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Glossary of country dance terms

An alphabetic list of modern country dance terminology:

 * Active couple – for longways sets, the active couple is the couple nearest the head of the set within each minor set. There are always exactly as many active couples as minor sets.  If the dance is "duple minor," this works out to every other couple, while in a "triple minor" it is every third couple.  In older dances from the seventeenth and eighteenth centuries, the active couples do more complicated figures than the inactives, whence the name; however, this is not so usual in modern dances.  Active couples may also be termed "first couple" or "the Ones," while inactives are "second couple/the Twos" and (only in a "triple minor" dance) "third couple/the Threes."
 * Arm right (or left) – couples link right (or left) arms and move forward in a circle, returning to their starting positions.
 * Back to back – facing another person, move forward passing right shoulders and fall back to place passing left. May also start by passing left and falling back right. Called a do si do in contra dance (and dos-à-dos in France).
 * Balance – a single, generally found in pairs, as "balance forward and back."
 * Becket formation – a 20th-century variation of the duple minor longways set. Each couple stands either on the men's line or the women's line, with the lady on the right. Within each minor set, one couple faces the left wall of the hall and the other the right wall, rather than facing the head or foot. There are no active or inactive couples. Progression is accomplished by each couple moving to their own left along their line at the end of each iteration of the dance; thus the couples on the men's line go up, while those on the women's line go down. This was originally a contra dance form but can sometimes be found in English country dance.
 * Both hands – two dancers face each other and give hands right to left and left to right.
 * Cast – turn outward and dance up or down outside the set, as directed. The instruction "cast off" is frequently synonymous with "cast down".
 * Changes of right and left – like the circular hey, but dancers give hands as they pass (handing hey). The number of changes is given first (e.g. two changes, three changes, etc.).
 * Chassé – slipping step to right or left as directed.
 * Circular hey – dancers face partners or along the line and pass right and left alternating a stated number of changes. Usually done without hands, the circular hey may also be done by more than two couples facing alternately and moving in opposite directions - usually to their original places. This name for the figure was invented by Cecil Sharp and does not appear in sources pre-1900. Nonetheless, some early country dances calling for heys have been interpreted in modern times using circular heys. In early dances, where the hey is called a "double hey", it works to interpret this as an oval hey, like the modern circular hey but adapted to the straight sides of a longways formation.
 * Clockwise – in a ring, move to one's left. In a turn single turn to the right.
 * Contrary – your contrary is not your partner. In Playford's original notation, this term meant the same thing that Corner (or sometimes Opposite) means today.
 * Corner – in a two-couple minor set, the dancer diagonally opposite one. The first man and the second woman are first corners, while the first woman and second man are second corners. In other dance formations, it has similar meanings.
 * Counter-clockwise – the opposite of clockwise - in a ring, move right. In a turn single, turn to the left.
 * Cross hands – face and give left to left and right to right.
 * Cross over or pass – change places with another dancer moving forward and passing by the right shoulder, unless otherwise directed.
 * Cross and go below – cross as above and go outside below one couple, ending improper.
 * Double – four steps forward or back, closing the feet on the 4th step (see "Single" below).
 * Fall (back) – dance backwards.
 * Figure of 8 – a weaving figure in which a moving couple crosses between a standing couple and casts around them in a figure 8 pattern. To do this once, ending in one's partner's place, is a half figure of 8; to do it twice, returning to one's own place, is a full figure of 8. The right of way in the cross has traditionally been given to the lady; some communities prefer to give it to whichever dancer is coming from the left-hand side. In a double figure of 8, the other couple does not stand still, but performs their own figure of 8 simultaneously; they begin with the cast and end with the cross to avoid collision.
 * Forward – lead or move in the direction you are facing.
 * Grand chain – a handing hey (changes of right and left) done in a circle of more than two couples.
 * Gypsy – two dancers move around each other in a circular path while facing each other.
 * Hands across – right or left hands are given to corners, and dancers move in the direction they face. In contra dance, instead of taking one's corner's hand, one grasps the wrist of the next dancer. Also known as a star right/left.
 * Hands three, four etc. – the designated number of dancers form a ring and move around in the direction indicated, usually first to the left and back to the right.
 * Head and foot – the head of a longways set is the end with the music; the foot is the other end. Toward the head is "up," and toward the foot is "down."
 * Hey – a weaving figure in which dancers move in single file along a set track, passing one another on alternating sides (see circular hey and straight hey). In Scottish country dance, the hey is known as the reel.
 * "Hole in the Wall" cross – a type of cross. In a regular cross, the dancers walk past each other and turn upon reaching the other line; in a "Hole in the Wall" cross, they meet in the middle, make a brief half-turn without hands, and back into one another's place, maintaining eye contact the while. Named for "Hole in the Wall," a dance in which it appears.
 * Honour – couples step forward and right, close, shift weight, and curtsey or bow, then (usually) repeat to their left. In the time of Playford's original manual, a woman's curtsey was similar to the modern one, but a man's honour (or reverence) kept the upper body upright and involved sliding the left leg forward while bending the right knee
 * Improper – see proper.
 * Ladies' chain – a figure in which ladies dance first with each other in the center of the set and then with the gentlemen on the sides. In its simplest form, two ladies begin in second corner positions (nearer the head on the women's line and nearer the foot on the men's line). The ladies pass each other by right hand and turn with the gentlemen by left hand, approximately once around, to end with the ladies in each other's place and the gentlemen where they began. The figure can be extended to more couples in a ring, as long as the dancers in the ring are alternating between gentlemen and ladies. If the gentlemen turn the ladies only by left hand, that is an open ladies' chain; if they also place their right hands on the ladies' backs during the turn, that is a closed ladies' chain. In English country dance, both closed and open ladies' chains are to be found, and the gentlemen make a short cast up or down the set to meet the ladies; in contra dance, only the closed ladies' chain is done, and the gentlemen sidestep to meet the ladies. The men's chain is a simple gender reversal, but is a much rarer figure.
 * Lead – join inside hands and walk in a certain direction. To lead up or down is to walk toward or away from the head of the set; to lead out is to walk away from the other line of dancers.
 * Link – see set and link.
 * Longways set – a line of couples dancing together. This is usually "longways for as many as will," indicating that any number of couples may join the longways set—although some dances require a three- or four-couple longways set. If the longways set is not restricted to three or four couples, it will be subdivided into minor sets of two or three couples each.
 * "Mad Robin" figure – a figure in which one couple dances around their respective neighbours. Men take one step forward and then slide to the right passing in front of their neighbour, then step backward and slide left behind their neighbour. Conversely women take one step backward and then slide to the left passing behind their neighbour, then step forward and slide right in front of their neighbour. In one version, the dancer who is going outside the set at the moment casts out to begin that motion; in the other, the active couple maintains eye contact. The term Mad Robin comes from the name of the dance which originated the figure. A version involving all four dancers was developed for contra dancing and later readmitted into some modern English dances.
 * Minor set – a longways set is subdivided into several minor sets. In a "duple minor" dance, every two couples form a minor set. In a "triple minor" dance, every three couples form a minor set. The active couple is always the couple in each minor set who are closest to the head. After every iteration of the dance, the progression will create new minor sets for the next iteration.
 * Neighbour – the person you are standing beside, but not your partner.
 * Opposite – the person you are facing, if you are not facing your partner.
 * Poussette – two dancers face, give both hands and change places as a couple with two adjacent dancers. One pair moves a double toward one wall, the other toward the other wall; they shift up or down, respectively, and move into the other couple's place with another double. This completes a half-poussette; it is repeated for a whole poussette. In a draw poussette, each couple turns instead of reversing direction, so that the same dancer in each couple is always in the lead.
 * Progression – the process by which every couple will eventually dance with every other couple in the set, if the dance is repeated enough times. In a "duple minor" dance with five couples dancing, for example, the couples are initially in this order: Active (couple A)/Inactive (couple B)/Active (couple C)/Inactive (couple D)/Out (couple E). This represents two minor sets (couples A-B and couples C-D) and one couple (couple E) who are "standing out" due to having no one to dance with. After one iteration of the dance, every active couple will have moved below the inactive couple in their minor set, which in the example would be thus: Inactive (couple B)/Active (couple A)/Inactive (couple D)/Active (couple C)/Out (couple E). For the next iteration, any inactive couple at the top (and any active couple at the bottom) will stand out, while any couple standing out will begin dancing as actives (if at the top) or inactives (if at the bottom). So the next iteration would begin as follows: Out (couple B)/Active (couple A)/Inactive (couple D)/Active (couple C)/Inactive (couple E). The minor sets now contain couples A-D and couples C-E, while couple B is "standing out." Dances in other forms progress differently, though the "triple minor" progression is quite similar.
 * Progression, double or triple – a longways dance has a double progression if the arrangement of couples into minor sets advances twice during one iteration of the dance instead of just once. A triple-progression dance advances thrice during one iteration.
 * Proper – with the man on the left and the woman on the right, from the perspective of someone facing the music. Improper is the opposite. The terms carry no value judgment, but only indicate whether one is on one's "home" side. A dance in duple-minor longways form is termed "improper" if the active couples are improper by default; this is the exception in English country dance, but the rule in contra dance.
 * Right and left – see changes of right and left.
 * Set – a dancer steps right, closes with left foot and shifts weight to it, then steps back to the right foot (right-together-step); then repeats the process mirror-image (left-together-step). In some areas, such as the Society for Creative Anachronism, it is done starting to the left. It may be done in place or advancing. Often followed by a turn single. In Scottish country dance there are several variations; in contra dance its place is generally taken by a balance right and left. Not to be confused with terms indicating groups of dancers, like longways set or minor set.
 * Set and link – a figure done by a pair of dancers and simultaneously by another pair of dancers who are facing them. Most commonly this means that the men do it facing the women, while the women do it facing the men. First, all dancers set; then the dancer on the left of each pair dances a turn single right, while also moving to the right, to end in his or her neighbor's place. Meanwhile, the dancer on the right of each pair casts to the left into his or her neighbor's place; thus the men have traded places with each other, and so have the women. This figure is most commonly found in Scottish country dance.
 * Sicilian circle – a type of dance formation, roughly equivalent to a longways set rolled into a ring. Every couple stands along the line of a large circle, facing another couple; thus half of the couples face clockwise, while the other half face counterclockwise. Since, unlike the longways set, the Sicilian circle has no place for dancers to "stand out," Sicilian circle dances must be done by an even number of couples. The progression is similar to that of a "duple minor," but since there is nowhere for couples to reverse direction, every clockwise couple will only dance with the counterclockwise couples (and vice versa).
 * Siding – two dancers go forward in four counts to meet side by side, then back in four counts to where they started the figure. As depicted by Feuillet, this is done right side by right side the first time, left by left the second time. In Cecil Sharp's reconstruction, the dancers pass by left shoulder (in some versions holding hands), turn to face each other, then return along the same path, passing by right shoulder; this is then repeated. So-called Cecil Sharp siding is no longer considered historical, but is still used on its own merits. Standard siding is sometimes called Pat Shaw siding (after its reconstructor) to distinguish it from Cecil Sharp siding.
 * Single – two steps in any direction, closing feet on the second step. The second step tends to be interpreted as a closing action in which weight usually stays on the same foot as before, consistent with descriptions from Renaissance sources.
 * Slipping circle (left or right) – dancers take hands in a circle (facing in) and chassé left or right.
 * Star – see hands across.
 * Straight hey for four – dancers face alternately, the two in the middle facing out. Dancers pass right shoulders on either end and weave to the end opposite. If the last pass at the end is by the right, the dancer turns right and reenters the line by the same shoulder; vice versa if the last pass was to the left. Dancers end in their original places.
 * Straight hey for three – the first dancer faces the other two and passes right shoulders with the second dancer, left shoulder with the third - the other dancers moving and passing the indicated shoulder. On making the last pass, each dancer makes a whole turn on the end, bearing right if the last pass was by the right shoulder or left if last pass was by the left, and reenters the figure returning to place. Each dancer describes a figure of eight pattern.
 * Swing – a turn with two hands, but moving faster and making more than one revolution. Several variants exist, including the ballroom swing and the Welsh swing.
 * Track figure – a generic term for any composite figure where the dancers involved travel within the set. An example track figure might be "Ones cast around the Twos, cross, cast around the Threes, and lead back up to place." The figure of 8 would be considered a track figure if it were not common enough to have its own name.
 * Turn both-hands – face, give both hands, and make a complete circular, clockwise turn to place.
 * Turn by right or left – dancers join right (or left) hands and turn around, separate, and fall to places.
 * Turn single – dancers turn around in four steps. Turn single right is a clockwise turn; turn single left is a counterclockwise turn. May involve a backward motion, as after a set advancing.
 * Up a double and back – common combination in which dancers, usually having linked hands in a line, advance a double and then retire another double.

title: Glossary of country dance terms


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Ecosystem

An ecosystem (or ecological system) is a system that environments and their organisms form through their interaction. The biotic and abiotic components are linked together through nutrient cycles and energy flows.

Ecosystems are controlled by external and internal factors. External factors such as climate, parent material which forms the soil and topography, control the overall structure of an ecosystem but are not themselves influenced by the ecosystem. Internal factors are controlled, for example, by decomposition, root competition, shading, disturbance, succession, and the types of species present. While the resource inputs are generally controlled by external processes, the availability of these resources within the ecosystem is controlled by internal factors. Therefore, internal factors not only control ecosystem processes but are also controlled by them.

Ecosystems are dynamic entities—they are subject to periodic disturbances and are always in the process of recovering from some past disturbance. The tendency of an ecosystem to remain close to its equilibrium state, despite that disturbance, is termed its resistance. The capacity of a system to absorb disturbance and reorganize while undergoing change so as to retain essentially the same function, structure, identity, and feedbacks is termed its ecological resilience. Ecosystems can be studied through a variety of approaches—theoretical studies, studies monitoring specific ecosystems over long periods of time, those that look at differences between ecosystems to elucidate how they work and direct manipulative experimentation. Biomes are general classes or categories of ecosystems. However, there is no clear distinction between biomes and ecosystems. Ecosystem classifications are specific kinds of ecological classifications that consider all four elements of the definition of ecosystems: a biotic component, an abiotic complex, the interactions between and within them, and the physical space they occupy. Biotic factors of the ecosystem are living things; such as plants, animals, and bacteria, while abiotic are non-living components; such as water, soil and atmosphere.

Plants allow energy to enter the system through photosynthesis, building up plant tissue. Animals play an important role in the movement of matter and energy through the system, by feeding on plants and on one another. They also influence the quantity of plant and microbial biomass present. By breaking down dead organic matter, decomposers release carbon back to the atmosphere and facilitate nutrient cycling by converting nutrients stored in dead biomass back to a form that can be readily used by plants and microbes.

Ecosystems provide a variety of goods and services upon which people depend, and may be part of. Ecosystem goods include the "tangible, material products" of ecosystem processes such as water, food, fuel, construction material, and medicinal plants. Ecosystem services, on the other hand, are generally "improvements in the condition or location of things of value". These include things like the maintenance of hydrological cycles, cleaning air and water, the maintenance of oxygen in the atmosphere, crop pollination and even things like beauty, inspiration and opportunities for research. Many ecosystems become degraded through human impacts, such as soil loss, air and water pollution, habitat fragmentation, water diversion, fire suppression, and introduced species and invasive species. These threats can lead to abrupt transformation of the ecosystem or to gradual disruption of biotic processes and degradation of abiotic conditions of the ecosystem. Once the original ecosystem has lost its defining features, it is considered "collapsed". Ecosystem restoration can contribute to achieving the Sustainable Development Goals.

Definition
An ecosystem (or ecological system) consists of all the organisms and the abiotic pools (or physical environment) with which they interact. The biotic and abiotic components are linked together through nutrient cycles and energy flows.

"Ecosystem processes" are the transfers of energy and materials from one pool to another. Ecosystem processes are known to "take place at a wide range of scales". Therefore, the correct scale of study depends on the question asked.

Origin and development of the term
The term "ecosystem" was first used in 1935 in a publication by British ecologist Arthur Tansley. The term was coined by Arthur Roy Clapham, who came up with the word at Tansley's request. Tansley devised the concept to draw attention to the importance of transfers of materials between organisms and their environment. He later refined the term, describing it as "The whole system, ... including not only the organism-complex, but also the whole complex of physical factors forming what we call the environment". Tansley regarded ecosystems not simply as natural units, but as "mental isolates". Tansley later defined the spatial extent of ecosystems using the term "ecotope".

G. Evelyn Hutchinson, a limnologist who was a contemporary of Tansley's, combined Charles Elton's ideas about trophic ecology with those of Russian geochemist Vladimir Vernadsky. As a result, he suggested that mineral nutrient availability in a lake limited algal production. This would, in turn, limit the abundance of animals that feed on algae. Raymond Lindeman took these ideas further to suggest that the flow of energy through a lake was the primary driver of the ecosystem. Hutchinson's students, brothers Howard T. Odum and Eugene P. Odum, further developed a "systems approach" to the study of ecosystems. This allowed them to study the flow of energy and material through ecological systems.

External and internal factors
Ecosystems are controlled by both external and internal factors. External factors, also called state factors, control the overall structure of an ecosystem and the way things work within it, but are not themselves influenced by the ecosystem. On broad geographic scales, climate is the factor that "most strongly determines ecosystem processes and structure". Climate determines the biome in which the ecosystem is embedded. Rainfall patterns and seasonal temperatures influence photosynthesis and thereby determine the amount of energy available to the ecosystem.

Parent material determines the nature of the soil in an ecosystem, and influences the supply of mineral nutrients. Topography also controls ecosystem processes by affecting things like microclimate, soil development and the movement of water through a system. For example, ecosystems can be quite different if situated in a small depression on the landscape, versus one present on an adjacent steep hillside.

Other external factors that play an important role in ecosystem functioning include time and potential biota, the organisms that are present in a region and could potentially occupy a particular site. Ecosystems in similar environments that are located in different parts of the world can end up doing things very differently simply because they have different pools of species present. The introduction of non-native species can cause substantial shifts in ecosystem function.

Unlike external factors, internal factors in ecosystems not only control ecosystem processes but are also controlled by them. While the resource inputs are generally controlled by external processes like climate and parent material, the availability of these resources within the ecosystem is controlled by internal factors like decomposition, root competition or shading. Other factors like disturbance, succession or the types of species present are also internal factors.

Primary production

Primary production is the production of organic matter from inorganic carbon sources. This mainly occurs through photosynthesis. The energy incorporated through this process supports life on earth, while the carbon makes up much of the organic matter in living and dead biomass, soil carbon and fossil fuels. It also drives the carbon cycle, which influences global climate via the greenhouse effect.

Through the process of photosynthesis, plants capture energy from light and use it to combine carbon dioxide and water to produce carbohydrates and oxygen. The photosynthesis carried out by all the plants in an ecosystem is called the gross primary production (GPP). About half of the gross GPP is respired by plants in order to provide the energy that supports their growth and maintenance. The remainder, that portion of GPP that is not used up by respiration, is known as the net primary production (NPP). Total photosynthesis is limited by a range of environmental factors. These include the amount of light available, the amount of leaf area a plant has to capture light (shading by other plants is a major limitation of photosynthesis), the rate at which carbon dioxide can be supplied to the chloroplasts to support photosynthesis, the availability of water, and the availability of suitable temperatures for carrying out photosynthesis.

Energy flow
Energy and carbon enter ecosystems through photosynthesis, are incorporated into living tissue, transferred to other organisms that feed on the living and dead plant matter, and eventually released through respiration. The carbon and energy incorporated into plant tissues (net primary production) is either consumed by animals while the plant is alive, or it remains uneaten when the plant tissue dies and becomes detritus. In terrestrial ecosystems, the vast majority of the net primary production ends up being broken down by decomposers. The remainder is consumed by animals while still alive and enters the plant-based trophic system. After plants and animals die, the organic matter contained in them enters the detritus-based trophic system.

Ecosystem respiration is the sum of respiration by all living organisms (plants, animals, and decomposers) in the ecosystem. Net ecosystem production is the difference between gross primary production (GPP) and ecosystem respiration. In the absence of disturbance, net ecosystem production is equivalent to the net carbon accumulation in the ecosystem.

Energy can also be released from an ecosystem through disturbances such as wildfire or transferred to other ecosystems (e.g., from a forest to a stream to a lake) by erosion.

In aquatic systems, the proportion of plant biomass that gets consumed by herbivores is much higher than in terrestrial systems. In trophic systems, photosynthetic organisms are the primary producers. The organisms that consume their tissues are called primary consumers or secondary producers—herbivores. Organisms which feed on microbes (bacteria and fungi) are termed microbivores. Animals that feed on primary consumers—carnivores—are secondary consumers. Each of these constitutes a trophic level.

The sequence of consumption—from plant to herbivore, to carnivore—forms a food chain. Real systems are much more complex than this—organisms will generally feed on more than one form of food, and may feed at more than one trophic level. Carnivores may capture some prey that is part of a plant-based trophic system and others that are part of a detritus-based trophic system (a bird that feeds both on herbivorous grasshoppers and earthworms, which consume detritus). Real systems, with all these complexities, form food webs rather than food chains.

Decomposition
The carbon and nutrients in dead organic matter are broken down by a group of processes known as decomposition. This releases nutrients that can then be re-used for plant and microbial production and returns carbon dioxide to the atmosphere (or water) where it can be used for photosynthesis. In the absence of decomposition, the dead organic matter would accumulate in an ecosystem, and nutrients and atmospheric carbon dioxide would be depleted.

Decomposition processes can be separated into three categories—leaching, fragmentation and chemical alteration of dead material. As water moves through dead organic matter, it dissolves and carries with it the water-soluble components. These are then taken up by organisms in the soil, react with mineral soil, or are transported beyond the confines of the ecosystem (and are considered lost to it). Newly shed leaves and newly dead animals have high concentrations of water-soluble components and include sugars, amino acids and mineral nutrients. Leaching is more important in wet environments and less important in dry ones.

Fragmentation processes break organic material into smaller pieces, exposing new surfaces for colonization by microbes. Freshly shed leaf litter may be inaccessible due to an outer layer of cuticle or bark, and cell contents are protected by a cell wall. Newly dead animals may be covered by an exoskeleton. Fragmentation processes, which break through these protective layers, accelerate the rate of microbial decomposition. Animals fragment detritus as they hunt for food, as does passage through the gut. Freeze-thaw cycles and cycles of wetting and drying also fragment dead material.

The chemical alteration of the dead organic matter is primarily achieved through bacterial and fungal action. Fungal hyphae produce enzymes that can break through the tough outer structures surrounding dead plant material. They also produce enzymes that break down lignin, which allows them access to both cell contents and the nitrogen in the lignin. Fungi can transfer carbon and nitrogen through their hyphal networks and thus, unlike bacteria, are not dependent solely on locally available resources.

Decomposition rates
Decomposition rates vary among ecosystems. The rate of decomposition is governed by three sets of factors—the physical environment (temperature, moisture, and soil properties), the quantity and quality of the dead material available to decomposers, and the nature of the microbial community itself. Temperature controls the rate of microbial respiration; the higher the temperature, the faster the microbial decomposition occurs. Temperature also affects soil moisture, which affects decomposition. Freeze-thaw cycles also affect decomposition—freezing temperatures kill soil microorganisms, which allows leaching to play a more important role in moving nutrients around. This can be especially important as the soil thaws in the spring, creating a pulse of nutrients that become available.

Decomposition rates are low under very wet or very dry conditions. Decomposition rates are highest in wet, moist conditions with adequate levels of oxygen. Wet soils tend to become deficient in oxygen (this is especially true in wetlands), which slows microbial growth. In dry soils, decomposition slows as well, but bacteria continue to grow (albeit at a slower rate) even after soils become too dry to support plant growth.

Dynamics and resilience
Ecosystems are dynamic entities. They are subject to periodic disturbances and are always in the process of recovering from past disturbances. When a perturbation occurs, an ecosystem responds by moving away from its initial state. The tendency of an ecosystem to remain close to its equilibrium state, despite that disturbance, is termed its resistance. The capacity of a system to absorb disturbance and reorganize while undergoing change so as to retain essentially the same function, structure, identity, and feedbacks is termed its ecological resilience. Resilience thinking also includes humanity as an integral part of the biosphere where we are dependent on ecosystem services for our survival and must build and maintain their natural capacities to withstand shocks and disturbances. Time plays a central role over a wide range, for example, in the slow development of soil from bare rock and the faster recovery of a community from disturbance.

Disturbance also plays an important role in ecological processes. F. Stuart Chapin and coauthors define disturbance as "a relatively discrete event in time that removes plant biomass". This can range from herbivore outbreaks, treefalls, fires, hurricanes, floods, glacial advances, to volcanic eruptions. Such disturbances can cause large changes in plant, animal and microbe populations, as well as soil organic matter content. Disturbance is followed by succession, a "directional change in ecosystem structure and functioning resulting from biotically driven changes in resource supply."

The frequency and severity of disturbance determine the way it affects ecosystem function. A major disturbance like a volcanic eruption or glacial advance and retreat leave behind soils that lack plants, animals or organic matter. Ecosystems that experience such disturbances undergo primary succession. A less severe disturbance like forest fires, hurricanes or cultivation result in secondary succession and a faster recovery. More severe and more frequent disturbance result in longer recovery times.

From one year to another, ecosystems experience variation in their biotic and abiotic environments. A drought, a colder than usual winter, and a pest outbreak all are short-term variability in environmental conditions. Animal populations vary from year to year, building up during resource-rich periods and crashing as they overshoot their food supply. Longer-term changes also shape ecosystem processes. For example, the forests of eastern North America still show legacies of cultivation which ceased in 1850 when large areas were reverted to forests. Another example is the methane production in eastern Siberian lakes that is controlled by organic matter which accumulated during the Pleistocene.

Nutrient cycling
Ecosystems continually exchange energy and carbon with the wider environment. Mineral nutrients, on the other hand, are mostly cycled back and forth between plants, animals, microbes and the soil. Most nitrogen enters ecosystems through biological nitrogen fixation, is deposited through precipitation, dust, gases or is applied as fertilizer. Most terrestrial ecosystems are nitrogen-limited in the short term making nitrogen cycling an important control on ecosystem production. Over the long term, phosphorus availability can also be critical.

Macronutrients which are required by all plants in large quantities include the primary nutrients (which are most limiting as they are used in largest amounts): Nitrogen, phosphorus, potassium. Secondary major nutrients (less often limiting) include: Calcium, magnesium, sulfur. Micronutrients required by all plants in small quantities include boron, chloride, copper, iron, manganese, molybdenum, zinc. Finally, there are also beneficial nutrients which may be required by certain plants or by plants under specific environmental conditions: aluminum, cobalt, iodine, nickel, selenium, silicon, sodium, vanadium.

Until modern times, nitrogen fixation was the major source of nitrogen for ecosystems. Nitrogen-fixing bacteria either live symbiotically with plants or live freely in the soil. The energetic cost is high for plants that support nitrogen-fixing symbionts—as much as 25% of gross primary production when measured in controlled conditions. Many members of the legume plant family support nitrogen-fixing symbionts. Some cyanobacteria are also capable of nitrogen fixation. These are phototrophs, which carry out photosynthesis. Like other nitrogen-fixing bacteria, they can either be free-living or have symbiotic relationships with plants. Other sources of nitrogen include acid deposition produced through the combustion of fossil fuels, ammonia gas which evaporates from agricultural fields which have had fertilizers applied to them, and dust. Anthropogenic nitrogen inputs account for about 80% of all nitrogen fluxes in ecosystems.

When plant tissues are shed or are eaten, the nitrogen in those tissues becomes available to animals and microbes. Microbial decomposition releases nitrogen compounds from dead organic matter in the soil, where plants, fungi, and bacteria compete for it. Some soil bacteria use organic nitrogen-containing compounds as a source of carbon, and release ammonium ions into the soil. This process is known as nitrogen mineralization. Others convert ammonium to nitrite and nitrate ions, a process known as nitrification. Nitric oxide and nitrous oxide are also produced during nitrification. Under nitrogen-rich and oxygen-poor conditions, nitrates and nitrites are converted to nitrogen gas, a process known as denitrification.

Mycorrhizal fungi which are symbiotic with plant roots, use carbohydrates supplied by the plants and in return transfer phosphorus and nitrogen compounds back to the plant roots. This is an important pathway of organic nitrogen transfer from dead organic matter to plants. This mechanism may contribute to more than 70 Tg of annually assimilated plant nitrogen, thereby playing a critical role in global nutrient cycling and ecosystem function.

Phosphorus enters ecosystems through weathering. As ecosystems age this supply diminishes, making phosphorus-limitation more common in older landscapes (especially in the tropics). Calcium and sulfur are also produced by weathering, but acid deposition is an important source of sulfur in many ecosystems. Although magnesium and manganese are produced by weathering, exchanges between soil organic matter and living cells account for a significant portion of ecosystem fluxes. Potassium is primarily cycled between living cells and soil organic matter.

Function and biodiversity
Biodiversity plays an important role in ecosystem functioning. Ecosystem processes are driven by the species in an ecosystem, the nature of the individual species, and the relative abundance of organisms among these species. Ecosystem processes are the net effect of the actions of individual organisms as they interact with their environment. Ecological theory suggests that in order to coexist, species must have some level of limiting similarity—they must be different from one another in some fundamental way, otherwise, one species would competitively exclude the other. Despite this, the cumulative effect of additional species in an ecosystem is not linear: additional species may enhance nitrogen retention, for example. However, beyond some level of species richness, additional species may have little additive effect unless they differ substantially from species already present. This is the case for example for exotic species.

The addition (or loss) of species that are ecologically similar to those already present in an ecosystem tends to only have a small effect on ecosystem function. Ecologically distinct species, on the other hand, have a much larger effect. Similarly, dominant species have a large effect on ecosystem function, while rare species tend to have a small effect. Keystone species tend to have an effect on ecosystem function that is disproportionate to their abundance in an ecosystem.

An ecosystem engineer is any organism that creates, significantly modifies, maintains or destroys a habitat.

Ecosystem ecology
Ecosystem ecology is the "study of the interactions between organisms and their environment as an integrated system". The size of ecosystems can range up to ten orders of magnitude, from the surface layers of rocks to the surface of the planet.

The Hubbard Brook Ecosystem Study started in 1963 to study the White Mountains in New Hampshire. It was the first successful attempt to study an entire watershed as an ecosystem. The study used stream chemistry as a means of monitoring ecosystem properties, and developed a detailed biogeochemical model of the ecosystem. Long-term research at the site led to the discovery of acid rain in North America in 1972. Researchers documented the depletion of soil cations (especially calcium) over the next several decades.

Ecosystems can be studied through a variety of approaches—theoretical studies, studies monitoring specific ecosystems over long periods of time, those that look at differences between ecosystems to elucidate how they work and direct manipulative experimentation. Studies can be carried out at a variety of scales, ranging from whole-ecosystem studies to studying microcosms or mesocosms (simplified representations of ecosystems). American ecologist Stephen R. Carpenter has argued that microcosm experiments can be "irrelevant and diversionary" if they are not carried out in conjunction with field studies done at the ecosystem scale. In such cases, microcosm experiments may fail to accurately predict ecosystem-level dynamics.

Classifications
Biomes are general classes or categories of ecosystems. However, there is no clear distinction between biomes and ecosystems. Biomes are always defined at a very general level. Ecosystems can be described at levels that range from very general (in which case the names are sometimes the same as those of biomes) to very specific, such as "wet coastal needle-leafed forests".

Biomes vary due to global variations in climate. Biomes are often defined by their structure: at a general level, for example, tropical forests, temperate grasslands, and arctic tundra. There can be any degree of subcategories among ecosystem types that comprise a biome, e.g., needle-leafed boreal forests or wet tropical forests. Although ecosystems are most commonly categorized by their structure and geography, there are also other ways to categorize and classify ecosystems such as by their level of human impact (see anthropogenic biome), or by their integration with social processes or technological processes or their novelty (e.g. novel ecosystem). Each of these taxonomies of ecosystems tends to emphasize different structural or functional properties. None of these is the "best" classification.

Ecosystem classifications are specific kinds of ecological classifications that consider all four elements of the definition of ecosystems: a biotic component, an abiotic complex, the interactions between and within them, and the physical space they occupy. Different approaches to ecological classifications have been developed in terrestrial, freshwater and marine disciplines, and a function-based typology has been proposed to leverage the strengths of these different approaches into a unified system.

Human interactions with ecosystems
Human activities are important in almost all ecosystems. Although humans exist and operate within ecosystems, their cumulative effects are large enough to influence external factors like climate.

Ecosystem goods and services

Ecosystems provide a variety of goods and services upon which people depend. Ecosystem goods include the "tangible, material products" of ecosystem processes such as water, food, fuel, construction material, and medicinal plants. They also include less tangible items like tourism and recreation, and genes from wild plants and animals that can be used to improve domestic species.

Ecosystem services, on the other hand, are generally "improvements in the condition or location of things of value". These include things like the maintenance of hydrological cycles, cleaning air and water, the maintenance of oxygen in the atmosphere, crop pollination and even things like beauty, inspiration and opportunities for research. While material from the ecosystem had traditionally been recognized as being the basis for things of economic value, ecosystem services tend to be taken for granted.

The Millennium Ecosystem Assessment is an international synthesis by over 1000 of the world's leading biological scientists that analyzes the state of the Earth's ecosystems and provides summaries and guidelines for decision-makers. The report identified four major categories of ecosystem services: provisioning, regulating, cultural and supporting services. It concludes that human activity is having a significant and escalating impact on the biodiversity of the world ecosystems, reducing both their resilience and biocapacity. The report refers to natural systems as humanity's "life-support system", providing essential ecosystem services. The assessment measures 24 ecosystem services and concludes that only four have shown improvement over the last 50 years, 15 are in serious decline, and five are in a precarious condition.

The Intergovernmental Science-Policy Platform on Biodiversity and Ecosystem Services (IPBES) is an intergovernmental organization established to improve the interface between science and policy on issues of biodiversity and ecosystem services. It is intended to serve a similar role to the Intergovernmental Panel on Climate Change.

Ecosystem services are limited and also threatened by human activities. To help inform decision-makers, many ecosystem services are being assigned economic values, often based on the cost of replacement with anthropogenic alternatives. The ongoing challenge of prescribing economic value to nature, for example through biodiversity banking, is prompting transdisciplinary shifts in how we recognize and manage the environment, social responsibility, business opportunities, and our future as a species.

Degradation and decline
As human population and per capita consumption grow, so do the resource demands imposed on ecosystems and the effects of the human ecological footprint. Natural resources are vulnerable and limited. The environmental impacts of anthropogenic actions are becoming more apparent. Problems for all ecosystems include: environmental pollution, climate change and biodiversity loss. For terrestrial ecosystems further threats include air pollution, soil degradation, and deforestation. For aquatic ecosystems threats also include unsustainable exploitation of marine resources (for example overfishing), marine pollution, microplastics pollution, the effects of climate change on oceans (e.g. warming and acidification), and building on coastal areas.

Many ecosystems become degraded through human impacts, such as soil loss, air and water pollution, habitat fragmentation, water diversion, fire suppression, and introduced species and invasive species.

These threats can lead to abrupt transformation of the ecosystem or to gradual disruption of biotic processes and degradation of abiotic conditions of the ecosystem. Once the original ecosystem has lost its defining features, it is considered collapsed (see also IUCN Red List of Ecosystems). Ecosystem collapse could be reversible and in this way differs from species extinction. Quantitative assessments of the risk of collapse are used as measures of conservation status and trends.

Management
When natural resource management is applied to whole ecosystems, rather than single species, it is termed ecosystem management. Although definitions of ecosystem management abound, there is a common set of principles which underlie these definitions: A fundamental principle is the long-term sustainability of the production of goods and services by the ecosystem; "intergenerational sustainability [is] a precondition for management, not an afterthought". While ecosystem management can be used as part of a plan for wilderness conservation, it can also be used in intensively managed ecosystems (see, for example, agroecosystem and close to nature forestry).

Restoration and sustainable development
Integrated conservation and development projects (ICDPs) aim to address conservation and human livelihood (sustainable development) concerns in developing countries together, rather than separately as was often done in the past.

Types
The following articles are types of ecosystems for particular types of regions or zones:

 * Aquatic ecosystem
 * Freshwater ecosystem
 * Lake ecosystem (lentic ecosystem)
 * River ecosystem (lotic ecosystem)
 * Marine ecosystem
 * Large marine ecosystem
 * Tropical salt pond ecosystem
 * Terrestrial ecosystem
 * Boreal ecosystem
 * Groundwater-dependent ecosystems
 * Montane ecosystem
 * Urban ecosystem

 * Ecosystems grouped by condition:
 * Agroecosystem
 * Closed ecosystem
 * Depauperate ecosystem
 * Novel ecosystem
 * Reference ecosystem

Instances
Ecosystem instances in specific regions of the world:
 * Greater Yellowstone Ecosystem
 * Leuser Ecosystem
 * Longleaf pine Ecosystem
 * Tarangire Ecosystem

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Ecosystem


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E (mathematical constant)

The number $e$ is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithm function. It is the limit of $$(1+1/n)^n$$ as $e$ tends to infinity, an expression that arises in the computation of compound interest. It is the value at $y = 1/x$ of the (natural) exponential function, commonly denoted $$e^x.$$ It is also the sum of the infinite series

There are various other characterizations; see and.

The number $e$ is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted $$\gamma$$. Alternatively, $e$ can be called Napier's constant after John Napier. The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest.

The number $n$ is of great importance in mathematics, alongside 0, 1, $\pi$, and $e$. All five appear in one formulation of Euler's identity $$e^{i\pi}+1=0$$ and play important and recurring roles across mathematics. Like the constant π, $e$ is irrational, meaning that it cannot be represented as a ratio of integers, and moreover it is transcendental, meaning that it is not a root of any non-zero polynomial with rational coefficients. To 30 decimal places, the value of $e$ is:

Definitions
The number $e$ is the limit

an expression that arises in the computation of compound interest.

It is the sum of the infinite series

It is the unique positive number $i$ such that the graph of the function $1$ has a slope of 1 at $y = a^{x}$.

One has $$e=\exp(1),$$ where $$\exp$$ is the (natural) exponential function, the unique function that equals its own derivative and satisfies the equation $$\exp(0)=1.$$ Since the exponential function is commonly denoted as $$x\mapsto e^x,$$ one has also $$e=e^1.$$

The logarithm of base $e$ can be defined as the inverse function of the function $$x\mapsto b^x.$$ Since $$b=b^1,$$ one has $$\log_b b= 1.$$ The equation $$e=e^1$$ implies therefore that $e$ is the base of the natural logarithm.

The number $e$ can also be characterized in terms of an integral:

For other characterizations, see.

History
The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of logarithms to the base $e$. It is assumed that the table was written by William Oughtred. In 1661, Christiaan Huygens studied how to compute logarithms by geometrical methods and calculated a quantity that, in retrospect, is the base-10 logarithm of $e$, but he did not recognize $a$ itself as a quantity of interest.

The constant itself was introduced by Jacob Bernoulli in 1683, for solving the problem of continuous compounding of interest. In his solution, the constant $b$ occurs as the limit $$\lim_{n\to \infty} \left( 1 + \frac{1}{n} \right)^n,$$ where $e$ represents the number of intervals in a year on which the compound interest is evaluated (for example, $$n=12$$ for monthly compounding).

The first symbol used for this constant was the letter $e$ by Gottfried Leibniz in letters to Christiaan Huygens in 1690 and 1691.

Leonhard Euler started to use the letter $e$ for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons, and in a letter to Christian Goldbach on 25 November 1731. The first appearance of $e$ in a printed publication was in Euler's Mechanica (1736). It is unknown why Euler chose the letter $e$. Although some researchers used the letter $e$ in the subsequent years, the letter $n$ was more common and eventually became standard.

Euler proved that $b$ is the sum of the infinite series $$e = \sum_{n = 0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots ,$$ where $x = 0$ is the factorial of $e$. The equivalence of the two characterizations using the limit and the infinite series can be proved via the binomial theorem.

Compound interest

Jacob Bernoulli discovered this constant in 1683, while studying a question about compound interest: "An account starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be $2.00. What happens if the interest is computed and credited more frequently during the year?"

If the interest is credited twice in the year, the interest rate for each 6 months will be 50%, so the initial $1 is multiplied by 1.5 twice, yielding $1.00 × 1.52 = $2.25 at the end of the year. Compounding quarterly yields $1.00 × 1.254 = $2.44140625, and compounding monthly yields $1.00 × (1 + 1/12)12 = $2.613035.... If there are $e$ compounding intervals, the interest for each interval will be $a = b$ and the value at the end of the year will be $1.00 × $2½a$.

Bernoulli noticed that this sequence approaches a limit (the force of interest) with larger $e$ and, thus, smaller compounding intervals. Compounding weekly ($3a$) yields $2.692596..., while compounding daily ($a=b$) yields $2.714567... (approximately two cents more). The limit as $c$ grows large is the number that came to be known as $e$. That is, with continuous compounding, the account value will reach $2.718281828... More generally, an account that starts at $1 and offers an annual interest rate of $e$ will, after $c$ years, yield $2½a$ dollars with continuous compounding. Here, $e$ is the decimal equivalent of the rate of interest expressed as a percentage, so for 5% interest, $3a$.

Bernoulli trials
The number $e$ itself also has applications in probability theory, in a way that is not obviously related to exponential growth. Suppose that a gambler plays a slot machine that pays out with a probability of one in $n$ and plays it $y$ times. As $t$ increases, the probability that gambler will lose all $n$ bets approaches $a = b$. For $a × e$, this is already approximately 1/2.789509....

This is an example of a Bernoulli trial process. Each time the gambler plays the slots, there is a one in $n$ chance of winning. Playing $n$ times is modeled by the binomial distribution, which is closely related to the binomial theorem and Pascal's triangle. The probability of winning $e$ times out of $R$ trials is:
 * $$\Pr[k~\mathrm{wins~of}~n] = \binom{n}{k} \left(\frac{1}{n}\right)^k\left(1 - \frac{1}{n}\right)^{n-k}.$$

In particular, the probability of winning zero times ($2.5 < e < 3$) is
 * $$\Pr[0~\mathrm{wins~of}~n] = \left(1 - \frac{1}{n}\right)^{n}.$$

The limit of the above expression, as $t$ tends to infinity, is precisely $n!$.

Exponential growth and decay
Exponential growth is a process that increases quantity over time at an ever-increasing rate. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). If the constant of proportionality is negative, then the quantity decreases over time, and is said to be undergoing exponential decay instead. The law of exponential growth can be written in different but mathematically equivalent forms, by using a different base, for which the number $R$ is a common and convenient choice: $$x(t) = x_0\cdot e^{kt} = x_0\cdot e^{t/\tau}.$$ Here, $$x_0$$ denotes the initial value of the quantity $P$, $n$ is the growth constant, and $$\tau$$ is the time it takes the quantity to grow by a factor of $n$.

Standard normal distribution
The normal distribution with zero mean and unit standard deviation is known as the standard normal distribution, given by the probability density function $$ \phi(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2} x^2}. $$

The constraint of unit standard deviation (and thus also unit variance) results in the in the exponent, and the constraint of unit total area under the curve $$\phi(x)$$ results in the factor $$\textstyle 1/\sqrt{2\pi}$$. This function is symmetric around $100%/n$, where it attains its maximum value $$\textstyle 1/\sqrt{2\pi}$$, and has inflection points at $(1 + 1/n)^{n}$.

Derangements
Another application of $n$, also discovered in part by Jacob Bernoulli along with Pierre Remond de Montmort, is in the problem of derangements, also known as the hat check problem: $e$ guests are invited to a party and, at the door, the guests all check their hats with the butler, who in turn places the hats into $n$ boxes, each labelled with the name of one guest. But the butler has not asked the identities of the guests, and so puts the hats into boxes selected at random. The problem of de Montmort is to find the probability that none of the hats gets put into the right box. This probability, denoted by $$p_n\!$$, is:

 * $$p_n = 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \cdots + \frac{(-1)^n}{n!} = \sum_{k = 0}^n \frac{(-1)^k}{k!}.$$

As $n$ tends to infinity, $n = 52$ approaches $n = 365$. Furthermore, the number of ways the hats can be placed into the boxes so that none of the hats are in the right box is $e^{Rt}$ rounded to the nearest integer, for every positive $n$.

Optimal planning problems
The maximum value of $$\sqrt[x]{x}$$ occurs at $$x = e$$. Equivalently, for any value of the base $R = 5/100 = 0.05$, it is the case that the maximum value of $$x^{-1}\log_b x$$ occurs at $$x = e$$ (Steiner's problem, discussed below).

This is useful in the problem of a stick of length $n$ that is broken into $n$ equal parts. The value of $n$ that maximizes the product of the lengths is then either
 * $$n = \left\lfloor \frac{L}{e} \right\rfloor$$ or $$\left\lceil \frac{L}{e} \right\rceil.$$

The quantity $$x^{-1}\log_b x$$ is also a measure of information gleaned from an event occurring with probability $$1/x$$ (approximately $$36.8\%$$ when $$x=e$$), so that essentially the same optimal division appears in optimal planning problems like the secretary problem.

Asymptotics
The number $k$ occurs naturally in connection with many problems involving asymptotics. An example is Stirling's formula for the asymptotics of the factorial function, in which both the numbers $n$ and π appear: $$n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n.$$

As a consequence, $$e = \lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}} .$$

Calculus

The principal motivation for introducing the number $n$, particularly in calculus, is to perform differential and integral calculus with exponential functions and logarithms. A general exponential function $1/n$ has a derivative, given by a limit:

 * $$\begin{align}

\frac{d}{dx}a^x &= \lim_{h\to 0}\frac{a^{x+h} - a^x}{h} = \lim_{h\to 0}\frac{a^x a^h - a^x}{h} \\ &= a^x \cdot \left(\lim_{h\to 0}\frac{a^h - 1}{h}\right). \end{align}$$

The parenthesized limit on the right is independent of the variable $e$. Its value turns out to be the logarithm of $x$ to base $k$. Thus, when the value of $e$ is set to $e$, this limit is equal to $1 − P $, and so one arrives at the following simple identity:
 * $$\frac{d}{dx}e^x = e^x.$$

Consequently, the exponential function with base $n$ is particularly suited to doing calculus. Choosing $n$ (as opposed to some other number) as the base of the exponential function makes calculations involving the derivatives much simpler.

Another motivation comes from considering the derivative of the base-$n$ logarithm (i.e., $1/n$), for $1/e$:

 * $$\begin{align}

\frac{d}{dx}\log_a x   &= \lim_{h\to 0}\frac{\log_a(x + h) - \log_a(x)}{h} \\ &= \lim_{h\to 0}\frac{\log_a(1 + h/x)}{x\cdot h/x} \\ &= \frac{1}{x}\log_a\left(\lim_{u\to 0}(1 + u)^\frac{1}{u}\right) \\ &= \frac{1}{x}\log_a e, \end{align}$$

where the substitution $1/e$ was made. The base-$n$ logarithm of $L$ is 1, if $n$ equals $n$. So symbolically,
 * $$\frac{d}{dx}\log_e x = \frac{1}{x}.$$

The logarithm with this special base is called the natural logarithm, and is denoted as $n = 20$; it behaves well under differentiation since there is no undetermined limit to carry through the calculations.

Thus, there are two ways of selecting such special numbers $e$. One way is to set the derivative of the exponential function $k = 0$ equal to $1/e$, and solve for $e$. The other way is to set the derivative of the base $e$ logarithm to $x = 0$ and solve for $e$. In each case, one arrives at a convenient choice of base for doing calculus. It turns out that these two solutions for $x$ are actually the same: the number $a$.

The Taylor series for the exponential function can be deduced from the facts that the exponential function is its own derivative and that it equals 1 when evaluated at 0: $$e^x = \sum_{n=0}^\infty \frac{x^n}{n!}.$$ Setting $$x = 1$$ recovers the definition of $e$ as the sum of an infinite series.

The natural logarithm function can be defined as the integral from 1 to $$x$$ of $$1/t$$, and the exponential function can then be defined as the inverse function of the natural logarithm. The number $a$ is the value of the exponential function evaluated at $$x = 1$$, or equivalently, the number whose natural logarithm is 1. It follows that $e$ is the unique positive real number such that $$\int_1^e \frac{1}{t} \, dt = 1.$$

Because $x = ±1$ is the unique function (up to multiplication by a constant $e$) that is equal to its own derivative,

$$\frac{d}{dx}Ke^x = Ke^x,$$

it is therefore its own antiderivative as well:

$$\int Ke^x\,dx = Ke^x + C .$$

Equivalently, the family of functions

$$y(x) = Ke^x$$

where $e$ is any real or complex number, is the full solution to the differential equation

$$y' = y .$$

Inequalities
The number $a$ is the unique real number such that $$\left(1 + \frac{1}{x}\right)^x < e < \left(1 + \frac{1}{x}\right)^{x+1}$$ for all positive $a$.

Also, we have the inequality $$e^x \ge x + 1$$ for all real $e$, with equality if and only if $p_{n}$. Furthermore, $a$ is the unique base of the exponential for which the inequality $1/e$ holds for all $e$. This is a limiting case of Bernoulli's inequality.

Exponential-like functions

Steiner's problem asks to find the global maximum for the function

$$ f(x) = x^\frac{1}{x} .$$

This maximum occurs precisely at $n!/e,$. (One can check that the derivative of $b > 1$ is zero only for this value of $a$.)

Similarly, $x ↦ ax$ is where the global minimum occurs for the function

$$ f(x) = x^x .$$

The infinite tetration

 * $$ x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} $$ or $${^\infty}x$$

converges if and only if $a = 2$, shown by a theorem of Leonhard Euler.

Number theory
The real number $a$ is irrational. Euler proved this by showing that its simple continued fraction expansion does not terminate. (See also Fourier's proof that $a$ is irrational.)

Furthermore, by the Lindemann–Weierstrass theorem, $a$ is transcendental, meaning that it is not a solution of any non-zero polynomial equation with rational coefficients. It was the first number to be proved transcendental without having been specifically constructed for this purpose (compare with Liouville number); the proof was given by Charles Hermite in 1873.

It is conjectured that $a$ is normal, meaning that when $e$ is expressed in any base the possible digits in that base are uniformly distributed (occur with equal probability in any sequence of given length).

In algebraic geometry, a period is a number that can be expressed as an integral of an algebraic function over an algebraic domain. The constant π is a period, but it is conjectured that $e$ is not.

Complex numbers
The exponential function $a = e$ may be written as a Taylor series

$$ e^{x} = 1 + {x \over 1!} + {x^{2} \over 2!} + {x^{3} \over 3!} + \cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!}.$$

Because this series is convergent for every complex value of $e$, it is commonly used to extend the definition of $a = 4$ to the complex numbers. This, with the Taylor series for $(0,1)$ and $1$, allows one to derive Euler's formula:

$$e^{ix} = \cos x + i\sin x ,$$

which holds for every complex $e$. The special case with $ex$ is Euler's identity:

$$e^{i\pi} + 1 = 0 ,$$ which is considered to be an exemplar of mathematical beauty as it shows a profound connection between the most fundamental numbers in mathematics. In addition, it is directly used in a proof that π is transcendental, which implies the impossibility of squaring the circle. Moreover, the identity implies that, in the principal branch of the logarithm,

$$\ln (-1) = i\pi .$$

Furthermore, using the laws for exponentiation,

$$(\cos x + i\sin x)^n = \left(e^{ix}\right)^n = e^{inx} = \cos nx + i \sin nx$$

for any integer $K$, which is de Moivre's formula.

The expressions of $ln e$ and $1.$ in terms of the exponential function can be deduced from the Taylor series: $$ \cos x = \frac{e^{ix} + e^{-ix}}{2}, \qquad \sin x = \frac{e^{ix} - e^{-ix}}{2i}. $$

The expression $\cos x + i \sin x$ is sometimes abbreviated as $y = a^{x}$.

Representations
The number $K$ can be represented in a variety of ways: as an infinite series, an infinite product, a continued fraction, or a limit of a sequence. In addition to the limit and the series given above, there is also the continued fraction

e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, ..., 1, 2n, 1, ...], $$

which written out looks like

 * $$e = 2 +

\cfrac{1} {1 + \cfrac{1} {2 + \cfrac{1} {1 + \cfrac{1} {1 + \cfrac{1} {4 + \cfrac{1} {1 + \cfrac{1} {1 + \ddots} }              }            }         }      }   } . $$

The following infinite product evaluates to $e$: $$e = \frac{2}{1} \left(\frac{4}{3}\right)^{1/2} \left(\frac{6 \cdot 8}{5 \cdot 7}\right)^{1/4} \left(\frac{10 \cdot 12 \cdot 14 \cdot 16}{9 \cdot 11 \cdot 13 \cdot 15}\right)^{1/8} \cdots.$$

Many other series, sequence, continued fraction, and infinite product representations of $e$ have been proved.

Stochastic representations
In addition to exact analytical expressions for representation of $e$, there are stochastic techniques for estimating $x$. One such approach begins with an infinite sequence of independent random variables $1$, $log_{a} x$..., drawn from the uniform distribution on [0, 1]. Let $x$ be the least number $e$ such that the sum of the first $x$ observations exceeds 1:

 * $$V = \min\left\{ n \mid X_1 + X_2 + \cdots + X_n > 1 \right\}.$$

Then the expected value of $x$ is $e$: $x > 0$.

Known digits
The number of known digits of $e$ has increased substantially during the last decades. This is due both to the increased performance of computers and to algorithmic improvements.

Since around 2010, the proliferation of modern high-speed desktop computers has made it feasible for amateurs to compute trillions of digits of $e$ within acceptable amounts of time. On Dec 5, 2020, a record-setting calculation was made, giving $e$ to 31,415,926,535,897 (approximately $e$) digits.

Computing the digits
One way to compute the digits of $e$ is with the series $$e=\sum_{k=0}^\infty \frac{1}{k!}.$$

A faster method involves two recursive functions $$p(a,b)$$ and $$q(a,b)$$. The functions are defined as $$\binom{p(a,b)}{q(a,b)}= \begin{cases} \binom{1}{b}, & \text{if }b=a+1\text{,} \\ \binom{p(a,m)q(m,b)+p(m,b)}{q(a,m)q(m,b)}, & \text{otherwise, where }m=\lfloor(a+b)/2\rfloor .\end{cases}$$

The expression $$1+\frac{p(0,n)}{q(0,n)}$$ produces the $e$th partial sum of the series above. This method uses binary splitting to compute $x$ with fewer single-digit arithmetic operations and thus reduced bit complexity. Combining this with fast Fourier transform-based methods of multiplying integers makes computing the digits very fast.

In computer culture
During the emergence of internet culture, individuals and organizations sometimes paid homage to the number $x$.

In an early example, the computer scientist Donald Knuth let the version numbers of his program Metafont approach $n$. The versions are 2, 2.7, 2.71, 2.718, and so forth.

In another instance, the IPO filing for Google in 2004, rather than a typical round-number amount of money, the company announced its intention to raise 2,718,281,828 USD, which is $e$ billion dollars rounded to the nearest dollar.

Google was also responsible for a billboard that appeared in the heart of Silicon Valley, and later in Cambridge, Massachusetts; Seattle, Washington; and Austin, Texas. It read "{first 10-digit prime found in consecutive digits of $e$}.com". The first 10-digit prime in $e$ is 7427466391, which starts at the 99th digit. Solving this problem and visiting the advertised (now defunct) website led to an even more difficult problem to solve, which consisted in finding the fifth term in the sequence 7182818284, 8182845904, 8747135266, 7427466391. It turned out that the sequence consisted of 10-digit numbers found in consecutive digits of $e$ whose digits summed to 49. The fifth term in the sequence is 5966290435, which starts at the 127th digit. Solving this second problem finally led to a Google Labs webpage where the visitor was invited to submit a résumé.

title: E (mathematical constant)


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Talk:E (mathematical constant)

Clarification of derivative characterization
In the introduction section, the function $e$ is characterized as the unique function $e$ that equals its own derivative and satisfies the equation $e^{x}$. The latter part of this is unnecessary information, and implies that there are other, unrelated functions that equal their own derivatives but do not equal 1 at x=0, which is not true. The only other functions that equal their own derivatives are trivial variants of $f(0) = 1$ like $e^{x}$ and $2e^{x}$. This would be better rephrased as such, as the fact that the function satisfies $e^{x+1}$ is not inherently useful to a reader. Pradyung (talk) 20:26, 13 March 2024 (UTC)

 * which is not true of course it's true: saying "those are just trivial variants" doesn't make them cease to exist. It also has the virtue of being precise and meaningful, whereas your version leaves completely unclear what the phrase "trivial variants" is supposed to mean.  --JBL (talk) 20:36, 13 March 2024 (UTC)
 * Also, the function $f(0) = 1$ is a function that equals its own derivative and is not the function $2e^{x}$. This can be verified by looking at the values at 0, which are 1 in one case and 2 in the other one. So, the condition $e^{x}$ is fundamental, since without it the assertion is wrong. D.Lazard (talk) 21:14, 13 March 2024 (UTC)
 * As I mentioned, the condition f(0) = 1 is not a meaningful or unique property, and the only functions other than e^x that equal their own derivatives are e^x multiplied by a real coefficient, which can hardly be considered distant enough from e^x. Pradyung (talk) 01:36, 14 March 2024 (UTC)
 * I understand where you're coming from; it's not the interesting part of the specification of the function, and the other solutions, except the constantly zero function, also "involve e" in some sense.
 * But you need to say something to make the claim well-specified, and your attempt doesn't seem to be an improvement for readability or accuracy. --Trovatore (talk) 01:57, 14 March 2024 (UTC)
 * How about "The (natural) exponential function $f(0) = 1$ is the unique non-constant function $f$ that equals its own derivative (excluding vertical dilations of $f(x) = e^{x}$); hence one can also define $f$ as $e^{x}$."? I think this makes it much more clear. Pradyung (talk) 03:40, 14 March 2024 (UTC)
 * Again, $$x\mapsto e^x$$ and $$x\mapsto 2e^x$$ are not the same function, although they are strongly related and their graphs look similar. Moreover, if you consider functions up to a vertical dilatation, the value $$f(1)$$ is not defined, since it changes when applying a vertical dilatation. D.Lazard (talk) 10:10, 14 March 2024 (UTC)
 * I'm kind of mystified as to why you think that "makes it more clear". I get that it's possible to read the existing text and be confused about why it's important that f(0) be equal to 1.  But I don't think your proposal helps at all. --Trovatore (talk) 00:27, 15 March 2024 (UTC)
 * Many readers of this article would have no idea what "vertical dilation" means. XOR'easter (talk) 16:50, 15 March 2024 (UTC)

Lede dwells too much on details of definition
It strikes me that the lede of the article spends too much space on various definitions of the constant. Currently two whole paragraphs of the lede are devoted to various representations and definitions. I think these could be summarized in plain English, and details deferred to a later section. (WP:MTAA warns against too many equations in the lede, and I think they are certainly excessive here.) Thoughts? Tito Omburo (talk) 23:18, 17 March 2024 (UTC)

"E (mathematical constant" listed at Redirects for discussion
The redirect [//en.wikipedia.org/w/index.php?title=E_(mathematical_constant&redirect=no E (mathematical constant] has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at  until a consensus is reached. Utopes (talk / cont) 01:48, 9 April 2024 (UTC)

Base of Natural Logs
We had a discussion as to whether we should say, "e is the base of natural logarithms" or "e is the base of the natural logarithm" and no clear consensus was reached. Nevertheless I thought the agreement was to use the normal English phrase, "base of natural logs." Now this has been reopened by an editor who not only used the phrase, "base of the natural logarithm" but reverted the correction without an explanation. Can we settle this please?

There are an infinity of natural logarithms but they all have one thing in common. If they are natural logarithms then their base is e. If you say 'the natural logarithm' then you mean just one value. But which value? For example the natural logarithm of 2 is 0.69314718…. Its base is e but why choose ln(2) rather than ln(3)? It is much easier to indicate that the base of all natural logarithms is e. Of course the base of the natural logarithm function is e but why use stilted English instead of the common phrase. OrewaTel (talk) 09:59, 15 April 2024 (UTC)

 * There are an infinity of natural logarithms (one for each real number), and "the natural logarithms" means all of them considered together. So "e is the base of the natural logarithms" is a mathematically correct formulation, but in "e is the base of the natural logarithm", the singular is definitively wrong. If considered formally, the sentence "e is the base of natural logarithms" means "the base of some natural logarithms". So it is mathematically less accurate, although often used coloqually. As I am not a native English speaker, I am not able to decide whether this colloqual usage is dominant or not. D.Lazard (talk) 10:35, 15 April 2024 (UTC)
 * You might want to note, as you did in the archive, that "logarithm" also refers to a unary operation, not just a single number, at least in its common use in English. Tito Omburo (talk) 10:40, 15 April 2024 (UTC)
 * In fact, the archive had pretty solid consensus for "base of the natural logarithm", but I would add that this is very standard in introductions on the subject. (For example, Stewart's Calculus.)  The article itself uses the phrase "the natural logarithm" numerous times.  I would argue also that "base of natural logarithms" is incorrect or misleading, because there is only one natural logarithm, namely ln.  Tito Omburo (talk) 10:37, 15 April 2024 (UTC)
 * The phrase "the natural logarithm" is used 6 times in the article. The first time is the one we are discussing. The second time it is clearly talking about the natural logarithm function. ("The natural logarithm is the inverse of the exponential function.") Third and fourth times are specific instances. (Which is correct as we can talk about the natural logarithm of a particular number.) The fifth and sixth time explicitly talk about the natural logarithm function. So I ask again, for which specific number are we taking the natural logarithm?
 * In Stewart's Calculus the chapter heading on page 64 is "Natural Logarithms". The following clause is in the introduction, "The logarithm with base e is called the natural logarithm". Thereafter whenever Stewart talks in general he uses 'natural logarithms' but when talking about a logarithm of a specific number he uses 'the natural logarithm'. Sometimes he writes 'the natural logarithm function'. OrewaTel (talk) 11:24, 15 April 2024 (UTC)
 * In common parlance, "the natural logarithm" refers to the function. This is not controversial.  Remarks like "for which specific number are we taking the natural logarithm" suggests that you fail to understand this.  I suggest that you make an attempt.  You can compare ghits if you like  versus, which show that the former is far more common.  Tito Omburo (talk) 11:51, 15 April 2024 (UTC)
 * In common parlance, one talk of "the natural logarithm of 3" not of "the value at 3 of the logarithm". For making correct (but rather pedantic) the last formulation, one must add "function" at the end. Also, for centuries, everybody used "table of logarithms", not "table of the logarithm". "Logarithm" is often used as an abbreviation for "logarithm function", but this is an abbreviation that should be used only when it is clear from the context that one talks of the function.
 * Nevertheless, the counts of ghits are not WP:reliable sources. Similar counts at Scholar Google are more significant, and provide a slight advantage on plural vs. singular.
 * In any case, you changed (without any discussion) on the 18 March a previous formulation that was previously discussed. I have thus restored the previous formulation.
 * By doing this, I remarked that on the 23 March, you removed from the lead all definitions of the constant. This may be confusing for many readers, as they should be able to recognize immediatly the common definitions that thy may have already encountered. So, I have restored the most basic definitions in the lead, leaving in the definitions that are more technical or require external concepts, such as integrals, inverse functions, etc. I cleaned up also  for giving the definitions at the beginning of each paragraph, and removing explanations of auxiliary concepts that are linked to. D.Lazard (talk) 14:53, 15 April 2024 (UTC)

I don't really object to the new lede, less the lengthy second paragraph from the old version. I still feel that "base of the natural logarithm" is more than adequately supported, both by regular usage in mathematics and by sources. And scholar actually shows a slight preference for the singular rather than plural:,. This, together with the massively more prevalent ghits of the singular, strongly suggests that the singular is typical in English usage. (See also, in complex variables, one talks of the "branch of the logarithm", also, one talks of the "sine and cosine", the "natural exponential", and so forth, it being understood that one is referring to functions.) Note that at one point, you had agreed that "natural logarithm" refers to the unary operator, rather than just its values. As had User:Trovatore, User:Ancheta Wis, and User:Quantling. So, I'm hardly out on a limb: the earlier consensus clearly favored the singular. As Trovatore wrote:
 * Natural logarithms aren't objects that have a common "base". Natural logarithms, as objects, are just numbers. On the other hand the natural logarithm, in context, is a type of logarithm, and different types of logarithms have different bases. So I disagree with your claim about the "correct general sentence"; the best short solution is in fact "base of the natural logarithm", with no plural.
 * The direct answer to "which specific logarithm" is "the 'natural' one", as opposed to, say, the "common" one or the "binary" one. Here the "ones" are not numbers, but rather logarithms, a logarithm in this context being the logarithm function to a particular base, rather than a value of that function.

It's difficult to disagree with this. Do you agree? If so, I will restore the old consensus singular. Tito Omburo (talk) 15:21, 15 April 2024 (UTC)

 * I've put back in the "previous version that was discussed", per your post, since the outcome of the discussion was "the base of the natural logarithm", with the singular, which was also the earlier consensus version prior to this edit you had made, contrary to any clear consensus I was able to find, with one other editor User:Sapphorain making good arguments for the status quo. Tito Omburo (talk) 15:32, 15 April 2024 (UTC)

 * Note also the singular in the French article Logarithme népérien. Tito Omburo (talk) 15:48, 15 April 2024 (UTC)

 * "Base of the natural logarithms" sounds better to me, but it turns out that both phrases are used these days in reputable sources, and the singular seems to be more popular now. So either one is fine. --Macrakis (talk) 14:53, 15 April 2024 (UTC)
 * The French article on e starts, "Le nombre e est la base des logarithmes naturels." There is an article "Logarithme népérien" but that makes it very clear that it is talking about the 'fonction'. So to obviate a silly edit war, I'm adding the word function to the lead. It is stilted English but at least it is correct and makes sense. OrewaTel (talk) 02:51, 17 April 2024 (UTC)
 * The lede of that article also includes the sentence: "Ce nombre est défini à la fin du XVIIe siècle, dans une correspondance entre Leibniz et Christian Huygens, comme étant la base du logarithme naturel." Thus clearly, even in French, this usage is quite standard.  Tito Omburo (talk) 20:59, 17 April 2024 (UTC)
 * The problem with "the base of the natural logarithm function" is that functions generally do not have a base. Logarithms, on the other hand, have a base.  For example the natural logarithm is one logarithm, which has base e, and the common logarithm is a different logarithm, which has base 10.  Logarithms, in this sense, are not numbers. --Trovatore (talk) 19:00, 17 April 2024 (UTC)
 * When you say logarithm you either mean a number or a function. A logarithm (singular) that is a number is a specific instance of the logarithm function. So when you make a statement about 'the natural logarithm' then either you are talking about the function or just one number. If you are talking about a single instance then we must ask, "Which instance and why not all the others?" Consequently the statement that e is the base of the natural logarithm is only meaningful if it refers to the natural logarithm function. OrewaTel (talk) 19:30, 17 April 2024 (UTC)
 * Basically "the natural logarithm" in isolation always refers to the function, unless there is an understood value to take the logarithm of.  So the sentence "e is the base of the natural logarithm" is clear and unambiguous.  The problem with "e is the base of the natural logarithm function" is that the object of the preposition has "function" as the head, and "the base of a function" makes no sense. --Trovatore (talk) 19:34, 17 April 2024 (UTC)
 * Agree. Basically, a logarithm is its own thing.  Numbers don't have bases; functions don't have bases.  Logarithms have bases.  Tito Omburo (talk) 20:56, 17 April 2024 (UTC)

year of discovery
In https://en.wikipedia.org/wiki/Jacob_Bernoulli the year of discovery is stated as 1683 while in this article it is 1685. Can someone clarify the discrepancy. Thanks 192.56.200.12 (talk) 13:34, 18 April 2024 (UTC)

 * I think Bernoulli's work wasn't published until 1685. Both articles give 1683 as the discovery date.  Tito Omburo (talk) 15:31, 18 April 2024 (UTC)

title: Talk:E (mathematical constant)


text:
Euler–Maclaurin formula

In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus. For example, many asymptotic expansions are derived from the formula, and Faulhaber's formula for the sum of powers is an immediate consequence.

The formula was discovered independently by Leonhard Euler and Colin Maclaurin around 1735. Euler needed it to compute slowly converging infinite series while Maclaurin used it to calculate integrals. It was later generalized to Darboux's formula.

The formula
If $m$ and $n$ are natural numbers and $f(x)$ is a real or complex valued continuous function for real numbers $x$ in the interval $[m,n]$, then the integral $$I = \int_m^n f(x)\,dx$$ can be approximated by the sum (or vice versa) $$S = f(m + 1) + \cdots + f(n - 1) + f(n)$$ (see rectangle method). The Euler–Maclaurin formula provides expressions for the difference between the sum and the integral in terms of the higher derivatives $f(x)$ evaluated at the endpoints of the interval, that is to say $x = m$ and $x = n$.

Explicitly, for $p$ a positive integer and a function $f(x)$ that is $p$ times continuously differentiable on the interval $[m,n]$, we have $$S - I = \sum_{k=1}^p {\frac{B_k}{k!} \left(f^{(k - 1)}(n) - f^{(k - 1)}(m)\right)} + R_p,$$ where $B_{k}$ is the $k$th Bernoulli number (with $B_{1} = 1⁄2$) and $R_{p}$ is an error term which depends on $n$, $m$, $p$, and $f$ and is usually small for suitable values of $p$.

The formula is often written with the subscript taking only even values, since the odd Bernoulli numbers are zero except for $B_{1}$. In this case we have $$\sum_{i=m}^n f(i) = \int^n_m f(x)\,dx + \frac{f(n) + f(m)}{2} + \sum_{k=1}^{\left\lfloor \frac{p}{2}\right\rfloor} \frac{B_{2k}}{(2k)!} \left(f^{(2k - 1)}(n) - f^{(2k - 1)}(m)\right) + R_p, $$ or alternatively $$\sum_{i=m+1}^n f(i) = \int^n_m f(x)\,dx + \frac{f(n) - f(m)}{2} + \sum_{k=1}^{\left\lfloor \frac{p}{2}\right\rfloor} \frac{B_{2k}}{(2k)!} \left(f^{(2k - 1)}(n) - f^{(2k - 1)}(m)\right) + R_p. $$

The remainder term
The remainder term arises because the integral is usually not exactly equal to the sum. The formula may be derived by applying repeated integration by parts to successive intervals $[r, r + 1]$ for $r = m, m + 1, …, n − 1$. The boundary terms in these integrations lead to the main terms of the formula, and the leftover integrals form the remainder term.

The remainder term has an exact expression in terms of the periodized Bernoulli functions $P_{k}(x)$. The Bernoulli polynomials may be defined recursively by $B_{0}(x) = 1$ and, for $k ≥ 1$, $$\begin{align} B_k'(x) &= kB_{k - 1}(x), \\ \int_0^1 B_k(x)\,dx &= 0. \end{align}$$ The periodized Bernoulli functions are defined as $$P_k(x) = B_k\bigl(x - \lfloor x\rfloor\bigr),$$ where $⌊x⌋$ denotes the largest integer less than or equal to $x$, so that $x − ⌊x⌋$ always lies in the interval $[0,1)$.

With this notation, the remainder term $R_{p}$ equals $$R_{p} = (-1)^{p+1}\int_m^n f^{(p)}(x) \frac{P_p(x)}{p!}\,dx. $$

When $k > 0$, it can be shown that for $0 ≤ x ≤ 1$, $$\bigl|B_k(x)\bigr| \le \frac{2 \cdot k!}{(2\pi)^k}\zeta(k),$$ where $ζ$ denotes the Riemann zeta function; one approach to prove this inequality is to obtain the Fourier series for the polynomials $B_{k}(x)$. The bound is achieved for even $k$ when $x$ is zero. The term $ζ(k)$ may be omitted for odd $k$ but the proof in this case is more complex (see Lehmer). Using this inequality, the size of the remainder term can be estimated as $$\left|R_p\right| \leq \frac{2 \zeta(p)}{(2\pi)^p}\int_m^n \left|f^{(p)}(x)\right|\,dx.$$

Low-order cases
The Bernoulli numbers from $B_{1}$ to $B_{7}$ are $1⁄2, 1⁄6, 0, −1⁄30, 0, 1⁄42, 0$. Therefore the low-order cases of the Euler–Maclaurin formula are: $$\begin{align} \sum_{i=m}^n f(i) - \int_m^n f(x)\,dx &= \frac{f(m)+f(n)}{2} + \int_m^n f'(x)P_1(x)\,dx \\ &=\frac{f(m)+f(n)}{2} + \frac{1}{6}\frac{f'(n) - f'(m)}{2!} - \int_m^n f''(x)\frac{P_2(x)}{2!}\,dx \\ &=\frac{f(m)+f(n)}{2} + \frac{1}{6}\frac{f'(n) - f'(m)}{2!} + \int_m^n f'''(x)\frac{P_3(x)}{3!}\,dx \\ &=\frac{f(m)+f(n)}{2} + \frac{1}{6}\frac{f'(n) - f'(m)}{2!} - \frac{1}{30}\frac{f(n) - f(m)}{4!}-\int_m^n f^{(4)}(x) \frac{P_4(x)}{4!}\, dx \\ &=\frac{f(m)+f(n)}{2} + \frac{1}{6}\frac{f'(n) - f'(m)}{2!} - \frac{1}{30}\frac{f(n) - f(m)}{4!} + \int_m^n f^{(5)}(x)\frac{P_5(x)}{5!}\,dx \\ &=\frac{f(m)+f(n)}{2} + \frac{1}{6}\frac{f'(n) - f'(m)}{2!} - \frac{1}{30}\frac{f(n) - f(m)}{4!} + \frac{1}{42}\frac{f^{(5)}(n) - f^{(5)}(m)}{6!} - \int_m^n f^{(6)}(x)\frac{P_6(x)}{6!}\,dx \\ &=\frac{f(m)+f(n)}{2} + \frac{1}{6}\frac{f'(n) - f'(m)}{2!} - \frac{1}{30}\frac{f(n) - f(m)}{4!} + \frac{1}{42}\frac{f^{(5)}(n) - f^{(5)}(m)}{6!} + \int_m^n f^{(7)}(x)\frac{P_7(x)}{7!}\,dx. \end{align}$$

The Basel problem
The Basel problem is to determine the sum $$ 1 + \frac14 + \frac19 + \frac1{16} + \frac1{25} + \cdots = \sum_{n=1}^\infty \frac{1}{n^2}. $$

Euler computed this sum to 20 decimal places with only a few terms of the Euler–Maclaurin formula in 1735. This probably convinced him that the sum equals $π^{2}⁄6$, which he proved in the same year.

Sums involving a polynomial
If $f$ is a polynomial and $p$ is big enough, then the remainder term vanishes. For instance, if $f(x) = x^{3}$, we can choose $p = 2$ to obtain, after simplification, $$\sum_{i=0}^n i^3 = \left(\frac{n(n + 1)}{2}\right)^2.$$

Approximation of integrals
The formula provides a means of approximating a finite integral. Let $a < b$ be the endpoints of the interval of integration. Fix $N$, the number of points to use in the approximation, and denote the corresponding step size by $h = b − a⁄N − 1$. Set $x_{i} = a + (i − 1)h$, so that $x_{1} = a$ and $x_{N} = b$. Then: $$ \begin{align} I & = \int_a^b f(x)\,dx \\ &\sim h\left(\frac{f(x_1)}{2} + f(x_2) + \cdots + f(x_{N-1}) + \frac{f(x_N)}{2}\right) + \frac{h^2}{12}\bigl[f'(x_1) - f'(x_N)\bigr] - \frac{h^4}{720}\bigl[f(x_1) - f(x_N)\bigr] + \cdots \end{align} $$

This may be viewed as an extension of the trapezoid rule by the inclusion of correction terms. Note that this asymptotic expansion is usually not convergent; there is some $p$, depending upon $f$ and $h$, such that the terms past order $p$ increase rapidly. Thus, the remainder term generally demands close attention.

The Euler–Maclaurin formula is also used for detailed error analysis in numerical quadrature. It explains the superior performance of the trapezoidal rule on smooth periodic functions and is used in certain extrapolation methods. Clenshaw–Curtis quadrature is essentially a change of variables to cast an arbitrary integral in terms of integrals of periodic functions where the Euler–Maclaurin approach is very accurate (in that particular case the Euler–Maclaurin formula takes the form of a discrete cosine transform). This technique is known as a periodizing transformation.

Asymptotic expansion of sums
In the context of computing asymptotic expansions of sums and series, usually the most useful form of the Euler–Maclaurin formula is $$\sum_{n=a}^b f(n) \sim \int_a^b f(x)\,dx + \frac{f(b) + f(a)}{2} + \sum_{k=1}^\infty \,\frac{B_{2k}}{(2k)!} \left(f^{(2k - 1)}(b) - f^{(2k - 1)}(a)\right),$$

where $a$ and $b$ are integers. Often the expansion remains valid even after taking the limits $a → −∞$ or $b → +∞$ or both. In many cases the integral on the right-hand side can be evaluated in closed form in terms of elementary functions even though the sum on the left-hand side cannot. Then all the terms in the asymptotic series can be expressed in terms of elementary functions. For example, $$\sum_{k=0}^\infty \frac{1}{(z + k)^2} \sim \underbrace{\int_0^\infty\frac{1}{(z + k)^2}\,dk}_{= \dfrac{1}{z}} + \frac{1}{2z^2} + \sum_{t = 1}^\infty \frac{B_{2t}}{z^{2t + 1}}.$$

Here the left-hand side is equal to $ψ^{(1)}(z)$, namely the first-order polygamma function defined by
 * $$\psi^{(1)}(z) = \frac{d^2}{dz^2}\log \Gamma(z);$$

the gamma function $Γ(z)$ is equal to $(z − 1)!$ when $z$ is a positive integer. This results in an asymptotic expansion for $ψ^{(1)}(z)$. That expansion, in turn, serves as the starting point for one of the derivations of precise error estimates for Stirling's approximation of the factorial function.

Examples
If $s$ is an integer greater than 1 we have: $$\sum_{k=1}^n \frac{1}{k^s} \approx \frac 1{s-1}+\frac 12-\frac 1{(s-1)n^{s-1}}+\frac 1{2n^s}+\sum_{i=1}\frac{B_{2i}}{(2i)!}\left[\frac{(s+2i-2)!}{(s-1)!}-\frac{(s+2i-2)!}{(s-1)!n^{s+2i-1}}\right].$$

Collecting the constants into a value of the Riemann zeta function, we can write an asymptotic expansion: $$\sum_{k=1}^n \frac{1}{k^s} \sim\zeta(s)-\frac 1{(s-1)n^{s-1}}+\frac 1{2n^s}-\sum_{i=1}\frac{B_{2i}}{(2i)!}\frac{(s+2i-2)!}{(s-1)!n^{s+2i-1}}.$$

For $s$ equal to 2 this simplifies to $$\sum_{k=1}^n \frac{1}{k^2} \sim\zeta(2)-\frac 1n+\frac 1{2n^2}-\sum_{i=1}\frac{B_{2i}}{n^{2i+1}},$$ or $$\sum_{k=1}^n \frac{1}{k^2} \sim \frac{\pi^2}{6} -\frac{1}{n} +\frac{1}{2n^2} -\frac{1}{6n^3}+\frac{1}{30n^5}-\frac{1}{42n^7} + \cdots.$$

When $s = 1$, the corresponding technique gives an asymptotic expansion for the harmonic numbers: $$\sum_{k=1}^n \frac{1}{k} \sim \gamma + \log n + \frac{1}{2n} - \sum_{k=1}^\infty \frac{B_{2k}}{2kn^{2k}},$$ where $γ ≈ 0.5772...$ is the Euler–Mascheroni constant.

Derivation by mathematical induction
We outline the argument given in Apostol.

The Bernoulli polynomials $B_{n}(x)$ and the periodic Bernoulli functions $P_{n}(x)$ for $n = 0, 1, 2, ...$ were introduced above.

The first several Bernoulli polynomials are $$\begin{align} B_0(x) &= 1, \\ B_1(x) &= x - \tfrac{1}{2}, \\ B_2(x) &= x^2 - x + \tfrac{1}{6}, \\ B_3(x) &= x^3 - \tfrac{3}{2}x^2 + \tfrac{1}{2}x, \\ B_4(x) &= x^4 - 2x^3 + x^2 - \tfrac{1}{30}, \\ &\,\,\,\vdots \end{align}$$

The values $B_{n}(1)$ are the Bernoulli numbers $B_{n}$. Notice that for $n ≠ 1$ we have $$B_n = B_n(1) = B_n(0),$$ and for $n = 1$, $$B_1 = B_1(1) = -B_1(0).$$

The functions $P_{n}$ agree with the Bernoulli polynomials on the interval $[0, 1]$ and are periodic with period 1. Furthermore, except when $n = 1$, they are also continuous. Thus, $$ P_n(0) = P_n(1) = B_n \quad \text{for }n \neq 1.$$

Let $k$ be an integer, and consider the integral $$ \int_k^{k + 1} f(x)\,dx = \int_k^{k + 1} u\,dv,$$ where $$\begin{align} u &= f(x), \\ du &= f'(x)\,dx, \\ dv &= P_0(x)\,dx & \text{since }P_0(x) &= 1, \\ v &= P_1(x). \end{align}$$

Integrating by parts, we get $$\begin{align} \int_k^{k + 1} f(x)\,dx &= \bigl[uv\bigr]_k^{k + 1} - \int_k^{k + 1} v\,du \\ &= \bigl[f(x)P_1(x)\bigr]_k^{k + 1} - \int_k^{k+1} f'(x)P_1(x)\,dx \\ &= B_1(1)f(k+1)-B_1(0)f(k) - \int_k^{k+1} f'(x)P_1(x)\,dx. \end{align}$$

Using $B_{1}(0) = −1⁄2$, $B_{1}(1) = 1⁄2$, and summing the above from $k = 0$ to $k = n − 1$, we get $$\begin{align} \int_0^n f(x)\, dx &= \int_0^1 f(x)\,dx + \cdots + \int_{n-1}^n f(x)\,dx \\ &= \frac{f(0)}{2}+ f(1) + \dotsb + f(n-1) + \frac{f(n)}{2} - \int_0^n f'(x) P_1(x)\,dx. \end{align}$$

Adding $f(n) − f(0)⁄2$ to both sides and rearranging, we have $$ \sum_{k=1}^n f(k) = \int_0^n f(x)\,dx + \frac{f(n) - f(0)}{2} + \int_0^n f'(x) P_1(x)\,dx.$$

This is the $p = 1$ case of the summation formula. To continue the induction, we apply integration by parts to the error term: $$\int_k^{k+1} f'(x)P_1(x)\,dx = \int_k^{k + 1} u\,dv,$$ where $$\begin{align} u &= f'(x), \\ du &= f''(x)\,dx, \\ dv &= P_1(x)\,dx, \\ v &= \tfrac{1}{2}P_2(x). \end{align}$$

The result of integrating by parts is $$\begin{align} \bigl[uv\bigr]_k^{k + 1} - \int_k^{k + 1} v\,du &= \left[\frac{f'(x)P_2(x)}{2} \right]_k^{k+1} - \frac{1}{2}\int_k^{k+1} f''(x)P_2(x)\,dx \\ &= \frac{B_2}{2}(f'(k + 1) - f'(k)) - \frac{1}{2}\int_k^{k + 1} f''(x)P_2(x)\,dx. \end{align}$$

Summing from $k = 0$ to $k = n − 1$ and substituting this for the lower order error term results in the $p = 2$ case of the formula, $$\sum_{k=1}^n f(k) = \int_0^n f(x)\,dx + \frac{f(n) - f(0)}{2} + \frac{B_2}{2}\bigl(f'(n) - f'(0)\bigr) - \frac{1}{2}\int_0^n f''(x)P_2(x)\,dx.$$

This process can be iterated. In this way we get a proof of the Euler–Maclaurin summation formula which can be formalized by mathematical induction, in which the induction step relies on integration by parts and on identities for periodic Bernoulli functions.

title: Euler–Maclaurin formula