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Medellín Cartel

The Medellín Cartel (Cártel de Medellín) was a powerful and highly organized Colombian drug cartel and terrorist organization originating in the city of Medellín, Colombia, that was founded and led by Pablo Escobar. It is often considered to be the first major "drug cartel" and was referred to as such; (a cartel) due to the organization's upper echelons and overall power-structure being built on a partnership between multiple Colombian traffickers operating alongside Escobar. Included were Jorge Luis Ochoa Vásquez, Fabio Ochoa Vásquez, Juan David Ochoa Vásquez, José Gonzalo Rodríguez Gacha and Carlos Lehder. Escobar's main partner in the organization however was his cousin Gustavo Gaviria who handled much of the cartel's shipping arrangements and the more general and detailed logistical aspects of the cocaine trafficking routes and international smuggling networks which were supplying at least 80% of the world's cocaine during its peak. Gustavo, also known as León seems to have also had a strong hand in the cartel's unprecedented acts of narcoterrorism, right alongside his cousin Pablo and was considered to be second in command of the cartel and therefore one of Colombia's most wanted men, with both him and Escobar having arrest warrants pending from other nations where their criminal activity had spread to, such as in Spain and the U.S. Meanwhile, Pablo Escobar's brother Roberto Escobar acted as the organization's accountant. The cartel operated from 1976 to 1993 in Colombia (Antioquia), Bolivia, Panama, Central America, Peru, the Bahamas, the United States (namely in cities such as Los Angeles, New York and Miami), as well as in Canada.

Although Escobar started profitably smuggling contraband by the early 1970s, the true beginnings of what would eventually become the mafia-like organization itself officially turned to trafficking cocaine as their main contraband product by 1976, (largely through the assistance of Carlos Lehder and George Jung) which greatly influenced the infamous sociocultural cocaine boom phenomenon of late 70s and early 80s in the United States. This "boom" was noticeably demonstrated by the impact of the violent street crimes which characterized the Miami drug war due to the cartel's trafficking operations significantly increasing the drug's overall availability and access through these newly enhanced markets as well as the further complexified and proliferated distribution networks. At the height of its operations, the Medellín Cartel smuggled multiple tons of cocaine each week into countries around the world and brought in an upwards of US$200 million daily in drug profits, and thus billions per year. Additionally, despite being well-known for once dominating the international illicit cocaine trade (along with expanding it) throughout the late 1970s and early 1980s, the organization, particularly in its later years, was also noted for its use of violence for political aims (mainly in protest of judicial extradition to the U.S.) as demonstrated by their societally straining and volatile asymmetric war against the Colombian state itself, primarily in the form of bombings, kidnappings, indiscriminate murder of law enforcement and political assassinations. Many of the victims included non-combatants or random citizens as attempts to negotiate with the government using fear through unambiguous acts of terror.

At its height, (during the early 80s) the Medellín Cartel was recognized as being the largest drug-trafficking syndicate in the world, estimated to have been smuggling three times as much cocaine as their main competitor, the Cali Cartel, an international drug-trafficking organization based in the Valle del Cauca department of Colombia. However, some experts and U.S. government officials have claimed the opposite, or said that most data compiled during this period was potentially skewed since most of the national security-based focus was mostly centered on the Medellín organization specifically due to its more ostentatious acts of violence and "vindictive" nature.

Relations between the Medellín and Cali trafficking organizations
Traditionally, Pablo and the Medellín Cartel originally had a strategic alliance and specific smuggling arrangement with the Cali Cartel "godfathers" Gilberto Rodríguez Orejuela and Miguel Rodríguez Orejuela early on. This arrangement resulted in Medellín controlling the cocaine trade in Los Angeles, Cali controlling New York City, while both of them agreed to share Miami and Houston. Eventually however, a personal dispute between Escobar and one of the Cali Cartel leaders named Pacho Herrera surfaced over a disagreement on territory in one or more of these distribution hubs. Although the two organizations had scrimmages beforehand; by 1988 (supposedly after the Monaco Apartment bombing) the Medellín and Cali organizations became perpetual enemies, a feud which continued until Medellín's eventual demise in 1993. A demise which was largely due to the rise of the anti-Escobar "vigilante" group known as Los Pepes, which was officially formed in January 1993. "Los Pepes" was a diminutive form of the phrase "Perseguidos por Pablo Escobar" (meaning: "[Those] Persecuted by Pablo Escobar"). The group was largely financed by the Cali Cartel and led by brothers Carlos and Fidel Castaño whom were right-wing paramilitary leading commanders that were actually once part of the Medellín Cartel. Los Pepe's tactics, which were stylized to reflect Escobar's own violent methods now turned against him, and which proved ultimately instrumental to the progressive defeat of the Medellín Cartel reign, was further accentuated by Pablo's accumulation of enemies within his own organization (helping to characterize the formation of Los Pepes), as well as Pablo's ongoing and relentless war against the Colombian government and American law-enforcement agencies; a combination of factors which cornered Escobar during the last year of his life, along with his quickly diminishing sphere of power as the now final and last-standing leader of his once dominant cartel.

During the initial days of their mutual alliance however, before these intense conflicts; the Cali and Medellín groups each showcased differing philosophies over how to "ameliorate" their situations with the authorities. While the more 'businesslike' Orejuela brothers of Cali wanted to solve such legal issues through corruption, Escobar instead sought to do so mostly through violence and intimidation. Particularly when his (often minimal) attempts to bribe officials were not immediately accepted. Notably, Escobar and the Medellín crew also garnered a unique reputation, even in their earlier days for murdering not only those who crossed them, but oftentimes their immediate family members or other relatives as well, unless otherwise stipulated such as when Pablo's enemies "turned themselves in" instead of fleeing the region or pleading for protection from the authorities (many of whom were already compromised). This method of "webbed" violence resulting from one person's mistakes against Escobar was reportedly explained directly to American smuggler George Jung when he first met Pablo in 1978 moments after witnessing a man get shot in the chest after he "voluntarily" gave himself up to Escobar upon the knowledge that the man's entire family would be killed if he did not.

While Pablo Escobar seemingly viewed fear through violence and extortion (threats) as an effective motivator and more straightforward measure, Cali leader Gilberto in particular thought violence should only be used as a last resort, considering it to be generally bad for business due to the practical understanding that; "dead people do not pay what they owe". The Cali Cartel leaders even required their employees to fill out "application forms"; elucidating their personal information, not necessarily for the sake of threats but rather for the purpose of thorough background checks and security. This included not just names and addresses of Cali Cartel hopefuls, but financial assets and personal possessions such as automobiles. The Orejuela brothers wanted to be seen as "the Kennedys of Colombia" while Escobar similarly dreamed of one day becoming president of the nation. When this failed however due to scandals over his criminal reputation, Escobar thereafter began a brutal; nearly decade-long terroristic campaign against the government from April of 1984 starting with the assassination of Rodrigo Lara Bonilla (Colombia's Minister of Justice who favored extradition), to April of 1993 with the massive 440 pound bomb detonation in Bogotá's busy Center 93 shopping mall on the city's popular 15th Avenue neighborhood.

1960s cocaine trade in Latin America
In the 1960s, Colombia was not yet viewed or stereotyped as being a conventional geophysical, narcotrafficking corridor for the world's cocaine trade like it seems to have been for marijuana trafficking at this time. By the late 1960s, Colombia's marijuana trade resulted from booming production in the Sierra Nevada de Santa Marta and the Urabá peninsula where it was then smuggled within hidden northbound shipments of bananas. Reportedly when it came to the international cocaine trade during the 60s; Argentina, Brazil and Chile were the countries with the biggest hand in the international transferring of cocaine. Rio de Janeiro (Brazil), Buenos Aires (Argentina) and Valparaiso (Chile) acted as some of the most common starting points for the 1960s maritime trafficking routes while Havana, Cuba was commonly used as an intermediary transshipment point north. Then it would go to locations like Panama City in Florida or Barcelona in Spain which both regularly functioned as final points of destination for the international transporting of cocaine or other related contraband from South America. Cartagena, in Colombia and Miami, Florida still seemed to play a modest role however. However, marijuana was still said to dominate the loads of shipments which traveled through these routes at the time. Therefore, cocaine's prevalence by then still went largely under the radar due to how in the United States, even though cocaine use was steadily rising by the 1960s, it was still only marginally - moderately more popular when compared to its relative scope of prevalence during the 1950s when the use of cocaine was considered to be at an all-time low and was essentially considered by many to mostly be "a problem of the past".

Colombian smugglers & Chileans (1972–1976)
In contrast to his law-abiding parents, Escobar began a life of crime early on; stealing tombstones from cemeteries, sanding down the headstones to make them appear unused, reselling them, stealing cars, kidnapping people for ransom; including a wealthy businessman at one point before eventually settling into the criminal occupation known as contraband smuggling which characterized a large swath and subcultural domain of Colombia's criminal underworld at this time. This is presumably around when his gang-like organizational abilities and his criminal racketeering predilections seems to have come together in what was now the early 1970s in Medellín, Colombia. Escobar took advantage of Colombia's fertile and expanding marijuana trade as well as the apparent smuggling and illegal trade of cigarettes. There are several claims that he was allegedly a noteworthy figure in the region's supposed "Marlboro wars"; a seemingly violent set of scrimmages or conflicts over the control of Colombia's illicit cigarette smuggling market. This implying that he was priming his sensibilities in street crime and violent black market schemes in what would come out in a much more organized and violent fashion not far down the line.

In 1973, there was a military coup in Chile which led to a strong crackdown on Chilean drug traffickers. This caused some drug traffickers to either flee the country and/or immediately start to use different trafficking routes. This frequently resulted in Chilean traffickers setting their eyes on Colombia to use as a trade route further north. Around the same time, the prevalence and social acceptance of contraband in Colombia was at an all-time high.

Background
In the mid to late 1970s, specifically around 1975 and 1976, the illegal cocaine trade started to become a growing and significant problem for American law enforcement as the drug's prevalence began to quickly takeoff in unanticipated ways. Initially, its first phase of substantial growth in popularity started out in the entertainment industry in Hollywood before quickly spreading to other major hotspots such as Miami.

Because of cocaine's socially prestigious reputation as a cultural symbol of high status, especially as it related to wealth and glamour among Americans during the late 1970s (mainly due to its high cost), Colombian criminals who were already accomplished smugglers of various goods through black market trade networks began to see cocaine as an irresistibly lucrative item for smuggling. A product like cocaine, which "packs tighter" than other types of drugs or contraband, indicated a greater, more eased potential for logistical transportability, with less of a chance for interdiction or complications during the product's long journey across borders and to various regional markets around the globe. This further maximized the drug's profit potential, since its volume and weight relative to its final monetary value was now exponentially greater compared to trafficking the same or similar weight and volume of a less valuable product such as marijuana. Also, unlike marijuana, which can be produced within close proximity to desirable North American drug markets, cocaine production was almost exclusive to South America at this time and therefore easier for regional criminal groups to control and monopolize.

During this era, many of the most well-established, entrepreneurial, and capable contraband smugglers or trafficking groups were composed of Colombian nationals. But even amongst them, the soon-to-be-massively-wealthy drug lord Pablo Escobar stood out from the rest. Escobar became characterized by his strategy of providing protection to other smugglers who partnered with him and helped further distribute the organization's cocaine into markets of high demand such as New York City and eventually Miami. Thus after forming a larger and more formidable web of suppliers and combined trade networks that continually empowered Escobar and his cousin's operations, enabled by a newly formed, mutually beneficial collaborative effort between different drug traffickers now working together (hence the term cartel), the organization was able to, at its height, move around 2,500-3,500 kilos per day.

The criminal power structure already present within Pablo and Gustavo's earlier, informal organization or smuggling ring rose to prominence as a result of cocaine's unusually high profit margins; even after just brief exposure to the North American market and its respective distribution channels, the organization's influence over the global cocaine business increased dramatically, which was naturally followed by unprecedented amounts of wealth. The then "cartel-style" model for Pablo and Gustavo's criminal organization began to reach fruition between 1976 and 1979, when it also incorporated the contraband smuggling networks of the Ochoas (Jorge Luis Ochoa Vásquez, Fabio Ochoa Vásquez and Juan David Ochoa Vásquez), as well as the trafficking enterprise and muscle (armed men) working under José Gonzalo Rodríguez Gacha; nicknamed "the Mexican", and the eccentric German Colombian figure Carlos Lehder, who initially partnered up with American smuggler George Jung or "Boston George", with whom he was bunkmates in prison at the Federal Correctional Institution, Danbury where Jung was incarcerated at the same time as Lehder on charges of marijuana trafficking.

Prior to this, Escobar's initial smuggling ring consisted of small gangs of hired gunmen on his payroll was just a precursor to the final and fully-fledged massive criminal organization that came to be known as the Medellín Cartel. Once these aforementioned drug lords stepped in to combine their trafficking power and capabilities, and after the ever-growing demand for their product in the North American market started to be recognized, the term cartel came into play, even if they did not originally refer to themselves as such. The term cartel, in reference to the mercantile nature of this now ever-expanding Colombian drug trafficking oligopoly, seems to have been first coined by the US State Department in order to distinguish these higher-level criminal organizations from the traditional, mid-sized gangs commonly associated with the drug trade. Eventually, with the cartel label becoming a household name, it was later embraced by the organization during its later years. This economic control, resulting from the unification of power structures of once separate groups, gave way to a sort of financial positive feedback loop, where further profits after the Medellín Cartel's initial expansion led to an even greater level of power and growth, leading to even more profits in the process.

Carlos Lehder and George Jung (1976–1978)
Jung described the Danbury prison facility located in Connecticut as a "medium-low security prison" that was "full of mob guys". He goes on to point out the "irony" (likely due to his marijuana charges) of him being assigned Colombian Carlos Lehder as a bunkmate who was serving a sentence for stealing cars and shipping them to South America. They casually discussed their charges, Jung mentioning he was caught flying marijuana out of Mexico; Lehder responded by asking him if he "knew anything about cocaine". He did not at the time, but became immediately entranced when Lehder went on to say; "you know it costs $60,000 a kilo". A proclamation that functioned as a compelling sales pitch for Jung; they both waited until getting out before attempting to get involved in the cocaine business.

While incarcerated together, Jung benefited Lehder by giving him valuable input and tips on how to smuggle drugs effectively. Jung was knowledgeable in this area due to his past experience flying shipments of marijuana across international borders—from Mexico into the US, and within the US, from California to New England. Lehder, however, would also be of benefit to Jung, once they were both released from prison, by introducing him to Colombia's most powerful drug smuggler: Pablo Escobar. This would prove useful to Jung in the future, when later he and Lehder had a falling out and were no longer in business together.

Around 1976, both men were out of prison (Jung being released slightly after Lehder). Lehder's behavior became increasingly erratic, supposedly "crazed" and "megalomaniacal" in the last two or three years of the 1970s, spiraling even further during the early 1980s. By decade's end (around 1978), Lehder had betrayed Jung. Lehder did this by trying to push Jung out of the business and began setting up his own major cocaine transshipment operation for Escobar and the Medellín organization on an island in the Bahamas called Norman's Cay. Here, Lehder would take up an unconventional, hedonistic, drug-fueled lifestyle after getting the island's residents to leave, either by force or by bribery. Jung from that point on just attempted to steer clear of Lehder, so he simply went back to Colombia to meet with Escobar directly, in order to establish a new, but now more modestly-sized, trafficking endeavor. This was to continue to operate under Escobar's network, since the two still had a good relationship, and Escobar could directly utilize Lehder as a reliable connection for acquiring sizable amounts of South American powdered cocaine.

Both Jung and Lehder made millions of dollars, not just for themselves, but for the cartel too, with their highly successful cocaine-distribution operation, that arguably may have been a factor in triggering America's cultural "cocaine era"; the take-off of this cultural phenomenon noticeably coincided with the beginning of Jung and Lehder's involvement in cocaine distribution. Despite their incredibly profitable smuggling operation, it only lasted from roughly 1976 to 1978. In this short period, however, they were able to achieve a historically unprecedented drug-trafficking operation that proved invaluable to Escobar's and the Medellín Cartel's overall initial rise to being the preeminent cocaine enterprise. Both Jung and Lehder were in the unique position of not only having been around during the very earliest days of the Medellín trafficking conglomerate which later birthed the cartel, but were also both in many ways directly responsible for the accelerated growth during its beginnings. Although Escobar and the others were highly capable and ingenious smugglers, they technically still did not have a major market in the United States or a major transportation system for their product. It was Jung and Lehder who effectively provided this.

From independent smuggling ring to collaborative cartel (1976–1979)
Pablo Escobar and his cousin Gustavo's mid-70s newly transformed trafficking operation aimed at moving cocaine powder into America after first fully ascertaining the indispensable distribution network capabilities and other logistical elements both Carlos Lehder and George Jung could facilitate for the sale of their product in the now major emerging markets was about to undergo another type of expansion. After carrying out their new cocaine operations with this augmented distribution system, this soon led the two cousins (Pablo and Gustavo) to amass a staggering amount of wealth in such a short timeframe (primarily due to Jung and Lehder), that their other fellow Colombian criminal acquaintances who were already noteworthy regional figures in the smuggling world, such as José Rodriguez Gacha and the Ochoas who were not yet part of the organization but began to more seriously consider getting involved with the two cousins and the cocaine business eventually came into the fold.

At this point, Pablo and Gustavo converged the increasing power and wealth of their single "independent" smuggling-ring towards a conglomerate made up of the multiple aforementioned smuggling rings coupled together in a collaborative fashion so as to give the overall organization a stronger backbone and structural apparatus by allowing Gacha and the Ochoas to pair up and synergize the relative power-spheres of their own trafficking networks with Escobar's, so as to maximize and concentrate their power instead of keeping it divided. After seeing the results of Pablo and Gustavo's impressive rates of profit and success (which now surpassed the Ochoas), they all soon began to prioritize the smuggling of cocaine over virtually all other forms of 'goods' and contraband, such as the items of commerce that had already served to profit them in years past.

1980s cocaine boom
This lucrative and newfound, expansive network of traffickers began to creatively utilize many novel methods of getting their cocaine product into the U.S. at that point; implementing increased ingenuity and novel strategies into the smuggling logistics. This ranged from cocaine-stuffed hollow pellets, multilayered condoms, or balloons which are then swallowed by mules to make it through customs or international borders (especially in commercial flight transportation) without detection, to hefty amounts loaded onto Cessnas; these now being recognized as a favorite method for smugglers throughout the years because of their ability to fly slowly which permit more accurate deliveries to drop zones. Some smuggling strategies also purportedly included diverting law-enforcement's attention away from one smuggled load of cocaine in order to smuggle in an even larger amount when authorities' attention and related interdiction resources were distracted or compromised. These rapid and clever innovations in illegal drug trafficking techniques by increasingly and well-funded organized criminal groups; such as Escobar and his Medellín associates proved to be too much for American law enforcement to effectively catch up with or curtail, especially with the price of cocaine dropping by 1980 and becoming yet even more popular and now among a much wider demographic instead of mostly just the affluent as it was in the late 70s. This would inevitably result in the manifestation of widespread cocaine use and cultural normalization, simultaneous crime waves in almost all major US cities, as well as small "drug wars" within the evermore relevant transshipment point areas or cities like Miami (see: Miami drug war) as well as in the ghettos of Los Angeles (see: Freeway Ricky Ross). Therefore, by the turn of the decade it was no longer just a drug of the sybaritic and wealthy despite still being a cultural status symbol of class and wealth in the early 80s.

The United States' "cocaine boom" era enriched the Colombian smugglers and catapulted them into massive levels of profit and wealth that now was in the billions of dollars, essentially all in the form of cash. During the cartel's peak, Escobar oversaw the import of large shipments of coca paste from Andean nations such as Peru and Bolivia into Colombia, where it was then processed into powdered cocaine in jungle labs such as in the case of Tranquilandia, before being flown into the United States in amounts of up to 15 tons per day at some points.

By 1982, cocaine surpassed coffee as the chief Colombian export (see: Coffee production in Colombia). Around this time in the early 1980s, kidnappings made by guerrilla groups led the State to collaborate with criminal groups like those formed by Escobar and the Ochoas. The abduction of Carlos Lehder as well as the 1981 kidnapping of the sister of the Ochoas; Martha Ochoa which led to the creation of cartel-funded private armies that were created to fight off guerrillas who were trying to either redistribute their lands to local peasants, kidnap them, or extort the gramaje money that the Revolutionary Armed Forces of Colombia (Fuerzas Armadas Revolucionarias de Colombia or FARC) attempted to steal.

"Death to Kidnappers" (1981)
At the end of 1981 and the beginning of 1982, members of the Medellín Cartel, Cali Cartel, the Colombian military, the U.S.-based corporation Texas Petroleum, the Colombian legislature, small industrialists, and wealthy cattle ranchers came together in a series of meetings in Puerto Boyacá and formed a paramilitary organization known as Muerte a Secuestradores ("Death to Kidnappers", MAS) to defend their economic interests, and to provide protection for local elites from kidnappings and extortion. By 1983, Colombian internal affairs had registered 240 political killings by MAS death squads, mostly community leaders, elected officials, and farmers.

The following year, the Asociación Campesina de Ganaderos y Agricultores del Magdalena Medio ("Association of Middle Magdalena Ranchers and Farmers", ACDEGAM) was created to handle both the logistics and the public relations of the organization, and to provide a legal front for various paramilitary groups. ACDEGAM worked to promote anti-labor policies, and threatened anyone involved with organizing for labor or peasants' rights. The threats were backed up by the MAS, which would attack or assassinate anyone who was suspected of being a "subversive". ACDEGAM also built schools whose stated purpose was the creation of a "patriotic and anti-Communist" educational environment, and built roads, bridges, and health clinics. Paramilitary recruiting, weapons storage, communications, propaganda, and medical services were all run out of ACDEGAM headquarters.

By the mid-1980s, ACDEGAM and MAS had undergone significant growth. In 1985, Pablo Escobar began funneling large amounts of cash into the organization to pay for equipment, training, and weaponry. Money for social projects was cut off and redirected towards strengthening the MAS. Modern battle rifles, such as the AKM, FN FAL, IMI Galil, and HK G3, were purchased from the military, INDUMIL, and drug-funded private sales. The organization had computers and ran a communications center that worked in coordination with the state telecommunications office. They had 30 pilots, and an assortment of fixed-wing aircraft and helicopters. British, Israeli, and U.S. military instructors were hired to teach at paramilitary training centers.

Middle–late 1980s

Following this time in the mid-80s, Escobar's hold on Medellín further increased when he founded a criminal debt collection service known as the "Oficina de Envigado" (The Office), which was considered their first "oficina de cobro" (collections service). This was an actual physical office located in the town hall of Envigado, a small municipality next to Medellín where Escobar grew up. Escobar used the municipal office to collect debts owed to him by other drug traffickers and set the "sicarios" or hired killers on those who refused. Escobar was known to flaunt his wealth and went on to make Forbes' Billionaires list for seven years straight, between 1987 and 1993. His luxurious multimillion-dollar "Hacienda Nápoles" estate had its own zoo, and he reportedly ate from solid gold dinner sets. Escobar was known for investing profits from the drug trade in luxury goods, property, and works of art. He is also reported to have stashed his cash in "hidden coves", allegedly burying it on his farms and under floors in many of his houses.

Pablo Escobar's death (December, 1993)

After a long, drawn-out manhunt that lasted many months; on 2 December, 1993 Escobar was finally found with only one bodyguard left by his side; Limón (Alvaro de Jesús Agudelo) who was also his longtime chauffeur (driver). They were found hiding out in a house in a middle-class residential area of his hometown of Medellín (Carrera 79B No. 45D – 94, Los Olivos neighborhood) by Colombian special forces supposedly using tracking technology provided by the United States. This was a direct consequence of Escobar the day before; on his 44th birthday (Wednesday) making a call from the hideout location to his family who were being protected by the government in a luxury hotel in the nation's capital of Bogota. This was also just after the family had returned from an unsuccessful attempt to travel by plane and take refuge in Germany. The authorities, going off the traceable data from the hotel which received Escobar's call began to utilize an electronic tracking system in order to pinpoint the geographical positioning that the call had originated from. At this time, Escobar who now had a long beard according to witnesses had apparently been hiding out there for almost six weeks. First after some initial troubleshooting, both the drug lord and Limón were eventually located the next day. Police tried to arrest Escobar upon arrival, but the situation quickly escalated to an exchange of gunfire. By the time Pablo and Limón escaped the inside of the building and tried to outrun the authorities on a tiled terracotta rooftop, Limón was soon shot and killed. Moments later Escobar himself was then shot and killed as he continued to try and make his way from the pursuing officers whilst still running across the top of the roof to escape. He was hit by bullets in the torso and feet, and a bullet which struck him in the ear, killing him.

This has sparked debate however about whether he killed himself or whether he was actually shot dead by the Colombian authorities, with his son Sebastián Marroquín (formerly named; Juan Pablo Escobar) still believing to this day that he shot himself due to the positioning of the bullet's entry wound behind his ear where he says his father would've apparently aimed for if he was going to attempt to kill himself. Escobar's primary objective in an end scenario such as this was simply to not allow himself to be captured alive and suffer for the rest of his life in the U.S. prison system via extradition or to be captured and tortured like he did to so many others. Escobar's legacy remains controversial; while many denounce the heinous nature of his crimes and terrorism, he was still seen as a "Robin Hood-like" figure for many in Colombia, as he provided many amenities such as the funding of infrastructure and residencies for the poor, especially in barrios throughout the areas surrounding Medellín. His death was mourned and his funeral attended by over 25,000 people. Additionally, his private estate, Hacienda Nápoles was thereafter eventually transformed into a theme park. His life as a major cocaine trafficker and incredibly wealthy criminal (bandido) has been dramatized widely in film and has served as inspiration for pop cultural references and the retelling of his gripping story as a prominent drug lord in both television, books and music.

Political relations
During the later years of the Cold War, the number of left-wing guerrillas in Latin America skyrocketed. The conflicts between them and the right-wing paramilitaries groups and dictatorships, mostly backed by CIA, made the Cartel's search for new allies while it was forced to be involved in corruption outside Colombia for political protection.

Relations with the Colombian government
Once U.S. authorities were made aware of "questionable activities", the group was put under Federal Drug Task Force surveillance. Evidence was gathered, compiled, and presented to a grand jury, resulting in indictments, arrests, and prison sentences for those convicted in the United States. However, very few Colombian cartel leaders were actually taken into custody as a result of these operations. Mostly, non-Colombians conspiring with the cartel were the "fruits" of these indictments in the United States.

Most Colombians targeted, as well as those named in such indictments, lived and stayed in Colombia, or fled before indictments were unsealed. However, by 1993 most, if not all, cartel fugitives had been either imprisoned, or located and shot dead, by the Colombian National Police trained and assisted by specialized military units and the CIA.

The last of Escobar's lieutenants to be assassinated was Juan Diego Arcila Henao, who had been released from a Colombian prison in 2002 and hidden in Venezuela to avoid the vengeance of "Los Pepes". However he was shot and killed in his Jeep Cherokee as he exited the parking area of his home in Cumaná, Venezuela, in April 2007.

While it is broadly believed that Los Pepes have been instrumental in the assassination of the cartel's members over the last 21 years, it is still in dispute whether the mantle is just a screen designed to deflect political repercussions from both the Colombian and United States governments' involvement in these assassinations.

Relations with Nicaraguan government
The Carlos Lehder's Norman's Cay strategy was shut down in 1982 after the Bahamian government started chasing the drug traffic. Hence, Central America was the path chosen by the Cartel to reach the US. In 1985, the DEA, knowing about Barry Seal ties with both the Medellin Cartel, made the pilot take pictures of the cartel's landing stripes in Nicaragua. The DEA prior investigation appointed that the cartel had the protection from the FSLN, the Sandinist party, to use Nicaragua as a "warehouse" for Matta-Ballesteros' logistic operation for either Medellin and Guadalajara cartels.

Knowing Seal's activity as a DEA informant, the cartel put a contract on him, having him murdered in February 1986. In March 16 the same year, during a TV national address, the president of the United States Ronald Reagan used surveillance pictures taken on Seal's undercover mission that showed Escobar, Gacha, the Nicaraguan government official Federico Vaughan and several other men loading a plane with cocaine.

Relations with Panamanian government
After Manuel Noriega arrest, the DEA and FBI got several tips linking the former dictator to the Medellin Cartel. The Nicaraguan and Colombian conflicts in remote areas could make both countries unsafe to operate, and the cartel was searching for new places to use as a stopover. Manuel Noriega, in exchange of bribes and share on profits, agreed using Panama as another stopover for the cartel logistics. Also, the links between him and the Cartel ensured that Panama could be a hideout for the group leadership and that the government would ignore their money laundering operations in the country.

The CIA turned a blind eye to the link between Medellin Cartel and Noriega, as he had an aggressive anticommunist policy. However, the agency stopped supporting Noriega after his ties with the Cartel came to public.

Relations with Cuban government
During the 1980 decade, the decrease and later end of Soviet Union subsidies almost wiped out the Cuban economy. During Manuel Noriega judgement, Carlos Lehder, the cartel member responsible for ensuring that the cocaine would reach Florida, testified that, as the drug smuggling to the USA was doing billions of dollars, Cuban intelligence helped manage Nicaraguan operations and the island's government agreed to using Cuba as one of the stopovers for the cartel. This was made with knowledge of the Castro brothers, having the younger, Raúl Castro, met with Lehder.

Fear of extradition
Perhaps the greatest threat posed to the Medellín Cartel and the other traffickers was the implementation of an extradition treaty between the United States and Colombia. It allowed Colombia to extradite to the US any Colombian suspected of drug trafficking and to be tried there for their crimes. This was a major problem for the cartel, since the drug traffickers had little access to their local power and influence in the US, and a trial there would most likely lead to imprisonment. Among the staunch supporters of the extradition treaty were Colombian Justice Minister Rodrigo Lara (who was pushing for more action against the drug cartels), Police Officer Jaime Ramírez, and numerous Colombian Supreme Court judges.

However, the cartel applied a "bend or break" strategy towards several of these supporters, using bribery, extortion, or violence. Nevertheless, when police efforts began to cause major losses, some of the major drug lords themselves were temporarily pushed out of Colombia, forcing them into hiding from which they ordered cartel members to take out key supporters of the extradition treaty.

The cartel issued death threats to the Supreme Court Judges, asking them to denounce the Extradition Treaty. The warnings were ignored. This led Escobar and the group he called Los Extraditables ("The Extraditables") to start a violent campaign to pressure the Colombian government by committing a series of kidnappings, murders, and narco-terrorist actions.

Alleged relation with the M-19
In November 1985, 35 heavily armed members of the M-19 guerrilla group stormed the Colombian Supreme Court in Bogotá, leading to the Palace of Justice siege. Some claimed at the time that the cartel's influence was behind the M-19's raid, because of its interest in intimidating the Supreme Court. Others state that the alleged cartel-guerrilla relationship was unlikely to occur at the time because the two organizations had been having several standoffs and confrontations, like the kidnappings by M-19 of drug lord Carlos Lehder and of Marta Nieves Ochoa, the sister of Juan David Ochoa. These kidnappings led to the creation of the MAS/Muerte a Secuestradores ("Death to Kidnappers") paramilitary group by Pablo Escobar. Former guerrilla members have also denied that the cartel had any part in this event. The issue continues to be debated inside Colombia.

Assassinations
As a means of intimidation, the cartel conducted thousands of assassinations throughout the country. Escobar and his associates made it clear that whoever stood against them would risk being killed along with their families. Some estimates put the total around 3,500 killed during the height of the cartel's activities, including over 500 police officers in Medellín, but the entire list is impossible to assemble, due to the limitation of the judiciary power in Colombia. The following is a brief list of the most notorious assassinations conducted by the cartel:

 * Luis Vasco and Gilberto Hernandez, two DAS agents who had arrested Pablo Escobar in 1976. Among the earliest assassinations of authority figures by the cartel.
 * Rodrigo Lara, Minister of Justice, killed on a Bogotá highway on April 30, 1984, when two gunmen riding a motorcycle approached his vehicle in traffic and opened fire.
 * Tulio Manuel Castro Gil, Superior Judge which investigating Escobar for the assassination of two DAS agents which in 1977 arrested Escobar and his cousin Gustavo Gaviria, killed by motorcycle gunmen in July 1985, shortly after indicting Escobar.
 * Hernando Baquero Borda, Supreme Court Justice; rapporteur and defender of the Extradition Treaty with the United States, killed by gunmen in Bogotá on July 31, 1986.
 * Jaime Ramírez Gómez, Police Colonel and head of the anti-narcotics unit of the National Police of Colombia. Killed near Fontibon on his way to Bogota on November 17, 1986, when assassins in a green Renault 18 drove beside his red Mitsubishi Montero and opened fire. Ramírez was killed instantly; his wife and two sons were unharmed.
 * Guillermo Cano Isaza, director of El Espectador who revealed publicly Escobar's criminal past, killed on December 17, 1986, in Bogotá by gunmen riding a motorcycle.
 * Jaime Pardo Leal, presidential candidate and head of the Patriotic Union party, killed by a gunman in October 1987.
 * Carlos Mauro Hoyos, Attorney General, kidnapped then killed by gunmen in Medellín in January 1988.
 * Antonio Roldan Betancur, governor of Antioquia, killed by a car bomb in July 1989.
 * Waldemar Franklin Quintero, Commander of the Antioquia police, killed by gunmen in Medellín in August 1989.
 * Luis Carlos Galán, presidential candidate, killed by gunmen during a rally in Soacha in August 1989. The assassination was carried out on the same day the commander of the Antioquia police was gunned down by the cartel.
 * Carlos Ernesto Valencia, Superior Judge, killed by gunmen shortly after indicting Escobar on the death of Guillermo Cano, in August 1989.
 * Jorge Enrique Pulido, journalist, director of Jorge Enrique Pulido TV, killed by gunmen in Bogotá in November 1989.
 * Diana Turbay, journalist, chief editor of the Hoy por Hoy magazine and former president Julio César Turbay Ayala's daughter, killed by Colombian military during a rescue attempt in January 1991. Actually, the bullet found in her body came from a police helicopter.
 * Enrique Low Murtra, Minister of Justice, killed by gunmen in downtown Bogotá in May 1991.
 * Myriam Rocio Velez, Superior Judge, killed by gunmen shortly before she was to sentence Escobar on the assassination of Galán, in September 1992.

Miguel Maza Márquez was targeted in the DAS Building Bombing, resulting in the death of 52 civilians caught in the blast. Miguel escaped unharmed.

In 1993, shortly before Escobar's death, the cartel lieutenants were also targeted by the vigilante group Los Pepes (or PEPES, People Persecuted by Pablo Escobar).

With the assassination of Juan Diego Arcila Henao in 2007, most if not all of Escobar's lieutenants who were not in prison had been killed by the Colombian National Police Search Bloc (trained and assisted by U.S. Delta Force and CIA operatives), or by the Los Pepes vigilantes.

DEA agents considered that their four-pronged "Kingpin Strategy", specifically targeting senior cartel figures, was a major contributing factor to the collapse of the organization.

Legacy
La Oficina de Envigado is believed to be a partial successor to the Medellín organization. It was founded by Don Berna as an enforcement wing for the Medellín Cartel. When Don Berna fell out with Escobar, La Oficina caused Escobar's rivals to oust Escobar. The organization then inherited the Medellín turf and its criminal connections in the US, Mexico, and the UK, and began to affiliate with the paramilitary United Self-Defense Forces of Colombia, organizing drug trafficking operations on their behalf.

In popular culture
The cartel has been both featured and referenced in numerous works of popular culture.
 * Kings of Cocaine: Inside the Medellín Cartel - An Astonishing True Story of Murder, Money and International Corruption, Book by Guy Gugliotta
 * Blow: 2001 film about drug smuggler George Jung, Carlos Lehder (named Diego Delgado in the film) and the Medellín Cartel
 * Narcos is a Netflix original television series (2015–2017) that chronicles the life of Pablo Escobar and the rise of the Medellín Cartel. The first and second season depict his rise to the status of a powerful drug lord as well his narcoterrorist acts, war against the Colombian government and finally, his death. The role of Escobar is played by Brazilian actor Wagner Moura. The Medellín organization is also briefly depicted and mentioned in the spinoff series Narcos: Mexico.
 * Cocaine Cowboys and Cocaine Cowboys 2: documentary series about the Miami Drug War and Griselda Blanco
 * American Desperado: a book by journalist Evan Wright and former Medellín Cartel trafficker Jon Roberts
 * The Two Escobars: an ESPN 30 for 30 film details the link between the Medellín Cartel and the rise of Colombian football
 * American Made: 2017 fictionalised film about drug smuggler Barry Seal (Tom Cruise) and the Medellín Cartel, led by Jorge Ochoa (Alejandro Edda)
 * Season 2 episode 7 of Deadliest Warrior pitted the Medellín Cartel against the Somali Pirates with Michael Corleone Blanco, Son Of Griselda Blanco and gangster turned informant Kenny "Kenji" Gallo testing the cartel's weapons.
 * Sicario: 2015 film follows the story of an FBI agent on a classified mission working with a representative for the Medellín cartel

title: Medellín Cartel


text:
Bus snooping

Bus snooping or bus sniffing is a scheme by which a coherency controller (snooper) in a cache (a snoopy cache) monitors or snoops the bus transactions, and its goal is to maintain a cache coherency in distributed shared memory systems. This scheme was introduced by Ravishankar and Goodman in 1983, under the name "write-once" cache coherency. A cache containing a coherency controller (snooper) is called a snoopy cache.

How it works
When specific data is shared by several caches and a processor modifies the value of the shared data, the change must be propagated to all the other caches which have a copy of the data. This change propagation prevents the system from violating cache coherency. The notification of data change can be done by bus snooping. All the snoopers monitor every transaction on a bus. If a transaction modifying a shared cache block appears on a bus, all the snoopers check whether their caches have the same copy of the shared block. If a cache has a copy of the shared block, the corresponding snooper performs an action to ensure cache coherency. The action can be a flush or an invalidation of the cache block. It also involves a change of cache block state depending on the cache coherence protocol.

Types of snooping protocols
There are two kinds of snooping protocols depending on the way to manage a local copy of a write operation:

Write-invalidate
When a processor writes on a shared cache block, all the shared copies in the other caches are invalidated through bus snooping. This method ensures that only one copy of a datum can be exclusively read and written by a processor. All the other copies in other caches are invalidated. This is the most commonly used snooping protocol. MSI, MESI, MOSI, MOESI, and MESIF protocols belong to this category.

Write-update
When a processor writes on a shared cache block, all the shared copies of the other caches are updated through bus snooping. This method broadcasts a write data to all caches throughout a bus. It incurs larger bus traffic than write-invalidate protocol. That is why this method is uncommon. Dragon and firefly protocols belong to this category.

Implementation
One of the possible implementations is as follows:

The cache would have three extra bits:

 * V – valid
 * D – dirty bit, signifies that data in the cache is not the same as in memory
 * S – shared

Each cache line is in one of the following states: "dirty" (has been updated by local processor), "valid", "invalid" or "shared". A cache line contains a value, and it can be read or written. Writing on a cache line changes the value. Each value is either in main memory (which is very slow to access), or in one or more local caches (which is fast). When a block is first loaded into the cache, it is marked as "valid".

On a read miss to the local cache, the read request is broadcast on the bus. All cache controllers monitor the bus. If one has cached that address and it is in the state "dirty", it changes the state to "valid" and sends the copy to requesting node. The "valid" state means that the cache line is current. On a local write miss (an attempt to write that value is made, but it's not in the cache), bus snooping ensures that any copies in other caches are set to "invalid". "Invalid" means that a copy used to exist in the cache, but it is no longer current.

For example, an initial state might look like this:

Tag | ID | V | D | S - 1111 | 00 | 1 | 0 | 0 0000 | 01 | 0 | 0 | 0 0000 | 10 | 1 | 0 | 1 0000 | 11 | 0 | 0 | 0

After a write of address 1111 00, it would change into this:

Tag | ID | V | D | S - 1111 | 00 | 1 | 1 | 0 0000 | 01 | 0 | 0 | 0 0000 | 10 | 1 | 0 | 1 0000 | 11 | 0 | 0 | 0

The caching logic monitors the bus and detects if any cached memory is requested. If the cache is dirty and shared and there is a request on the bus for that memory, a dirty snooping element will supply the data to the requester. At that point either the requester can take on responsibility for the data (marking the data as dirty), or memory can grab a copy (the memory is said to have "snarfed" the data) and the two elements go to the shared state.

When invalidating an address marked as dirty (i.e. one cache would have a dirty address and the other cache is writing) then the cache will ignore that request. The new cache will be marked as dirty, valid and exclusive and that cache will now take responsibility for the address.

Benefit
The advantage of using bus snooping is that it is faster than directory based coherency mechanism. The data being shared is placed in a common directory that maintains the coherence between caches in a directory-based system. Bus snooping is normally faster if there is enough bandwidth, because all transactions are a request/response seen by all processors.

Drawback
The disadvantage of bus snooping is limited scalability. Frequent snooping on a cache causes a race with an access from a processor, thus it can increase cache access time and power consumption. Each of the requests has to be broadcast to all nodes in a system. It means that the size of the (physical or logical) bus and the bandwidth it provides must grow, as the system becomes larger. Since the bus snooping does not scale well, larger cache coherent NUMA (ccNUMA) systems tend to use directory-based coherence protocols.

Snoop filter
When a bus transaction occurs to a specific cache block, all snoopers must snoop the bus transaction. Then the snoopers look up their corresponding cache tag to check whether it has the same cache block. In most cases, the caches do not have the cache block since a well optimized parallel program doesn’t share much data among threads. Thus the cache tag lookup by the snooper is usually an unnecessary work for the cache who does not have the cache block. But the tag lookup disturbs the cache access by a processor and incurs additional power consumption.

One way to reduce the unnecessary snooping is to use a snoop filter. A snoop filter determines whether a snooper needs to check its cache tag or not. A snoop filter is a directory-based structure and monitors all coherent traffic in order to keep track of the coherency states of cache blocks. It means that the snoop filter knows the caches that have a copy of a cache block. Thus it can prevent the caches that do not have the copy of a cache block from making the unnecessary snooping. There are three types of filters depending on the location of the snoop filters. One is a source filter that is located at a cache side and performs filtering before coherence traffic reaches the shared bus. Another is a destination filter that is located at receiver caches and prevents unnecessary cache-tag look-ups at the receiver core, but this type of filtering fails to prevent the initial coherence message from the source. Lastly, in-network filters prune coherence traffic dynamically inside the shared bus. The snoop filter is also categorized as inclusive and exclusive. The inclusive snoop filter keeps track of the presence of cache blocks in caches. However, the exclusive snoop filter monitors the absence of cache blocks in caches. In other words, a hit in the inclusive snoop filter means that the corresponding cache block is held by caches. On the other hand, a hit in the exclusive snoop filter means that no cache has the requested cache block.

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

title: Bus snooping


text:
Duke of Burgundy

Duke of Burgundy (duc de Bourgogne) was a title used by the rulers of the Duchy of Burgundy, from its establishment in 843 to its annexation by the French crown in 1477, and later by members of the House of Habsburg, including Holy Roman Emperors and kings of Spain, who claimed Burgundy proper and ruled the Burgundian Netherlands.

The Duchy of Burgundy was a small portion of the traditional lands of the Burgundians west of the river Saône which, in 843, was allotted to Charles the Bald's kingdom of the West Franks. Under the Ancien Régime, the duke of Burgundy was the premier lay peer of the Kingdom of France. Beginning with Robert II of France ((r. 996 – 1031)), the title was held by the Capetians, the French royal family. In 1032 King Henry I of France granted the duchy to his younger brother, Robert, who founded the House of Burgundy. When the senior line of the House of Burgundy became extinct in 1361, the title was inherited by King John II of France through proximity of blood. John granted the duchy to his younger son, Philip the Bold, in 1363. The Valois dukes gradually came to rule over a vast complex of territories known as the Burgundian State, and became dangerous rivals to the senior French royal line of the House of Valois.

When the male line of the Valois dukes of Burgundy became extinct in 1477, the Duchy of Burgundy was confiscated by Louis XI of France. The title "duke of Burgundy" passed to Habsburg monarchs after Mary of Burgundy married Maximilian I of Austria in 1477. The Habsburgs used this connection to claim Burgundy proper and to rule their Burgundian inheritance until the Napoleonic era. The title was subsequently revived for several younger sons of the House of Bourbon and since 1975, branches of it have used "duke of Burgundy" as a revived courtesy title.

Bosonid dynasty (880–956)
The first margrave (marchio), later duke (dux), of Burgundy was Richard of the House of Ardennes, whose duchy was created from the merging of several regional counties of the kingdom of Provence which had belonged to his brother Boso. Richard was the son of Eccard of Macon and Richildis of Arles, Boso the son of Bivin of Gorze and Richildis of Arles.

His descendants and their relatives by marriage ruled the duchy until its annexation over a century later by the French crown, their suzerain.

 * Richard the Justiciar (880–921)
 * Rudolph (921–923), then King of the Franks
 * Hugh the Black (923–952)
 * Gilbert (952–956)

Robertian dynasty (956–1002)

 * Otto (956–965)
 * Eudes Henry the Great (965–1002)

House of Ivrea (1002–1004)

 * Otto William (1002–1004)

House of Capet (1004–1032)
In 1004, Burgundy was annexed by the king, of the House of Capet. Otto William continued to rule what would come to be called the Free County of Burgundy. His descendants formed another House of Ivrea.

 * Robert (1004–1016) (also king of the Franks as Robert II)
 * Henry (1016–1032) (also king of the Franks as Henry I)

House of Burgundy (1032–1361)
Robert, son of Robert II of France, received the Duchy as a peace settlement, having disputed the succession to the throne of France with his brother Henry.

House of Valois-Burgundy (1363–1482)
John II of France, the second Valois king, successfully claimed the duchy after the death of Philip, the last Capet duke. John then passed the duchy to his youngest son Philip as an apanage.

House of Habsburg (1482–1700)
In 1477, the territory of the Duchy of Burgundy was annexed by France. In the same year, Mary married Maximilian, Archduke of Austria, giving the Habsburgs control of the remainder of the Burgundian Inheritance.

Although the territory of the Duchy of Burgundy itself remained in the hands of France, the Habsburgs remained in control of the title of Duke of Burgundy and the other parts of the Burgundian inheritance, notably the Low Countries and the Free County of Burgundy in the Holy Roman Empire as well as the County of Charolais in France. They often used the term Burgundy to refer to it (e.g. in the name of the Imperial Circle it was grouped into), until the late 18th century, when the Austrian Netherlands were lost to the French Republic. The Habsburgs also continued to claim Burgundy proper until the Treaty of Cambrai in 1529, when they surrendered their claim in exchange for French recognition of Imperial sovereignty over Flanders and Artois.

 * Maximilian I (1477–1482 with his wife; regent 1482–1494)
 * Philip IV the Handsome (Philipp der Schöne; Philippe le Beau), titular Duke of Burgundy as Philip IV (1482–1506)
 * Charles II (Holy Roman Emperor Charles V and King Charles I of Spain) 1506–1555

 * Philip V (King Philip II of Spain) 1556–1598
 * Isabella I (infanta Isabella Clara of Spain) and Albert I (Albert VII of Austria) (jure uxoris) 1598–1621
 * Philip VI (King Philip IV of Spain) 1621–1665
 * Charles III (King Charles II of Spain) 1665–1700

House of Bourbon, claimants of the title (1682–1761)

 * Louis, Duke of Burgundy (1682–1712)
 * Louis, Duke of Burgundy (1751–1761)

House of Habsburg (1713–1918)

 * Charles IV (Emperor Charles VI) 1713–1740
 * Maria Theresa 1740–1780
 * Francis I (Emperor Francis I) (1740–1765 with his wife)
 * Joseph (Emperor Joseph II) 1780–1790
 * Leopold (Emperor Leopold II) 1790–1792
 * Francis II (Emperor Francis II) 1792–1795/1835
 * Ferdinand (Emperor Ferdinand I) (1835–1848 titular only)
 * Franz Joseph (Emperor Franz Joseph I) (1848–1916 titular only)
 * Charles V (Emperor Charles I) (1916–1918 titular only later renounced)

House of Bourbon, revived title (1975–present)

 * King Juan Carlos I of Spain (1975–2014)
 * King Felipe VI of Spain (2014–present) – the title is one of the titles of the Spanish Crown
 * Prince Sixtus Henry of Bourbon-Parma (Carlist claimant to the throne of Spain as "Enrique V", 1977–present) – the title is one of the titles of the Spanish Crown. He is considered the legitimate successor to the Catholic Monarchy of the Spains as heir to Infante Carlos María Isidro ("Carlos V").
 * Louis, Dauphin of France (2010–present) – the title is used by eldest son of the Legitimist claimant to the French throne, Prince Louis, Duke of Anjou ("Louis XX").

title: Duke of Burgundy


text:
Talk:Improper integral

Structure
I think this page is reasonably 'mature' now in its content; but that the topics could usefully be re-ordered.

Charles Matthews 09:07, 5 Nov 2003 (UTC)

Move to Cauchy principal value?
There is now a free-standing Cauchy principal value page: example here -> there?

Charles Matthews 10:45, 14 May 2004 (UTC)

The term "improper integral" seems more widely known that "Cauchy principal value" or "principal value", so if the pages get merged, I think it would be better to keep this one and make the other one a redirect page. Michael Hardy 01:02, 20 Sep 2004 (UTC)

Introduction
The introduction on this article needs a lot of work. I can hardly understand it as it currently is, so somebody unfamiliar with the material could be hopelessly lost.
 * 1) The definition of an improper integral is vague and as stated and is either not well defined or a super class of all definite Riemann integrals. (depending on how you read specified real number)
 * 2) There are multiple topics addressed with little or no transition making reading it misleading (as just before the third integral)

 * 1) There's too many specific details that should be in the body of the article, not in the intro (as three fourths of it talks of Lebesgue integration) GromXXVII 19:51, 7 November 2007 (UTC)

You added a "definition" that referred to "with integrals of functions that could possibly have infinite area", and you complain of vagueness? How vague can you get?

Can you be specific about what it is that you can't understand about the definition? Michael Hardy 21:26, 7 November 2007 (UTC)

 * In calculus, an improper integral is the limit of a definite integral, as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits.
 * Lack of explanation as to the concept, it jumps right into the explicit limit definition (without explicit definitions)
 * approaches either a specified real number could mean any real number, which would imply all definite integrals are improper integrals, or some unknown specified real number the article never tells the reader about.

 * Essentially, the statement doesn’t tell the reader what an improper integral is, or why we need it. Then it leads into a bunch of cases without an explicit definition.

 * I was trying to link the two types (unbounded function, unbounded interval) into one cohesive, concise expression. Then it can be further refined into the different types and explicit definitions using limits. Sure it wasn’t the best possible, but with the other changes it made the article more readable and explanatory than it currently is and can always be improved upon. GromXXVII 21:41, 7 November 2007 (UTC)

...OK, I've looked over the article from top to bottom. I've expanded the first sentence a bit to include some expressions in mathematical notation for anyone who finds that easier to follow than the words preceeding it. But you really do need to be specific about each of your complaints. (1) What specifically do you find hard to understand about the initial sentence? (2) In which cases is the concept "not well defined"? (3) Are you saying it is a "super class of all definite Riemann integrals", and you object to that, or that it is not a "super class of all definite Riemann integrals", and you object to that? (4) What sort of "transition" do you have in mind "just before the third integral"? The point of that sentence is that even in cases where an integral can be defined without taking a limit, nonetheless it can't quickly be computed without taking a limit, i.e., in one sense it may not need to be considered improper, but for purposes of computation one treats it as improper. It seems to me that that needs to be given prominence early in the article. (5) Which specific details are you saying should not be in the introduction. Michael Hardy 21:46, 7 November 2007 (UTC)

 * 1) “specified real number” is ambiguous because there are restrictions on what the "number" can be to yield an improper integral. For instance, it implies that $$\int_0^1 1\,dx = \lim_{t\to 1} \int_0^t 1\,dx$$ is an improper integral. It’s not standard to use a limit in such case, but is certainly possible to do.
 * This is if the reader assumes “specified real number” could be anything real.

 * 2) It’s not well defined if the reader assumes there are restrictions on the specified number, but they’re not stated. I didn’t mean well defined in the mathematical sense, I meant it as in “part of the definition is not available to the reader”

 * 3) I’m saying it’s not a super class of all Riemann integrals, but the intro makes it seem so. (Also note it certainly doesn’t need to address this, just to be written so that it doesn’t have that tone)

 * 4) I was referring to this, where the integral appears to be trying to help explain the paragraph, but doesn’t quite have that purpose
 * “In some cases, the integral from a to c is not even defined, because the integrals of the positive and negative parts of f(x) dx from a to c are both infinite, but nonetheless the limit may exist. Such cases are "properly improper" integrals, i.e. their values cannot be defined except as such limits.

 * This should be reworded, but also in a different section.

 * The integral
 * $$\int_0^\infty\frac{dx}{1+x^2}$$
 * can be interpreted as
 * $$\lim_{b\to\infty}\int_0^b\frac{dx}{1+x^2}=\lim_{b\to\infty}\arctan{b}=\frac{\pi}{2},$$”

 * 5) I think that everything after “(see Figures 1 and 2)” can be moved somewhere else.

 * Ultimately I think it’s most important for the introduction to give a clear idea of what an improper integral is (which would address that a limit or other advanced techniques are used because standard Riemann integration does not work), and when an expression is an improper integral (which would address the issues of the interval the function is integrated over, and the issues of the range of the function)
 * There are perhaps other things that merit prominence to be in the introduction, but most of what is there doesn’t, especially because there isn’t a clear description currently, so all the stuff about computation, Lesbegue integration, and properly improper integrals serve to confuse instead of inform - even though they should be in the article.

 * It seems necessary to me that the article quickly:
 * Motivate why improper integrals are used
 * Explain when they’re used

 * Define the symbols used for them (in at least the two simplest cases of an unbounded function and interval). I had four explicit definitions to cover the most common cases so I put them all in a different section.

 * I think that the version I had would be a better place to continue: with fixing the intro and figuring out what else should have a prominent role. (because as is the introduction seems to have no structure, coherence, and doesn’t give the reader a good idea of what an improper integral is; and most of what is there doesn’t help with that)
 * I’ll go ahead and work on that unless if you’re just going to revert everything again… —Preceding unsigned comment added by GromXXVII (talk • contribs) 23:00, 7 November 2007 (UTC)

 * I've created more sections. A standard way to work on articles is to take one section at a time and deal with its issues. Charles Matthews 07:15, 8 November 2007 (UTC)

A suggestion
The article is currently written so as to accomodate both the Lebesgue and Riemann integrals. This has the effect that it would be, I think, completely impenetrable to someone who is not familiar with the Lebesgue integral. Since the article begins "in calculus", I think this is undesirable.

At the same time, the article is arguably not general enough. Whoever wrote that the improper integral is "not a kind of integral" was not familiar with the Kurzweil-Henstock, or generalized Riemann integral. Among the virtues of this integral is that it integrates all Lebesgue integrable functions and improperly Riemann integrable functions "automatically". Coming back down to earth, it seems clear from linguistic use, interpretation in terms of area and so forth that the improper (Riemann) integral is "some kind" of integral, right??

I recommend that the body of the article be written explictly in terms of the Riemann integral. Here a key fact is that any Riemann integrable function on [a,b] is bounded, which explains why a new definition is needed when the function blows up at one of the endpoints. It seems more clear to discuss separately the case of an infinite interval and an unbounded function. More examples would also be nice.

There can then be a section discussing the difference between the improper Riemann integral, and the Lebesgue integral, and ideally a final section discussing the Kurzweil-Henstock integral.

Again, remember that this is an article that second semester calculus students will try to read. Even the brighest calculus students do not know measure theory... Plclark (talk) 11:35, 20 November 2007 (UTC)Plclark

sigh....
OK, I'm going to return to this article soon. Right now we find this sentence:
 * The Riemann integral as commonly defined in calculus texts is only defined for a continuous function $$f$$ on a closed and bounded interval $$[a,b]$$,
 * The Riemann integral as commonly defined in calculus texts is only defined for a continuous function $$f$$ on a closed and bounded interval $$[a,b]$$,

That is not true. What is true is that introductory calculus texts often consider only the case where the function is continuous. But the definition they state works for many discontinuous functions as well, including piecewise continuous functions of a sort often dealt with in calculus texts. I think it's appropriate to consider in the article the manner in which the concept is first introduced in calculus texts, but this continuity issue is really not the essence of the reason why there is such a concept as that of improper integrals. That's one of a number of reasons I've flagged this for attention. Michael Hardy (talk) 21:07, 4 June 2008 (UTC)

 * Actually, I changed the phrasing because I was prompted by your "Attention" flag. You are right that the statement is incorrect: it should say that the Riemann integral is commonly defined in calculus texts for continuous functions.  (Not that this definition only applies to continuous functions, as is suggested by the current wording.)  Unfortunately, there is no elementary characterization of Riemann integrable functions suitable for an introductory section.  A function is Riemann integrable iff it is bounded and continuous almost everywhere, but this is quite a bit to put in the first sentence of the motivation.   siℓℓy rabbit  (  talk  ) 00:36, 5 June 2008 (UTC)

Maybe it would be better to avoid any discussion of different definitions of integral (e.g. Riemann versus Lebesgue) in the early parts of the article. There seem to be two essential issues: (1) some integrals can't be defined except as limits as a bound approaches some limit, and (2) some integrals, even if they can be defined, cannot conveniently be computed except as limits as a bound approaches some limit. In either case, an improper integral is a limit as one or more of the bounds of integration approaches some limit. Michael Hardy (talk) 00:54, 5 June 2008 (UTC)

 * Yes, that sounds reasonable. In fact, it is probably easier to simply define an improper integral from the outset as the limit of an integral rather than to attempt to motivate it by considering unbounded functions/domains.  The motivation is currently more of a potential source of confusion than anything else.  The situations in which one would want to (or need to) use an improper integral should still be mentioned, but perhaps after a definition has been properly formulated.   siℓℓy rabbit  (  talk  ) 03:26, 5 June 2008 (UTC)

Much improved
The article looks enormously improved. Here are some further quick comments (because silly rabbit has worked so hard recently on the edits, I would rather suggest them than change them myself):

1) I am not wild about the "first kind / second kind" dichotomy. This is old-fashioned language that has been all but abandoned nowadays, probably because it is so opaque: speaking of an object of the Nth kind gives no clues as to what sort of X the author might have in mind.  Ask a working complex geometer what a differential of the third kind is, and if you are lucky, they will tell you, "Well, I think it's this, but maybe that's the second kind...I can't quite remember."

Also this dichotomy is too simple, because combinations of both types are possible.

Anyway, you don't seem to use the terminology in the remainder of the article.

2) Similarly, what you say in the first sentence as the definition of an improper integral is a little too simplified, because there could be multiple (maybe infinitely many) "improprieties." Admittedly it makes it quite challenging to summarize the situation in a single sentence that is not technically wrong; I think the only way to go here is to be a bit more vague.

3) You don't seem to say in the introduction that an improper integral [of a non-negative function...] is an area under a curve. Isn't this a good motivating intuition?

4) You speak of the (proper) Riemann integral "not converging" for unbounded functions. As far as I know, one instead says that the integral is "not defined" or the function is not "integrable."  (It would make perfectly good sense to say that it is not convergent, since a Riemann integral is a Moore-Smith limit, but people don't seem to say it...)

5) The issue of improper integrability on the real line is a pedagogical minefield; this is one of the few topics in first year calculus for which I longer even expect that my explanation will make sense to most students. At a slightly higher level though -- say that of undergraduate analysis -- I have found it helpful to introduce a Cauchy criterion for convergence on the real line: a function f: R -> R which is Riemann integrable on each closed bounded subinterval I of R is improperly Riemann integrable if: for every positive epsilon, there exists a closed bounded interval I such that for any closed bounded intervals J_1 and J_2 containing I, |int_{J_1} f - int_{J_2} f| < epsilon.  The point is that this definition looks (to a sufficiently sophisticated eye) natural rather than ad hoc.

6) The subject heading "Types of integrals" doesn't seem quite right to me, although I'm not sure exactly what alternative to suggest.

7) Riemann and Darboux integrals are equivalent, so why not just discuss them together, or even omit mention of Darboux entirely?

"The Lebesgue theory does not see this as a deficiency..." I don't think the theory of Lebesgue integration has an opinion on this, or on anything! Anyway, this sounds like propaganda: the important point (which you have made) is that there is no containment between the class of improperly Riemann integrable and the class of Lebesgue integrable functions. This is just a fact: it doesn't need to be "spun". (It reminds me of an algebraic topology class when my instructor said that one of the great merits of homology / homotopy groups is that they are invariants of the homotopy equivalence class of the space, suggesting that it would be somehow uncouth to want to distinguish between non-homeomorphic but homotopically equivalent spaces.)

Similarly, the bit about the Kurzweil-Henstock theory can be reworded to be less POV: in a purely mathematical sense that theory _is_ stronger than the other two theories, since it properly includes them.

8) Concerning summability: some reference to the (simpler and more widely known) theory of summability of series seems to be in order here.

Anyway, like I said, it is a good-looking article. Plclark (talk) 17:39, 19 June 2008 (UTC)Plclark

 * Thank you for the insightful feedback. These are all very good points, and definitely should be addressed.  Regards,  siℓℓy rabbit  (  talk  ) 11:30, 20 June 2008 (UTC)

weird identity
Why is the improper integral of f(x)dx the same as it is for f(x-(1/x))dx? Wouldn't this also mean it would still work for f(x-(2/x)), f(x-(3/x))... f(x-(x/x)) which equals f(x-1)? srn347 —Preceding unsigned comment added by 68.7.25.121 (talk) 21:00, 12 November 2008 (UTC)

In fact, this identity is true for f(x-c/x) where c is a positive real number. --27.32.122.196 (talk) 04:30, 10 January 2016 (UTC)

Identities
I removed the following because it does not improve the article:

 * Identities
 * $$\int_{-\infty}^{\infty} f(x) \, dx = \int_{-\infty}^{\infty} f(x - 1/x) \, dx$$

There also seems to be some consensus that the above "section" does not belong. Please discuss merits for inclusion here. 71.182.247.220 (talk) 11:17, 27 October 2009 (UTC)

 * I should also add that the above identity is true for suitable inputs ƒ, but in spite of being true it is not suitable for the article (WP:WEIGHT and WP:INDISCRIMINATE being the policies that codify this impression). This integral follows essentially from an elementary change of variables u = x − 1/x, while appropriately accounting for the branch of the inverse function.  So it really is nothing more than a "fun fact", of which Wikipedia is WP:NOT an indiscriminate collection. 71.182.247.220 (talk) 11:39, 27 October 2009 (UTC)

 * Actually, I can only rigorously prove this for "suitable inputs" consisting of absolutely integrable functions (Lebesgue integrable), or if the integral is taken in a Cauchy principal value sense. Since the whole point of improper integrals is to regularize non-integrable functions at infinity, this example is particularly unsuited for the article (especially given that no hypotheses are imposed on the function ƒ).  71.182.247.220 (talk) 12:08, 27 October 2009 (UTC)
 * Concerning this "whole point": I agree except for the part about "at infinity". It's not only at infinity, but a finite points as well. Michael Hardy (talk) 15:47, 29 October 2009 (UTC)
 * Agreed. But what is needed in the transformed integrand in this case is regularization at infinity, not at finite points, which was my point. 71.182.210.215 (talk) 23:36, 29 October 2009 (UTC)
 * 71.182.247.220 asked me to comment here since I added the identity a year or so ago. If I recall correctly I added it as a "fun fact" as 71 suggested, but I have no problems with it being removed. Ben (talk) 15:23, 29 October 2009 (UTC)

At one point I deleted the proposed identity from the article. It seemed to me that it had been put into a part of the article where it was given great prominence at the expense of essential facts that needed to be there, distracting attention from those. Michael Hardy (talk) 15:50, 29 October 2009 (UTC)

Multivariable improper integrals?
This page seems to be talking only about one variable improper integrals. A section on multivariable improper integrals, such as $$\iint_{\mathbb{R}^2} e^{x^2+y^2} \, d(x,y)$$ would be nice. 75.165.3.78 (talk) 17:35, 24 December 2009 (UTC)
 * Not sure, but I think the specific integral you mention doesn't exist as an improper integral. It looks to me like its Cauchy principal value doesn't exist either. Am I missing something?--75.80.43.80 (talk) 05:25, 9 December 2010 (UTC)
 * Oh... did you mean $$\iint_{\mathbb{R}^2} e^{-(x^2+y^2)} \, d(x,y)$$? In that case, just do it as an iterated integral. As long as you keep the limit calculations straight, it is the same definition as in the article.--75.80.43.80 (talk) 05:37, 9 December 2010 (UTC)
 * Oops! Just noticed this was posted LAST December. Reading the year helps. Oh well.--75.80.43.80 (talk) 05:39, 9 December 2010 (UTC)

 * I just created such section, though it should be much longer... Boris Tsirelson (talk) 20:52, 22 June 2015 (UTC)

 * Thanks to Sławomir Biały this section is now quite informative. Boris Tsirelson (talk) 05:44, 30 June 2015 (UTC)

 * "A function on an arbitrary domain A in $$\mathbb R^n$$ is extended to a function on $$\mathbb R^n$$ in a standard way by multiplying by the indicator function of A." — Yes, but Riemann integrability may be lost (even for a constant function) if A is not Jordan measurable; either require Jordan measurability (which is stronger than boundedness), or use exhaustion, or alternatively, take the supremum over Riemann integrable functions under the product by the indicator function. Boris Tsirelson (talk) 05:56, 30 June 2015 (UTC)

 * Perhaps I was unclear that this is how the Riemann integral is defined for arbitrary domains, by extending by zero outside the domain. In particular, in your example of constant functions, a non-zero constant is integrable over a domain if and only if the domain is Jordan measurable.  I will try to clarify this presently.   Sławomir Biały  (talk) 11:19, 30 June 2015 (UTC)

 * Ah, really? I did not know. I would prefer the more permissive definition... Boris Tsirelson (talk) 11:49, 30 June 2015 (UTC)

 * One can also define an integral, in an extended sense, by taking a partition of unity with rectangular supports, integrating over the supports, and summing. This at least works for open sets, or manifolds with smooth charts.  But it doesn't deal with arbitrary subsets of $$\mathbb R^n$$.  This doesn't seem to be a very popular approach for dealing with the Riemann integral (unless dealing with manifolds).  Presumably, once one gets to the point of invoking partitions of unity, it probably makes more sense to use the Lebesgue integral anyway, which renders such quibbles moot.   Sławomir Biały  (talk) 12:06, 30 June 2015 (UTC)

 * Also interesting. But I wonder, do you mean finite partitions of unity, or countable?
 * In fact, I teach these things (and there OR is not prohibited).
 * I believe that improper Riemann integration is still useful for (quite arbitrary) open sets, but beyond sets that are open (modulo sets of locally zero Jordan measure) it is indeed better to use Lebesgue integral. Boris Tsirelson (talk) 12:54, 30 June 2015 (UTC)

 * I am thinking of countable partitions of unity. There are several different notions of integral that are in play here.  One is the usual Riemann integral over a bounded set A.  This is defined by extending by zero off A.  The other is an extended integral, defined by partitions of unity (which we might call the "de Rham integral").  Then there are improper integrals, where the set A is allowed to be unbounded.  The ordinary Riemann integral is not equivalent to the de Rham integral since, as you pointed out, constant functions are de Rham integrable over any bounded open set, but may not be Riemann integrable.   I'm not sure what the "correct" perspective is for the article, but I've used the Riemann version, thinking that to be the most familiar.  But, as you say, improper integrals of the de Rham type can also be considered.   Sławomir Biały  (talk) 14:10, 30 June 2015 (UTC)

This recently overwrote what I wrote "In elementary calculus, the term improper integral is usually taken to mean either the limit of an integral as at least one of the two bounds of integration approaches either + ∞  {\displaystyle +\infty }  +\infty  or − ∞   {\displaystyle -\infty }  -\infty  a as at least one of the two bounds approaches a number at which the function being integrated has a vertical asymptote." This is mathematically incorrect. You don't need and asymptote at at infinity. You don't even need an asymptote. If I change it back someone will just over write it again. So it's useless to fight this. You need higher level thinking to solve this issue globally. Solution 1): Hire a math Ph.D. to verify valuable changes (expensive) 2) when users go to a math or physics or tech page ask them if it was useful and if not ask them to compare it to a previous version and rate both (I like this one) or 3) Use deep learning (machine learning) to compare your definition to similar pages from other sources.

Good luck. 1 love wikipedia — Preceding unsigned comment added by JimFoit (talk • contribs) 22:03, 23 September 2016 (UTC)

missing the point
I did this edit because whoever wrote that sentence seemed to miss the point. "Bad behavior at infinity" is not involved, and what is involved was not mentioned. Michael Hardy (talk) 04:59, 24 September 2016 (UTC)

Last sentence in the multivariable section
The present phrasing "In order to exist in this sense, the improper integral necessarily converges absolutely," obfuscates the nature of the implication, to wit: https://www.quora.com/Why-is-the-multivariable-improper-integral-usually-defined-only-when-it-converges-absolutely-Is-there-a-consistent-notion-of-conditionally-convergent-improper-integrals-in-mathbb-R-n?share=2252ca03&srid=8TIx. Is there some reason not to write this simply as: "Thus" (or "Note that") "if the improper integral exists in this sense, then it converges absolutely: convergence implies absolute convergence"? OlyDLG (talk) 19:43, 5 September 2018 (UTC)

 * Obfuscates? I do not feel so. As for me, it says just the same (as your version). Anyway, I do not object to such rewording. But anyway, rewording does not really answer the (nontrivial) question asked on quora: why such seemingly bad definition? Boris Tsirelson (talk) 08:28, 6 September 2018 (UTC)
 * A short answer is in the article, see quotes in the notes: "We need to make this stronger definition of convergence in terms of |f(x)| because cancellation in the integrals can occur in so many different ways in higher dimensions", and "In fact, for improper integrals of such functions, unconditional convergence turns out to be equivalent to absolute convergence". A detailed explanation is available in my course, Sect. 1b "Dimension 2 is a different story". Boris Tsirelson (talk) 08:34, 6 September 2018 (UTC)

Improper Henstock–Kurzweil integrals
Is the section on Types of integral correct to say: "For the Henstock–Kurzweil integral, improper integration is not necessary"?

This seems to be true for finite bounds but not infinite ones; or else it would contradict the Henstock–Kurzweil integral article which implicitly defines it on a finite interval $[a, b]$. The same article goes on to say "To study an improper Henstock–Kurzweil integral with finite bounds would not be meaningful. However, it does make sense to consider improper Henstock–Kurzweil integrals with infinite bounds". NeilOnWiki (talk) 17:07, 2 January 2022 (UTC)

title: Talk:Improper integral


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