interval is the integral of the magnitude of the wave function squared over the interval. A set {|φn} of wave functions is orthonormal if they are normalized by

where δ is the Kronecker delta. A set of orthonormal wave functions is complete in the space of square-integrable functions if any wave function |ψ⟩ can be expressed as a linear combination of the {|φn} with complex coefficients:

with cn = φn|ψ. Complete orthonormal systems of wave functions appear naturally as the eigenfunctions of the Hamiltonian (of a bound system) in quantum mechanics that measures the energy levels, which are called the eigenvalues. The set of eigenvalues, in this case, is known as the spectrum of the Hamiltonian. In bra–ket notation, as above, this equality implies the resolution of the identity:

Here the eigenvalues are assumed to be discrete, but the set of eigenvalues of an observable may be continuous rather than discrete. An example is the position observable, (x) = xψ(x). The spectrum of the position (in one dimension) is the entire real line and is called a continuous spectrum. However, unlike the Hamiltonian, the position operator lacks proper eigenfunctions. The conventional way to overcome this shortcoming is to widen the class of available functions by allowing distributions as well: that is, to replace the Hilbert space of quantum mechanics with an appropriate rigged Hilbert space. In this context, the position operator has a complete set of eigen-distributions, labeled by the points y of the real line, given by

The eigenfunctions of position are denoted by φy = |y in Dirac notation, and are known as position eigenstates.

Similar considerations apply to the eigenstates of the momentum operator, or indeed any other self-adjoint unbounded operator P on the Hilbert space, provided the spectrum of P is continuous and there are no degenerate eigenvalues. In that case, there is a set Ω of real numbers (the spectrum), and a collection φy of distributions indexed by the elements of Ω, such that

That is, φy are the eigenvectors of P. If the eigenvectors are normalized so that

in the distribution sense, then for any test function ψ,

where c(y) = ψ, φy. That is, as in the discrete case, there is a resolution of the identity

where the operator-valued integral is again understood in the weak sense. If the spectrum of P has both continuous and discrete parts, then the resolution of the identity involves a summation over the discrete spectrum and an integral over the continuous spectrum.

The delta function also has many more specialized applications in quantum mechanics, such as the delta potential models for a single and double potential well.

Structural mechanics

The delta function can be used in structural mechanics to describe transient loads or point loads acting on structures. The governing equation of a simple mass–spring system excited by a sudden force impulse I at time t = 0 can be written

where m is the mass, ξ is the deflection, and k is the spring constant.

As another example, the equation governing the static deflection of a slender beam is, according to Euler–Bernoulli theory,

where EI is the bending stiffness of the beam, w is the deflection, x is the spatial coordinate, and q(x) is the load distribution. If a beam is loaded by a point force F at x = x0, the load distribution is written

As the integration of the delta function results in the Heaviside step function, it follows that the static deflection of a slender beam subject to multiple point loads is described by a set of piecewise polynomials.

Also, a point moment acting on a beam can be described by delta functions. Consider two opposing point forces F at a distance d apart. They then produce a moment M = Fd acting on the beam. Now, let the distance d approach the limit zero, while M is kept constant. The load distribution, assuming a clockwise moment acting at x = 0, is written

Point moments can thus be represented by the derivative of the delta function. Integration of the beam equation again results in piecewise polynomial deflection.

Inhalant

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