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Nitrox

Nitrox refers to any gas mixture composed (excepting trace gases) of nitrogen and oxygen. This includes atmospheric air, which is approximately 78% nitrogen, 21% oxygen, and 1% other gases, primarily argon. In the usual application, underwater diving, nitrox is normally distinguished from air and handled differently. The most common use of nitrox mixtures containing oxygen in higher proportions than atmospheric air is in scuba diving, where the reduced partial pressure of nitrogen is advantageous in reducing nitrogen uptake in the body's tissues, thereby extending the practicable underwater dive time by reducing the decompression requirement, or reducing the risk of decompression sickness (also known as the bends).

Nitrox is used to a lesser extent in surface-supplied diving, as these advantages are reduced by the more complex logistical requirements for nitrox compared to the use of simple low-pressure compressors for breathing gas supply. Nitrox can also be used in hyperbaric treatment of decompression illness, usually at pressures where pure oxygen would be hazardous. Nitrox is not a safer gas than compressed air in all respects; although its use can reduce the risk of decompression sickness, it increases the risks of oxygen toxicity and fire.

Though not generally referred to as nitrox, an oxygen-enriched air mixture is routinely provided at normal surface ambient pressure as oxygen therapy to patients with compromised respiration and circulation.

Decompression benefits
Reducing the proportion of nitrogen by increasing the proportion of oxygen reduces the risk of decompression sickness for the same dive profile, or allows extended dive times without increasing the need for decompression stops for the same risk. The significant aspect of extended no-stop time when using nitrox mixtures is reduced risk in a situation where breathing gas supply is compromised, as the diver can make a direct ascent to the surface with an acceptably low risk of decompression sickness. The exact values of the extended no-stop times vary depending on the decompression model used to derive the tables, but as an approximation, it is based on the partial pressure of nitrogen at the dive depth. This principle can be used to calculate an equivalent air depth (EAD) with the same partial pressure of nitrogen as the mix to be used, and this depth is less than the actual dive depth for oxygen enriched mixtures. The equivalent air depth is used with air decompression tables to calculate decompression obligation and no-stop times. The Goldman decompression model predicts a significant risk reduction by using nitrox (more so than the PADI tables suggest).

Nitrogen narcosis
Controlled tests have not shown breathing nitrox to reduce the effects of nitrogen narcosis, as oxygen seems to have similarly narcotic properties under pressure to nitrogen; thus one should not expect a reduction in narcotic effects due only to the use of nitrox. Nonetheless, there are people in the diving community who insist that they feel reduced narcotic effects at depths breathing nitrox. This may be due to a dissociation of the subjective and behavioural effects of narcosis. Although oxygen appears chemically more narcotic at the surface, relative narcotic effects at depth have never been studied in detail, but it is known that different gases produce different narcotic effects as depth increases. Helium has no narcotic effect, but results in HPNS when breathed at high pressures, which does not happen with gases that have greater narcotic potency. However, because of risks associated with oxygen toxicity, divers do not usually use nitrox at greater depths where more pronounced narcosis symptoms are more likely to occur. For deep diving, trimix or heliox gases are typically used; these gases contain helium to reduce the amount of narcotic gases in the mixture.

Oxygen toxicity
Diving with and handling nitrox raise a number of potentially fatal dangers due to the high partial pressure of oxygen (ppO2). Nitrox is not a deep-diving gas mixture owing to the increased proportion of oxygen, which becomes toxic when breathed at high pressure. For example, the maximum operating depth of nitrox with 36% oxygen, a popular recreational diving mix, is 29 m to ensure a maximum ppO2 of no more than 1.4 bar. The exact value of the maximum allowed ppO2 and maximum operating depth varies depending on factors such as the training agency, the type of dive, the breathing equipment and the level of surface support, with professional divers sometimes being allowed to breathe higher ppO2 than those recommended to recreational divers.

To dive safely with nitrox, the diver must learn good buoyancy control, a vital part of scuba diving in its own right, and a disciplined approach to preparing, planning and executing a dive to ensure that the ppO2 is known, and the maximum operating depth is not exceeded. Many dive shops, dive operators, and gas blenders (individuals trained to blend gases) require the diver to present a nitrox certification card before selling nitrox to divers.

Some training agencies, such as PADI and Technical Diving International, teach the use of two depth limits to protect against oxygen toxicity. The shallower depth is called the "maximum operating depth" and is reached when the partial pressure of oxygen in the breathing gas reaches 1.4 bar. The deeper depth, called the "contingency depth", is reached when the partial pressure reaches 1.6 bar. Diving at or beyond this level exposes the diver to a greater risk of central nervous system (CNS) oxygen toxicity. This can be extremely dangerous since its onset is often without warning and can lead to drowning, as the regulator may be spat out during convulsions, which occur in conjunction with sudden unconsciousness (general seizure induced by oxygen toxicity).

Divers trained to use nitrox may memorise the acronym VENTID-C or sometimes ConVENTID, (which stands for Vision (blurriness), Ears (ringing sound), Nausea, Twitching, Irritability, Dizziness, and Convulsions). However, evidence from non-fatal oxygen convulsions indicates that most convulsions are not preceded by any warning symptoms at all. Further, many of the suggested warning signs are also symptoms of nitrogen narcosis, and so may lead to misdiagnosis by a diver. A solution to either is to ascend to a shallower depth.

Carbon dioxide retention
Use of nitrox may cause a reduced ventilatory response, and when breathing dense gas at the deeper limits of the usable range, this may result in carbon dioxide retention when exercise levels are high, with an increased risk of loss of consciousness.

Other effects
There is anecdotal evidence that the use of nitrox reduces post-dive fatigue, particularly in older and or obese divers; however a double-blind study to test this found no statistically significant reduction in reported fatigue. There was, however, some suggestion that post-dive fatigue is due to sub-clinical decompression sickness (DCS) (i.e. micro bubbles in the blood insufficient to cause symptoms of DCS); the fact that the study mentioned was conducted in a dry chamber with an ideal decompression profile may have been sufficient to reduce sub-clinical DCS and prevent fatigue in both nitrox and air divers. In 2008, a study was published using wet divers at the same depth no statistically significant reduction in reported fatigue was seen.

Further studies with a number of different dive profiles, and also different levels of exertion, would be necessary to fully investigate this issue. For example, there is much better scientific evidence that breathing high-oxygen gases increases exercise tolerance, during aerobic exertion. Though even moderate exertion while breathing from the regulator is a relatively uncommon occurrence in recreational scuba, as divers usually try to minimize it in order to conserve gas, episodes of exertion while regulator-breathing do occasionally occur in recreational diving. Examples are surface-swimming a distance to a boat or beach after surfacing, where residual "safety" cylinder gas is often used freely, since the remainder will be wasted anyway when the dive is completed, and unplanned contingencies due to currents or buoyancy problems. It is possible that these so-far un-studied situations have contributed to some of the positive reputation of nitrox.

A 2010 study using critical flicker fusion frequency and perceived fatigue criteria found that diver alertness after a dive on nitrox was significantly better than after an air dive.

Underwater diving
Enriched Air Nitrox, nitrox with an oxygen content above 21%, is mainly used in scuba diving to reduce the proportion of nitrogen in the breathing gas mixture. The main benefit is reduced decompression risk. To a considerably lesser extent it is also used in surface supplied diving, where the logistics are relatively complex, similar to the use of other diving gas mixtures like heliox and trimix.

Training and certification
Recreational nitrox certification (Nitrox diver) allows the diver to use a single nitrox gas mixture with 40% or less oxygen by volume on a dive without obligatory decompression. The reason for using nitrox on this type of dive profile can be to extend the no-decompression limit, and for shorter dives, to reduce the decompression stress. The course is short, with a theory module on the risks of oxygen toxicity and the calculation of maximum operating depth, and a practical module of generally two dives using nitrox. It is one of the most popular further training programmes for entry level divers as it makes longer dives possible at a large number of popular sites.

Advanced nitrox certification requires competence to carry two nitrox mixtures in separate scuba sets, and to use the richer mix for accelerated decompression at the end of the dive, switching gases underwater at the correct planned depth and selecting the new gas on the dive computer if one is carried. For the purposes of the certification any mixture from air to nominally 100% oxygen may be used, though at least one agency prefers to limit oxygen fraction to 80% as they consider this has a lower risk for oxygen toxicity.

Therapeutic recompression
Nitrox50 is used as one of the options in the first stages of therapeutic recompression using the Comex therapeutic table CX 30 for treatment of vestibular or general decompression sickness. Nitrox is breathed at 30 msw and 24 msw and the ascents from these depths to the next stop. At 18 m the gas is switched to oxygen for the rest of the treatment.

Medicine, mountaineering and unpressurised aircraft
The use of oxygen at high altitudes or as oxygen therapy may be as supplementary oxygen, added to the inspired air, which would technically be a use of nitrox, blended on site, but this is not normally referred to as such, as the gas provided for the purpose is oxygen.

Terminology
Nitrox is known by many names: Enriched Air Nitrox, Oxygen Enriched Air, Nitrox, EANx or Safe Air. Since the word is a compound contraction or coined word and not an acronym, it should not be written in all upper case characters as "NITROX", but may be initially capitalized when referring to specific mixtures such as Nitrox32, which contains 68% nitrogen and 32% oxygen. When one figure is stated, it refers to the oxygen percentage, not the nitrogen percentage. The original convention, Nitrox68/32 became shortened as the first figure is redundant.

The term "nitrox" was originally used to refer to the breathing gas in a seafloor habitat where the oxygen has to be kept to a lower fraction than in air to avoid long term oxygen toxicity problems. It was later used by Dr Morgan Wells of NOAA for mixtures with an oxygen fraction higher than air, and has become a generic term for binary mixtures of nitrogen and oxygen with any oxygen fraction, and in the context of recreational and technical diving, now usually refers to a mixture of nitrogen and oxygen with more than 21% oxygen. "Enriched Air Nitrox" or "EAN", and "Oxygen Enriched Air" are used to emphasize richer than air mixtures. In "EANx", the "x" was originally the x of nitrox, but has come to indicate the percentage of oxygen in the mix and is replaced by a number when the percentage is known; for example, a 40% oxygen mix is called EAN40. The two most popular blends are EAN32 and EAN36, developed by NOAA for scientific diving, and also named Nitrox I and Nitrox II, respectively, or Nitrox68/32 and Nitrox64/36. These two mixtures were first utilized to the depth and oxygen limits for scientific diving designated by NOAA at the time.

The term Oxygen Enriched Air (OEN) was accepted by the (American) scientific diving community, but although it is probably the most unambiguous and simply descriptive term yet proposed, it was resisted by the recreational diving community, sometimes in favour of less appropriate terminology.

In its early days of introduction to non-technical divers, nitrox has occasionally also been known by detractors by less complimentary terms, such as "devil gas" or "voodoo gas" (a term now sometimes used with pride).

American Nitrox Divers International (ANDI) uses the term "SafeAir", which they define as any oxygen-enriched air mixture with O2 concentrations between 22% and 50% that meet their gas quality and handling specifications, and specifically claim that these mixtures are safer than normally produced breathing air for the end user not envolved to the mix production which. Considering the complexities and hazards of mixing, handling, analyzing, and using oxygen-enriched air, this name is considered inappropriate by those who consider that it is not inherently "safe", but merely has decompression advantages.

The constituent gas percentages are what the gas blender aims for, but the final actual mix may vary from the specification, and so a small flow of gas from the cylinder must be measured with an oxygen analyzer, before the cylinder is used underwater.

MOD
Maximum Operating Depth (MOD) is the maximum safe depth at which a given nitrox mixture can be used. MOD depends on the allowed partial pressure of oxygen, which is related to exposure time and the acceptable risk assumed for central nervous system oxygen toxicity. Acceptable maximum ppO2 varies depending on the application: Higher values are used by commercial and military divers in special circumstances, often when the diver uses surface supplied breathing apparatus, or for treatment in a chamber, where the airway is relatively secure.
 * 1.2 is often used in closed circuit rebreathers.
 * 1.4 is recommended by several recreational training agencies for ordinary scuba diving.
 * 1.5 is allowed for commercial diving in some jurisdictions.
 * 1.6 is allowed for technical diving decompression stops, and is the recommended maximum according to NOAA

Choice of mixture
The two most common recreational diving nitrox mixes contain 32% and 36% oxygen, which have maximum operating depths (MODs) of 34 m and 29 m respectively when limited to a maximum partial pressure of oxygen of 1.4 bar. Divers may calculate an equivalent air depth to determine their decompression requirements or may use nitrox tables or a nitrox-capable dive computer.

Nitrox with more than 40% oxygen is uncommon within recreational diving. There are two main reasons for this: the first is that all pieces of diving equipment that come into contact with mixes containing higher proportions of oxygen, particularly at high pressure, need special cleaning and servicing to reduce the risk of fire. The second reason is that richer mixes extend the time the diver can stay underwater without needing decompression stops far further than the duration permitted by the capacity of typical diving cylinders. For example, based on the PADI nitrox recommendations, the maximum operating depth for EAN45 would be 21 m and the maximum dive time available at this depth even with EAN36 is nearly 1 hour 15 minutes: a diver with a breathing rate of 20 litres per minute using twin 10-litre, 230-bar (about double 85 cu. ft.) cylinders would have completely emptied the cylinders after 1 hour 14 minutes at this depth.

Use of nitrox mixtures containing 50% to 80% oxygen is common in technical diving as decompression gas, which by virtue of its lower partial pressure of inert gases such as nitrogen and helium, allows for more efficient (faster) elimination of these gases from the tissues than leaner oxygen mixtures.

In deep open circuit technical diving, where hypoxic gases are breathed during the bottom portion of the dive, a Nitrox mix with 50% or less oxygen called a "travel mix" is sometimes breathed during the beginning of the descent in order to avoid hypoxia. Normally, however, the most oxygen-lean of the diver's decompression gases would be used for this purpose, since descent time spent reaching a depth where bottom mix is no longer hypoxic is normally small, and the distance between this depth and the MOD of any nitrox decompression gas is likely to be very short, if it occurs at all.

Best mix
The composition of a nitrox mix can be optimized for a given planned dive profile. This is termed "Best mix", for the dive, and provides the maximum no-decompression time compatible with acceptable oxygen exposure. An acceptable maximum partial pressure of oxygen is selected based on depth and planned bottom time, and this value is used to calculate the oxygen content of the best mix for the dive:

 * $$f_{\text{O}_2,\text{max}} = \frac{p_{\text{O}_2,\text{max}}}{p} = \frac{\text{Maximum acceptable partial pressure of oxygen}}{\text{Maximum ambient pressure of the dive}}$$

Production
There are several methods of production: A limited range of mixes is possible, but the equipment is quick and easy to operate and relatively safe, as there is never high partial pressure oxygen involved. A supply of clean low-pressure air at a constant temperature is required for consistent results. This may be supplied from a low pressure compressor or a regulated supply from high pressure storage or compressor. The air must be free of contaminants that could clog the membrane, and at a constant inlet temperature and pressure to produce a consistent delivered partial pressure of oxygen. The air must be of breathing quality, other contaminants must be filtered out independently. The input air pressure is regulated and pressure over the membrane controlled to adjust the product oxygen fraction. CGA Grade D or E air quality is suitable for supply gas, and is commonly heated to a constant inlet temperature. Heating also reduces the chance of high humidity causing wetting of the membrane. In a typical system supply air enters the thousands of hollow fibres of the membrane at one end, and oxygen preferentially permeates the fibre walls, leaving mostly nitrogen at the discharge end, which is vented from the system as waste.
 * Mixing by partial pressure: a measured pressure of oxygen is decanted into the cylinder and cylinder is "topped up" with air from the diving air compressor. This method is very versatile and requires relatively little additional equipment if a suitable compressor is available, but it is labour-intensive, and high partial pressures of oxygen are relatively hazardous.
 * Pre-mix decanting: the gas supplier provides large cylinders with popular mixes such as 32% and 36%. These may be further diluted with air to provide a larger range of mixtures.
 * Mixing by continuous blending: measured quantities of oxygen are introduced to air and mixed with it before it reaches the compressor inlet. Concentration of oxygen is commonly monitored as partial pressure using an oxygen cell. The compressor and particularly the compressor oil, must be suitable for this service. If the resulting oxygen fraction is less than 40%, the cylinder and valve may not be required to be cleaned for oxygen service. Relatively efficient and quick compared to partial pressure blending, but requires a suitable compressor, and the range of mixes may be limited by the compressor specification.
 * Mixing by mass fraction: oxygen and air or nitrogen are added to a cylinder that is accurately weighed until the required mix is achieved. This method requires fairly large and highly accurate scales, otherwise it is similar to partial pressure blending, but insensitive to temperature variations.
 * Mixing by gas separation: a nitrogen permeable membrane is used to remove some of the nitrogen molecules from air until the required mix is achieved. The resulting low pressure nitrox is then pumped into cylinders by a compressor.
 * Pressure swing adsorption requires relatively complex equipment, otherwise the advantages are similar to membrane separation. PSA is a technology used to separate gases from a mixture under pressure according to the molecular characteristics and affinity for an adsorbent material of the gases at near-ambient temperatures. Specific adsorbent materials are used as a trap, preferentially adsorbing the target gases at high pressure. The process then swings to low pressure to desorb the adsorbed material and flush the adsorbent container so that it can be reused.

Cylinder markings to identify contents
Any diving cylinder containing a blend of gasses other than standard air is required by most diver training organizations, and some national governments, to be clearly marked to indicate the current gas mixture. In practice it is common to use a printed adhesive label to indicate the type of gas (in this case nitrox), and to add a temporary label to specify the analysis of the current mix.

Training standards for nitrox certification suggest the composition must be verified by the diver by using an oxygen analyzer before use.

European Union
Within the EU, valves with M26x2 outlet thread are recommended for cylinders with increased oxygen content. Regulators for use with these cylinders require compatible connectors, and are not directly connectable with cylinders for compressed air.

Germany
A nitrox cylinder is specially cleaned and identified. According to EN 144-3 the cylinder colour is overall white with the letter N on opposite sides of the cylinder. The fraction of oxygen in the bottle is checked after filling and marked on the cylinder.

South Africa
South African National Standard 10019:2008 specifies the colour of all scuba cylinders as Golden yellow with French gray shoulder. This applies to all underwater breathing gases except medical oxygen, which must be carried in cylinders that are Black with a White shoulder. Nitrox cylinders must be identified by a transparent, self-adhesive label with green lettering, fitted below the shoulder. In effect this is green lettering on a yellow cylinder, with a gray shoulder. The composition of the gas must also be specified on the label. In practice this is done by a small additional self-adhesive label marked with the measured oxygen fraction, which is changed when a new mix is filled.

The 2021 revision of SANS 10019 changed the colour specification to Light navy grey for the shoulder, and a different label specification which includes hazard symbols for high pressure and oxidising materials.

United States
Every nitrox cylinder should also have a sticker stating whether or not the cylinder is oxygen clean and suitable for partial pressure blending. Any oxygen-clean cylinder may have any mix up to 100% oxygen inside. If by some accident an oxygen-clean cylinder is filled at a station that does not supply gas to oxygen-clean standards it is then considered contaminated and must be re-cleaned before a gas containing more than 40% oxygen may again be added. Cylinders marked as 'not oxygen clean' may only be filled with oxygen-enriched air mixtures from membrane or stick blending systems where the gas is mixed before being added to the cylinder, and to an oxygen fraction not exceeding 40% by volume.

Hazards
Nitrox can be a hazard to the blender and to the user, for different reasons.

Fire and toxic cylinder contamination from oxygen reactions
Partial pressure blending using pure oxygen decanted into the cylinder before topping up with air may involve very high oxygen fractions and oxygen partial pressures during the decanting process, which constitute a relatively high fire hazard. This procedure requires care and precautions by the operator, and decanting equipment and cylinders which are clean for oxygen service, but the equipment is relatively simple and inexpensive. Partial pressure blending using pure oxygen is often used to provide nitrox on live-aboard dive boats, but it is also used in some dive shops and clubs.

Any gas which contains a significantly larger percentage of oxygen than air is a fire hazard, and such gases can react with hydrocarbons or lubricants and sealing materials inside the filling system to produce toxic gases, even if a fire is not apparent. Some organisations exempt equipment from oxygen-clean standards if the oxygen fraction is limited to 40% or less.

Among recreational training agencies, only ANDI subscribes to the guideline of requiring oxygen cleaning for equipment used with more than 23% oxygen fraction. The USCG, NOAA, U.S. Navy, OSHA, and the other recreational training agencies accept the limit as 40% as no accident or incident has been known to occur when this guideline has been properly applied. Tens of thousands of recreational divers are trained each year and the overwhelming majority of these divers are taught the "over 40% rule". Most nitrox fill stations which supply pre-mixed nitrox will fill cylinders with mixtures below 40% without certification of cleanliness for oxygen service. Luxfer cylinders specify oxygen cleaning for all mixtures exceeding 23.5% oxygen.

The following references for oxygen cleaning specifically cite the "over 40%" guideline that has been in widespread use since the 1960s, and consensus at the 1992 Enriched Air Workshop was to accept that guideline and continue the status quo.
 * Code of Federal Regulations, Part 1910.430 (i) – Commercial Diving Operations
 * OSHA Oxygen Specifications 1910.420 (1)
 * NOAA Oxygen Specifications (appendix D)
 * U.S. Navy Oxygen Specifications U.S. MIL-STD-777E (SH) Note K-6-4, Cat. K.6
 * U.S. Coast Guard Oxygen Specifications Title 46: Shipping, revisions through 10-1-92. 197.452 Oxygen Cleaning 46 CFR 197.451

Much of the confusion appears to be a result of misapplying PVHO (pressure vessel for human occupancy) guidelines which prescribe a maximum ambient oxygen content of 25% when a human is sealed into a pressure vessel (chamber). The concern here is for a fire hazard to a living person who could be trapped in an oxygen-rich burning environment.

Of the three commonly applied methods of producing enriched air mixes – continuous blending, partial pressure blending, and membrane separation systems – only partial pressure blending would require the valve and cylinder components to be oxygen cleaned for mixtures with less than 40% oxygen. The other two methods ensure that the equipment is never subjected to greater than 40% oxygen content.

In a fire, the pressure in a gas cylinder rises in direct proportion to its absolute temperature. If the internal pressure exceeds the mechanical limitations of the cylinder and there are no means to safely vent the pressurized gas to the atmosphere, the vessel will fail mechanically. If the vessel contents are ignitable or a contaminant is present this event may result in a "fireball".

Incorrect gas mix
Use of a gas mix that differs from the planned mix introduces an increased risk of decompression sickness or an increased risk of oxygen toxicity, depending on the error. It may be possible to simply recalculate the dive plan or set the dive computer accordingly, but in some cases the planned dive may not be practicable.

Many training agencies such as PADI, CMAS, SSI and NAUI train their divers to personally check the oxygen percentage content of each nitrox cylinder before every dive. If the oxygen percentage deviates by more than 1% from the planned mix, the diver must either recalculate the dive plan with the actual mix, or else abort the dive to avoid increased risk of oxygen toxicity or decompression sickness. Under IANTD and ANDI rules for use of nitrox, which are followed by dive resorts around the world, filled nitrox cylinders are signed out personally in a blended gas records book, which contains, for each cylinder and fill, the cylinder number, the measured oxygen fraction by percentage, the calculated maximum operating depth for that mix, and the signature of the receiving diver, who should have personally measured the oxygen fraction before taking delivery. All of these steps reduce risk but increase complexity of operations as each diver must use the specific cylinder they have checked out. In South Africa, the national standard for handling and filling portable cylinders with pressurised gases (SANS 10019) requires that the cylinder be labelled with a sticker identifying the contents as nitrox, and specifying the oxygen fraction. Similar requirements may apply in other countries.

History
In 1874, Henry Fleuss made what was possibly the first Nitrox dive using a rebreather.

In 1911, Draeger of Germany tested an injector operated rebreather backpack for a standard diving suit. This concept was produced and marketed as the DM20 oxygen rebreather system and the DM40 nitrox rebreather system, in which air from one cylinder and oxygen from a second cylinder were mixed during injection through a nozzle which circulated the breathing gas through the scrubber and the rest of the loop. The DM40 was rated for depths up to 40m.

Christian J. Lambertsen proposed calculations for nitrogen addition to prevent oxygen toxicity in divers utilizing nitrogen–oxygen rebreather diving.

In World War II or soon after, British commando frogmen and clearance divers started occasionally diving with oxygen rebreathers adapted for semi-closed-circuit nitrox (which they called "mixture") diving by fitting larger cylinders and carefully setting the gas flow rate using a flow meter. These developments were kept secret until independently duplicated by civilians in the 1960s.

Lambertson published a paper on nitrox in 1947.

In the 1950s, the United States Navy (USN) documented enriched oxygen gas procedures for military use of what we today call nitrox, in the US Navy Diving Manual.

In 1955, E. Lanphier described the use of nitrogen–oxygen diving mixtures, and the equivalent air depth method for calculating decompression from air tables.

In the 1960s, A. Galerne used on-line blending for commercial diving.

In 1970, Morgan Wells, who was the first director of the National Oceanographic and Atmospheric Administration (NOAA) Diving Center, began instituting diving procedures for oxygen-enriched air. He introduced the concept of Equivalent Air Depth (EAD). He also developed a process for mixing oxygen and air which he called a continuous blending system. For many years Wells' invention was the only practical alternative to partial pressure blending. In 1979 NOAA published Wells' procedures for the scientific use of nitrox in the NOAA Diving Manual.

In 1985 Dick Rutkowski, a former NOAA diving safety officer, formed IAND (International Association of Nitrox Divers) and began teaching nitrox use for recreational diving. This was considered dangerous by some, and met with heavy skepticism by the diving community.

In 1989, the Harbor Branch Oceanographic institution workshop addressed blending, oxygen limits and decompression issues.

In 1991, Bove, Bennett and Skin Diver magazine took a stand against nitrox use for recreational diving. Skin Diver editor Bill Gleason dubbed nitrox the "Voodoo Gas". The annual DEMA show (held in Houston, Texas that year) banned nitrox training providers from the show. This caused a backlash, and when DEMA relented, a number of organizations took the opportunity to present nitrox workshops outside the show.

In 1992, the Scuba Diving Resources Group organised a workshop where some guidelines were established, and some misconceptions addressed.

In 1992, BSAC banned its members from using nitrox during BSAC activities. IAND's name was changed to the International Association of Nitrox and Technical Divers (IANTD), the T being added when the European Association of Technical Divers (EATD) merged with IAND. In the early 1990s, these agencies were teaching nitrox, but the main scuba agencies were not. Additional new organizations, including the American Nitrox Divers International (ANDI) – which invented the term "Safe Air" for marketing purposes – and Technical Diving International (TDI) were begun. NAUI became the first existing major recreational diver training agency to sanction nitrox.

In 1993, the Sub-Aqua Association was the first UK recreational diving training agency to acknowledge and endorse the Nitrox training their members had undertaken with one of the tech agencies. The SAA's first recreational Nitrox qualification was issued in April 1993. The SAA's first Nitrox instructor was Vic Bonfante and he was certified in September 1993.

Meanwhile, diving stores were finding a purely economic reason to offer nitrox: not only was an entire new course and certification needed to use it, but instead of cheap or free tank fills with compressed air, dive shops found they could charge premium amounts of money for custom-gas blending of nitrox to their ordinary, moderately experienced divers. With the new dive computers which could be programmed to allow for the longer bottom-times and shorter residual nitrogen times that nitrox gave, the incentive for the sport diver to use the gas increased.

In 1993, Skin Diver magazine, the leading recreational diving publication at the time, published a three-part series arguing that nitrox was unsafe for sport divers. DiveRite manufactured the first nitrox-compatible dive computer, called the Bridge, the aquaCorps TEK93 conference was held in San Francisco, and a practicable oil limit of 0.1 mg/m3 for oxygen compatible air was set. The Canadian armed forces issued EAD tables with an upper PO2 of 1.5 ATA.

In 1994, John Lamb and Vandagraph launched the first oxygen analyser built specifically for Nitrox and mixed-gas divers, at the Birmingham Dive Show.

In 1994, BSAC reversed its policy on Nitrox and announced BSAC nitrox training to start in 1995.

In 1996, the Professional Association of Diving Instructors (PADI) announced full educational support for nitrox. While other mainline scuba organizations had announced their support of nitrox earlier, it was PADI's endorsement that established nitrox as a standard recreational diving option.

In 1997, ProTec started with Nitrox 1 (recreational) and Nitrox 2 (technical). A German ProTec Nitrox manual (ref to the 6th edition) has been published.

In 1999, a survey by R.W. Hamilton showed that over hundreds of thousands of nitrox dives, the DCS record is good. Nitrox had become popular with recreational divers, but not used much by commercial divers who tend to use surface supplied breathing apparatus. The OSHA accepted a petition for a variance from the commercial diving regulations for recreational scuba instructors.

The 2001 edition of the NOAA Diving Manual included a chapter intended for Nitrox training.

In nature
At times in the geological past, the Earth's atmosphere contained much more than 20% oxygen: e.g. up to 35% in the Upper Carboniferous period. This let animals absorb oxygen more easily and influenced their evolutionary patterns.

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title: Nitrox


text:
Erik Satie

Eric Alfred Leslie Satie (17 May 1866 – 1 July 1925), who signed his name Erik Satie after 1884, was a French composer and pianist. He was the son of a French father and a British mother. He studied at the Paris Conservatoire, but was an undistinguished student and obtained no diploma. In the 1880s he worked as a pianist in café-cabaret in Montmartre, Paris, and began composing works, mostly for solo piano, such as his Gymnopédies and Gnossiennes. He also wrote music for a Rosicrucian sect to which he was briefly attached.

After a spell in which he composed little, Satie entered Paris's second music academy, the Schola Cantorum, as a mature student. His studies there were more successful than those at the Conservatoire. From about 1910 he became the focus of successive groups of young composers attracted by his unconventionality and originality. Among them were the group known as Les Six. A meeting with Jean Cocteau in 1915 led to the creation of the ballet Parade (1917) for Serge Diaghilev, with music by Satie, sets and costumes by Pablo Picasso, and choreography by Léonide Massine.

Satie's example guided a new generation of French composers away from post-Wagnerian impressionism towards a sparer, terser style. Among those influenced by him during his lifetime were Maurice Ravel, Claude Debussy, and Francis Poulenc, and he is seen as an influence on more recent composers such as John Cage and John Adams. His harmony is often characterised by unresolved chords, he sometimes dispensed with bar-lines, as in his Gnossiennes, and his melodies are generally simple and often reflect his love of old church music. He gave some of his later works absurd titles, such as Veritables Preludes flasques (pour un chien) ("True Flabby Preludes (for a Dog)", 1912), Croquis et agaceries d'un gros bonhomme en bois ("Sketches and Exasperations of a Big Wooden Man", 1913) and Sonatine bureaucratique ("Bureaucratic Sonatina", 1917). Most of his works are brief, and the majority are for solo piano. Exceptions include his "symphonic drama" Socrate (1919) and two late ballets Mercure and Relâche (1924).

Satie never married, and his home for most of his adult life was a single small room, first in Montmartre and, from 1898 to his death, in Arcueil, a suburb of Paris. He adopted various images over the years, including a period in quasi-priestly dress, another in which he always wore identically coloured velvet suits, and is known for his last persona, in neat bourgeois costume, with bowler hat, wing collar, and umbrella. He was a lifelong heavy drinker, and died of cirrhosis of the liver at the age of 59.

Early years
Satie was born on 17 May 1866 in Honfleur, Normandy, the first child of Alfred Satie and his wife Jane Leslie (née Anton). Jane Satie was an English Protestant of Scottish descent; Alfred Satie, a shipping broker, was a Roman Catholic anglophobe. A year later, the Saties had a daughter, Olga, and in 1869 a second son, Conrad. The children were baptised in the Anglican church.

After the Franco-Prussian War, Alfred Satie sold his business and the family moved to Paris, where he eventually set up as a music publisher. In 1872 Jane Satie died and Eric and his brother were sent back to Honfleur to be brought up by Alfred's parents. The boys were rebaptised as Roman Catholics and educated at a local boarding school, where Satie excelled in history and Latin but nothing else. In 1874 he began taking music lessons with a local organist, Gustave Vinot, a former pupil of Louis Niedermeyer. Vinot stimulated Satie's love of old church music, and in particular Gregorian chant.

In 1878 Satie's grandmother died, and the two boys returned to Paris to be informally educated by their father. Satie did not attend a school, but his father took him to lectures at the Collège de France and engaged a tutor to teach Eric Latin and Greek. Before the boys returned to Paris from Honfleur, Alfred had met a piano teacher and salon composer, Eugénie Barnetche, whom he married in January 1879, to the dismay of the twelve-year-old Satie, who did not like her.

Eugénie Satie resolved that her elder stepson should become a professional musician, and in November 1879 enrolled him in the preparatory piano class at the Paris Conservatoire. Satie strongly disliked the Conservatoire, which he described as "a vast, very uncomfortable, and rather ugly building; a sort of district prison with no beauty on the inside – nor on the outside, for that matter". He studied solfeggio with Albert Lavignac and piano with Émile Decombes, who had been a pupil of Frédéric Chopin. In 1880 Satie took his first examinations as a pianist: he was described as "gifted but indolent". The following year Decombes called him "the laziest student in the Conservatoire". In 1882 he was expelled from the Conservatoire for his unsatisfactory performance. In 1884 Satie wrote his first known composition, a short Allegro for piano, written while on holiday in Honfleur. He signed himself "Erik" on this and subsequent compositions, though continuing to use "Eric" on other documents until 1906. In 1885, he was readmitted to the Conservatoire, in the intermediate piano class of his stepmother's former teacher, Georges Mathias. He made little progress: Mathias described his playing as "Insignificant and laborious" and Satie himself "Worthless. Three months just to learn the piece. Cannot sight-read properly". Satie became fascinated by aspects of religion. He spent much time in Notre-Dame de Paris contemplating the stained glass windows and in the National Library examining obscure medieval manuscripts. His friend Alphonse Allais later dubbed him "Esotérik Satie". From this period comes Ogives, a set of four piano pieces inspired by Gregorian chant and Gothic church architecture.

Keen to leave the Conservatoire, Satie volunteered for military service, and joined the 33rd Infantry Regiment in November 1886. He quickly found army life no more to his liking than the Conservatoire, and deliberately contracted acute bronchitis by standing in the open, bare-chested, on a winter night. After three months' convalescence he was invalided out of the army.

Montmartre
In 1887, at the age of 21, Satie moved from his father's residence to lodgings in the 9th arrondissement. By this time he had started what was to be an enduring friendship with the romantic poet Contamine de Latour, whose verse he set in some of his early compositions, which Satie senior published. His lodgings were close to the popular Chat Noir cabaret on the southern edge of Montmartre where he became an habitué and then a resident pianist. The Chat Noir was known as the "temple de la 'convention farfelue'" – the temple of zany convention, and as the biographer Robert Orledge puts it, Satie, "free from his restrictive upbringing … enthusiastically embraced the reckless bohemian lifestyle and created for himself a new persona as a long-haired man-about-town in frock coat and top hat". This was the first of several personas that Satie invented for himself over the years. In the late 1880s Satie styled himself on at least one occasion "Erik Satie – gymnopédiste", and his works from this period include the three Gymnopédies (1888) and the first Gnossiennes (1889 and 1890). He earned a modest living as pianist and conductor at the Chat Noir, before falling out with the proprietor and moving to become second pianist at the nearby Auberge du Clou. There he became a close friend of Claude Debussy, who proved a kindred spirit in his experimental approach to composition. Both were bohemians, enjoying the same café society and struggling to survive financially. At the Auberge du Clou Satie first encountered the flamboyant, self-styled "Sâr" Joséphin Péladan, for whose mystic sect, the Ordre de la Rose-Croix Catholique du Temple et du Graal, he was appointed composer. This gave him scope for experiment, and Péladan's salons at the fashionable Galerie Durand-Ruel gained Satie his first public hearings. Frequently short of money, Satie moved from his lodgings in the 9th arrondissement to a small room in the rue Cortot not far from Sacre-Coeur, so high up the Butte Montmartre that he said he could see from his window all the way to the Belgian border.

By mid-1892, Satie had composed the first pieces in a compositional system of his own making (Fête donnée par des Chevaliers Normands en l'honneur d'une jeune demoiselle), provided incidental music to a chivalric esoteric play (two Préludes du Nazaréen), had a hoax published (announcing the premiere of his non-existent Le bâtard de Tristan, an anti-Wagnerian opera), and broken away from Péladan, starting that autumn with the "Uspud" project, a "Christian Ballet", in collaboration with Latour. He challenged the musical establishment by proposing himself – unsuccessfully – for the seat in the Académie des Beaux-Arts made vacant by the death of Ernest Guiraud. Between 1893 and 1895, Satie, affecting a quasi-priestly dress, was the founder and only member of the Eglise Métropolitaine d'Art de Jésus Conducteur. From his "Abbatiale" in the rue Cortot, he published scathing attacks on his artistic enemies. In 1893 Satie had what is believed to be his only love affair, a five-month liaison with the painter Suzanne Valadon. After their first night together, he proposed marriage. The two did not marry, but Valadon moved to a room next to Satie's at the rue Cortot. Satie became obsessed with her, calling her his Biqui and writing impassioned notes about "her whole being, lovely eyes, gentle hands, and tiny feet". During their relationship Satie composed the Danses gothiques as a means of calming his mind, and Valadon painted his portrait, which she gave him. After five months she moved away, leaving him devastated. He said later that he was left with "nothing but an icy loneliness that fills the head with emptiness and the heart with sadness".

In 1895 Satie attempted to change his image once again: this time to that of "the Velvet Gentleman". From the proceeds of a small legacy he bought seven identical dun-coloured suits. Orledge comments that this change "marked the end of his Rose+Croix period and the start of a long search for a new artistic direction".

Move to Arcueil
In 1898, in search of somewhere cheaper and quieter than Montmartre, Satie moved to a room in the southern suburbs, in the commune of Arcueil-Cachan, eight kilometres (five miles) from the centre of Paris. This remained his home for the rest of his life. No visitors were ever admitted. He joined a radical socialist party (he later switched his membership to the Communist Party), but adopted a thoroughly bourgeois image: the biographer Pierre-Daniel Templier, writes, "With his umbrella and bowler hat, he resembled a quiet school teacher. Although a Bohemian, he looked very dignified, almost ceremonious".

Satie earned a living as a cabaret pianist, adapting more than a hundred compositions of popular music for piano or piano and voice, adding some of his own. The most popular of these were Je te veux, text by Henry Pacory; Tendrement, text by Vincent Hyspa; Poudre d'or, a waltz; La Diva de l'Empire, text by Dominique Bonnaud/Numa Blès; Le Picadilly, a march; Légende californienne, text by Contamine de Latour (lost, but the music later reappears in La belle excentrique); and others. In his later years Satie rejected all his cabaret music as vile and against his nature. Only a few compositions that he took seriously remain from this period: Jack in the Box, music to a pantomime by Jules Depaquit (called a "clownerie" by Satie); Geneviève de Brabant, a short comic opera to a text by "Lord Cheminot" (Latour); Le poisson rêveur (The Dreamy Fish), piano music to accompany a lost tale by Cheminot, and a few others that were mostly incomplete. Few were presented, and none published at the time.

A decisive change in Satie's musical outlook came after he heard the premiere of Debussy's opera Pelléas et Mélisande in 1902. He found it "absolutely astounding", and he re-evaluated his own music. In a determined attempt to improve his technique, and against Debussy's advice, he enrolled as a mature student at Paris's second main music academy, the Schola Cantorum in October 1905, continuing his studies there until 1912. The institution was run by Vincent d'Indy, who emphasised orthodox technique rather than creative originality. Satie studied counterpoint with Albert Roussel and composition with d'Indy, and was a much more conscientious and successful student than he had been at the Conservatoire in his youth.

It was not until 1911, when he was in his mid-forties, that Satie came to the notice of the musical public in general. In January of that year Maurice Ravel played some early Satie works at a concert by the Société musicale indépendante, a forward-looking group set up by Ravel and others as a rival to the conservative Société nationale de musique. Satie was suddenly seen as "the precursor and apostle of the musical revolution now taking place"; he became a focus for young composers. Debussy, having orchestrated the first and third Gymnopédies, conducted them in concert. The publisher Demets asked for new works from Satie, who was finally able to give up his cabaret work and devote himself to composition. Works such as the cycle Sports et divertissements (1914) were published in de luxe editions. The press began to write about Satie's music, and a leading pianist, Ricardo Viñes, took him up, giving celebrated first performances of some Satie pieces.

Last years
Satie became the focus of successive groups of young composers, whom he first encouraged and then distanced himself from, sometimes rancorously, when their popularity threatened to eclipse his or they otherwise displeased him. First were the "jeunes" – those associated with Ravel – and then a group known at first as the "nouveaux jeunes", later called Les Six, including Georges Auric, Louis Durey, Arthur Honegger, and Germaine Tailleferre, joined later by Francis Poulenc and Darius Milhaud. Satie dissociated himself from the second group in 1918, and in the 1920s he became the focal point of another set of young composers including Henri Cliquet-Pleyel, Roger Désormière, Maxime Jacob and Henri Sauguet, who became known as the "Arcueil School". In addition to turning against Ravel, Auric and Poulenc in particular, Satie quarrelled with his old friend Debussy in 1917, resentful of the latter's failure to appreciate the more recent Satie compositions. The rupture lasted for the remaining months of Debussy's life, and when he died the following year, Satie refused to attend the funeral. A few of his protégés escaped his displeasure, and Milhaud and Désormière were among those who remained friends with him to the last. The First World War restricted concert-giving to some extent, but Orledge comments that the war years brought "Satie's second lucky break", when Jean Cocteau heard Viñes and Satie perform the Trois morceaux in 1916. This led to the commissioning of the ballet Parade, premiered in 1917 by Sergei Diaghilev's Ballets Russes, with music by Satie, sets and costumes by Pablo Picasso, and choreography by Léonide Massine. This was a succès de scandale, with jazz rhythms and instrumentation including parts for typewriter, steamship whistle and siren. It firmly established Satie's name before the public, and thereafter his career centred on the theatre, writing mainly to commission.

In October 1916 Satie received a commission from the Princesse de Polignac that resulted in what Orledge rates as the composer's masterpiece, Socrate, two years later. Satie set translations from Plato's Dialogues as a "symphonic drama". Its composition was interrupted in 1917 by a libel suit brought against him by a music critic, Jean Poueigh, which nearly resulted in a jail sentence for Satie. When Socrate was premiered, Satie called it "a return to classical simplicity with a modern sensibility", and among those who admired the work was Igor Stravinsky, a composer whom Satie regarded with awe.

In his later years Satie became known for his prose. He was in demand as a journalist, making contributions to the Revue musicale, Action, L'Esprit nouveau, the Paris-Journal and other publications from the Dadaist 391 to the English-language magazines Vanity Fair and The Transatlantic Review. As he contributed anonymously or under pen names to some publications it is not certain how many titles he wrote for, but Grove's Dictionary of Music and Musicians lists 25. Satie's habit of embellishing the scores of his compositions with all kinds of written remarks became so established that he had to insist that they must not be read out during performances.

In 1920 there was a festival of Satie's music at the Salle Erard in Paris. In 1924 the ballets Mercure (with choreography by Massine and décor by Picasso) and Relâche ("Cancelled") (in collaboration with Francis Picabia and René Clair), both provoked headlines with their first night scandals. Despite being a musical iconoclast, and encourager of modernism, Satie was uninterested to the point of antipathy about innovations such as the telephone, the gramophone and the radio. He made no recordings, and as far as is known heard only a single radio broadcast (of Milhaud's music) and made only one telephone call. Although his personal appearance was customarily immaculate, his room at Arcueil was in Orledge's word "squalid", and after his death the scores of several important works believed lost were found among the accumulated rubbish. He was incompetent with money. Having depended to a considerable extent on the generosity of friends in his early years, he was little better off when he began to earn a good income from his compositions, as he spent or gave away money as soon as he received it. He liked children, and they liked him, but his relations with adults were seldom straightforward. One of his last collaborators, Picabia, said of him: "Satie's case is extraordinary. He's a mischievous and cunning old artist. At least, that's how he thinks of himself. Myself, I think the opposite! He's a very susceptible man, arrogant, a real sad child, but one who is sometimes made optimistic by alcohol. But he's a good friend, and I like him a lot." Throughout his adult life Satie was a heavy drinker, and in 1925 his health collapsed. He was taken to the Hôpital Saint-Joseph in Paris, diagnosed with cirrhosis of the liver. He died there at 8.00 p.m. on 1 July, at the age of 59. He was buried in the cemetery at Arcueil.

Music
In the view of the Oxford Dictionary of Music, Satie's importance lay in "directing a new generation of French composers away from Wagner‐influenced impressionism towards a leaner, more epigrammatic style". Debussy christened him "the precursor" because of his early harmonic innovations. Satie summed up his musical philosophy in 1917: "To have a feeling for harmony is to have a feeling for tonality… the melody is the Idea, the outline; as much as it is the form and the subject matter of a work. The harmony is an illumination, an exhibition of the object, its reflection." Among his earliest compositions were sets of three Gymnopédies (1888) and his Gnossiennes (1889 onwards) for piano. They evoke the ancient world by what the critics Roger Nichols and Paul Griffiths describe as "pure simplicity, monotonous repetition, and highly original modal harmonies". It is possible that their simplicity and originality were influenced by Debussy; it is also possible that it was Satie who influenced Debussy. During the brief spell when Satie was composer to Péladan's sect he adopted a similarly austere manner.

While Satie was earning his living as a café pianist in Montmartre he contributed songs and little waltzes. After moving to Arcueil he began to write works with quirky titles, such as the seven-movement suite Trois morceaux en forme de poire ("Three Pear-shaped Pieces") for piano four-hands (1903), simply-phrased music that Nichols and Griffiths describe as "a résumé of his music since 1890" – reusing some of his earlier work as well as popular songs of the time. He struggled to find his own musical voice. Orledge writes that this was partly because of his "trying to ape his illustrious peers … we find bits of Ravel in his miniature opera Geneviève de Brabant and echoes of both Fauré and Debussy in the Nouvelles pièces froides of 1907".

After concluding his studies at the Schola Cantorum in 1912 Satie composed with greater confidence and more prolifically. Orchestration, despite his studies with d'Indy, was never his strongest suit, but his grasp of counterpoint is evident in the opening bars of Parade, and from the outset of his composing career he had original and distinctive ideas about harmony. In his later years he composed sets of short instrumental works with absurd titles, including Veritables Preludes flasques (pour un chien) ("True Flabby Preludes (for a Dog)", 1912), Croquis et agaceries d'un gros bonhomme en bois ("Sketches and Exasperations of a Big Wooden Man", 1913) and Sonatine bureaucratique ("Bureaucratic Sonata", 1917).

In his neat, calligraphic hand, Satie would write extensive instructions for his performers, and although his words appear at first sight to be humorous and deliberately nonsensical, Nichols and Griffiths comment, "a sensitive pianist can make much of injunctions such as 'arm yourself with clairvoyance' and 'with the end of your thought'". His Sonatine bureaucratique anticipates the neoclassicism soon adopted by Stravinsky. Despite his rancorous falling out with Debussy, Satie commemorated his long-time friend in 1920, two years after Debussy's death, in the anguished "Elégie", the first of the miniature song cycle Quatre petites mélodies. Orledge rates the cycle as the finest, though least known, of the four sets of short songs of Satie's last decade.

Satie invented what he called Musique d'ameublement – "furniture music" – a kind of background not to be listened to consciously. Cinéma, composed for the René Clair film Entr'acte, shown between the acts of Relâche (1924), is an example of early film music designed to be unconsciously absorbed rather than carefully listened to.

Satie is regarded by some writers as an influence on minimalism, which developed in the 1960s and later. The musicologist Mark Bennett and the composer Humphrey Searle have said that John Cage's music shows Satie's influence, and Searle and the writer Edward Strickland have used the term "minimalism" in connection with Satie's Vexations, which the composer implied in his manuscript should be played over and over again 840 times. John Adams included a specific homage to Satie's music in his 1996 Century Rolls.

Writings
Satie wrote extensively for the press, but unlike his professional colleagues such as Debussy and Dukas he did not write primarily as a music critic. Much of his writing is connected to music tangentially if at all. His biographer Caroline Potter describes him as "an experimental creative writer, a blagueur who provoked, mystified and amused his readers". He wrote jeux d'esprit claiming to eat dinner in four minutes with a diet of exclusively white food (including bones and fruit mould), or to drink boiled wine mixed with fuchsia juice, or to be woken by a servant hourly throughout the night to have his temperature taken; he wrote in praise of Beethoven's non-existent but "sumptuous" Tenth Symphony, and the family of instruments known as the cephalophones, "which have a compass of thirty octaves and are absolutely unplayable".

Satie grouped some of these writings under the general headings Cahiers d'un mammifère (A Mammal's Notebook) and Mémoires d'un amnésique (Memoirs of an Amnesiac), indicating, as Potter comments, that "these are not autobiographical writings in the conventional manner". He claimed the major influence on his humour was Oliver Cromwell, adding "I also owe much to Christopher Columbus, because the American spirit has occasionally tapped me on the shoulder and I have been delighted to feel its ironically glacial bite".

His published writings include:
 * A Mammal's Notebook: Collected Writings of Erik Satie (Serpent's Tail; Atlas Arkhive, No 5, 1997) ISBN 0-947757-92-9 (with introduction and notes by Ornella Volta, translations by Anthony Melville, contains several drawings by Satie)
 * Correspondence presque complète: Réunie, établie et présentée par Ornella Volta (Paris: Fayard/Imes, 2000; 1265 pages) ISBN 2-213-60674-9 (an almost complete edition of Satie's letters, in French)
 * Nigel Wilkins, The Writings of Erik Satie, London, 1980.

title: Erik Satie


text:
Talk:Eurovision Song Contest

A Commons file used on this page or its Wikidata item has been nominated for deletion
The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for deletion: Participate in the deletion discussion at the. —Community Tech bot (talk) 04:37, 30 May 2022 (UTC)
 * ABBA - TopPop 1974 5.png

Theme
The discussion is extracted from this talk. Should the main ESC page display the permanent slogan in the infobox like the annual pages do? Smthngnw (talk) 09:25, 17 November 2023 (UTC)
 * Agree. The slogans for the ESC had been a significant part of the shows like the logo in terms of representation, broadcasting and marketing. Since the introduction of the permanent slogan from now on the whole show keeps the same slogan. I don't see a problem in keeping the consistency between an annual contest page and the main page when they both have a permanent theme. Smthngnw (talk) 09:26, 17 November 2023 (UTC)
 * Oppose. I don't see the need, which is also what the other participant felt in the discussion you linked above. Choosing a slogan moving forward does not override past branding. Please also see here, which is discussing individual year pages and templates. Grk1011 (talk) 12:59, 17 November 2023 (UTC)
 * And the current ESC logo in the infobox doesn't override the past branding. I don't understand your point. Smthngnw (talk) 15:02, 17 November 2023 (UTC)

Sponsors
Moroccanoil has been a main sponsor and presenting partner for the ESC since 2020 (see links here and here). Shouldn't that be mentioned somewhere? Especially with the current criticisms of Israel's participation, and the fact that Moroccanoil has its production in Israel (link). No matter your opinion on the ongoing conflict, I think that information should be included for transparency. EiffelHenry (talk) 13:03, 29 April 2024 (UTC)

 * Unless the link between Moroccanoil and Israel's participation in the contest is explicitly mentioned in relevant sources, this would be a WP:SYNTH issue and violate Wikipedia's WP:OR policies. Additionally, I'm not sure how the contest's commercial interests and sponsorships are relevant to the article on the contest itself. Sims2aholic8 (talk) 10:05, 30 April 2024 (UTC)
 * Fair points. Here are some newspaper articles mentioning the potential link between Moroccanoil and Israel's participation: source (Swedish), source (English), and source (English).
 * I think you're right that this information doesn't fit here. Perhaps it would go under "Incidents and controversies" of Eurovision Song Contest 2024, or somewhere in Israel in the Eurovision Song Contest 2024? EiffelHenry (talk) 17:07, 15 May 2024 (UTC)

French name in the opening paragraph
If every country has a different name for ESC, why is the French one specified? 2A02:1810:C419:4700:65B9:C390:C431:4697 (talk) 19:04, 11 May 2024 (UTC)
 * The Naming sub-section probably guides us here. It tells us that the event first had both an English and a French name, and that on only four occasions has the name used not been in English or French. So the French language has a kind of special status when it comes to naming. HiLo48 (talk) 02:41, 12 May 2024 (UTC)
 * The official languages of the European Broadcasting Union, the contest organisers, are English and French, therefore it has dual official names. Sims2aholic8 (talk) 09:04, 16 May 2024 (UTC)

title: Talk:Eurovision Song Contest


text:
Elliptic integral

In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (c. 1750). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse.

Modern mathematics defines an "elliptic integral" as any function $f$ which can be expressed in the form

$$ f(x) = \int_{c}^{x} R{\left({\textstyle t, \sqrt{ P(t)} }\right)} \, dt,$$

where $R$ is a rational function of its two arguments, $P$ is a polynomial of degree 3 or 4 with no repeated roots, and $c$ is a constant.

In general, integrals in this form cannot be expressed in terms of elementary functions. Exceptions to this general rule are when $P$ has repeated roots, or when $R(x, y)$ contains no odd powers of $y$ or if the integral is pseudo-elliptic. However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legendre canonical forms (i.e. the elliptic integrals of the first, second and third kind).

Besides the Legendre form given below, the elliptic integrals may also be expressed in Carlson symmetric form. Additional insight into the theory of the elliptic integral may be gained through the study of the Schwarz–Christoffel mapping. Historically, elliptic functions were discovered as inverse functions of elliptic integrals.

Argument notation
Incomplete elliptic integrals are functions of two arguments; complete elliptic integrals are functions of a single argument. These arguments are expressed in a variety of different but equivalent ways (they give the same elliptic integral). Most texts adhere to a canonical naming scheme, using the following naming conventions.

For expressing one argument:
 * $α$, the modular angle
 * $k = sin α$, the elliptic modulus or eccentricity
 * $m = k^{2} = sin^{2} α$, the parameter

Each of the above three quantities is completely determined by any of the others (given that they are non-negative). Thus, they can be used interchangeably.

The other argument can likewise be expressed as $φ$, the amplitude, or as $x$ or $u$, where $x = sin φ = sn u$ and $sn$ is one of the Jacobian elliptic functions.

Specifying the value of any one of these quantities determines the others. Note that $u$ also depends on $m$. Some additional relationships involving $u$ include $$\cos \varphi = \operatorname{cn} u, \quad \textrm{and} \quad \sqrt{1 - m \sin^2 \varphi} = \operatorname{dn} u.$$

The latter is sometimes called the delta amplitude and written as $Δ(φ) = dn u$. Sometimes the literature also refers to the complementary parameter, the complementary modulus, or the complementary modular angle. These are further defined in the article on quarter periods.

In this notation, the use of a vertical bar as delimiter indicates that the argument following it is the "parameter" (as defined above), while the backslash indicates that it is the modular angle. The use of a semicolon implies that the argument preceding it is the sine of the amplitude: $$ F(\varphi, \sin \alpha) = F\left(\varphi \mid \sin^2 \alpha\right) = F(\varphi \setminus \alpha) = F(\sin \varphi ; \sin \alpha).$$ This potentially confusing use of different argument delimiters is traditional in elliptic integrals and much of the notation is compatible with that used in the reference book by Abramowitz and Stegun and that used in the integral tables by Gradshteyn and Ryzhik.

There are still other conventions for the notation of elliptic integrals employed in the literature. The notation with interchanged arguments, $F(k, φ)$, is often encountered; and similarly $E(k, φ)$ for the integral of the second kind. Abramowitz and Stegun substitute the integral of the first kind, $F(φ, k)$, for the argument $φ$ in their definition of the integrals of the second and third kinds, unless this argument is followed by a vertical bar: i.e. $E(F(φ, k) | k^{2})$ for $E(φ | k^{2})$. Moreover, their complete integrals employ the parameter $k^{2}$ as argument in place of the modulus $k$, i.e. $K(k^{2})$ rather than $K(k)$. And the integral of the third kind defined by Gradshteyn and Ryzhik, $Π(φ, n, k)$, puts the amplitude $φ$ first and not the "characteristic" $n$.

Thus one must be careful with the notation when using these functions, because various reputable references and software packages use different conventions in the definitions of the elliptic functions. For example, Wolfram's Mathematica software and Wolfram Alpha define the complete elliptic integral of the first kind in terms of the parameter $m$, instead of the elliptic modulus $k$.

Incomplete elliptic integral of the first kind
The incomplete elliptic integral of the first kind $F$ is defined as

$$ F(\varphi,k) = F\left(\varphi \mid k^2\right) = F(\sin \varphi ; k) = \int_0^\varphi \frac {d\theta}{\sqrt{1 - k^2 \sin^2 \theta}}.$$

This is the trigonometric form of the integral; substituting $t = sin θ$ and $x = sin φ$, one obtains the Legendre normal form:

$$ F(x ; k) = \int_{0}^{x} \frac{dt}{\sqrt{\left(1 - t^2\right)\left(1 - k^2 t^2\right)}}.$$

Equivalently, in terms of the amplitude and modular angle one has: $$ F(\varphi \setminus \alpha) = F(\varphi, \sin \alpha) = \int_0^\varphi \frac{d\theta}{\sqrt{1-\left(\sin \theta \sin \alpha\right)^2}}.$$

With $x = sn(u, k)$ one has: $$F(x;k) = u;$$ demonstrating that this Jacobian elliptic function is a simple inverse of the incomplete elliptic integral of the first kind.

The incomplete elliptic integral of the first kind has following addition theorem: $$F\bigl[\arctan(x),k\bigr] + F\bigl[\arctan(y),k\bigr] = F\left[\arctan\left(\frac{x\sqrt{k'^2y^2+1}}{\sqrt{y^2+1}}\right) + \arctan\left(\frac{y\sqrt{k'^2x^2+1}}{\sqrt{x^2+1}}\right),k\right] $$

The elliptic modulus can be transformed that way: $$F\bigl[\arcsin(x),k\bigr] = \frac{2}{1+\sqrt{1-k^2}}F\left[\arcsin\left(\frac{\left(1+\sqrt{1-k^2}\right)x}{1+\sqrt{1-k^2x^2}}\right),\frac{1-\sqrt{1-k^2}}{1+\sqrt{1-k^2}}\right] $$

Incomplete elliptic integral of the second kind
The incomplete elliptic integral of the second kind $E$ in trigonometric form is

$$ E(\varphi,k) = E\left(\varphi \,|\,k^2\right) = E(\sin\varphi;k) = \int_0^\varphi \sqrt{1-k^2 \sin^2\theta}\, d\theta.$$

Substituting $t = sin θ$ and $x = sin φ$, one obtains the Legendre normal form:

$$ E(x;k) = \int_0^x \frac{\sqrt{1-k^2 t^2} }{\sqrt{1-t^2}}\,dt.$$

Equivalently, in terms of the amplitude and modular angle: $$ E(\varphi \setminus \alpha) = E(\varphi, \sin \alpha) = \int_0^\varphi \sqrt{1-\left(\sin \theta \sin \alpha\right)^2} \, d\theta.$$

Relations with the Jacobi elliptic functions include $$\begin{align} E{\left(\operatorname{sn}(u ; k) ; k\right)} = \int_0^u \operatorname{dn}^2 (w ; k) \, dw &= u - k^2 \int_0^u \operatorname{sn}^2 (w ; k) \, dw \\[1ex] &= \left(1-k^2\right) u + k^2 \int_0^u \operatorname{cn}^2 (w ; k) \,dw. \end{align}$$

The meridian arc length from the equator to latitude $φ$ is written in terms of $E$: $$m(\varphi) = a\left(E(\varphi,e)+\frac{d^2}{d\varphi^2}E(\varphi,e)\right),$$ where $a$ is the semi-major axis, and $e$ is the eccentricity.

The incomplete elliptic integral of the second kind has following addition theorem: $$E{\left[\arctan(x), k\right]} + E{\left[\arctan(y), k\right]} = E{\left[\arctan\left(\frac{x\sqrt{k'^2y^2+1}}{\sqrt{y^2+1}}\right) + \arctan\left(\frac{y\sqrt{k'^2x^2+1}}{\sqrt{x^2+1}}\right),k\right]} + \frac{k^2xy}{k'^2x^2y^2+x^2+y^2+1}\left(\frac{x\sqrt{k'^2y^2+1}}{\sqrt{y^2+1}}+\frac{y\sqrt{k'^2x^2+1}}{\sqrt{x^2+1}}\right) $$

The elliptic modulus can be transformed that way: $$E{\left[\arcsin(x),k\right]} = \left(1+\sqrt{1-k^2}\right) E{\left[\arcsin\left(\frac{\left(1+\sqrt{1-k^2}\right)x}{1+\sqrt{1-k^2x^2}}\right),\frac{1-\sqrt{1-k^2}}{1+\sqrt{1-k^2}}\right]} - \sqrt{1-k^2} F{\left[\arcsin(x),k\right]} + \frac{k^2x\sqrt{1-x^2}}{1+\sqrt{1-k^2x^2}} $$

Incomplete elliptic integral of the third kind
The incomplete elliptic integral of the third kind $Π$ is $$ \Pi(n ; \varphi \setminus \alpha) = \int_0^\varphi \frac{1}{1-n\sin^2 \theta} \frac{d\theta}{\sqrt{1-\left(\sin\theta\sin \alpha\right)^2}}$$

or

$$ \Pi(n ; \varphi \,|\,m) = \int_{0}^{\sin \varphi} \frac{1}{1-nt^2} \frac{dt}{\sqrt{\left(1-m t^2\right)\left(1-t^2\right) }}.$$

The number $n$ is called the characteristic and can take on any value, independently of the other arguments. Note though that the value $Π(1; π⁄2 | m)$ is infinite, for any $m$.

A relation with the Jacobian elliptic functions is $$ \Pi\bigl(n; \,\operatorname{am}(u;k); \,k\bigr) = \int_0^u \frac{dw} {1 - n \,\operatorname{sn}^2 (w;k)}.$$

The meridian arc length from the equator to latitude $φ$ is also related to a special case of $Π$:

$$m(\varphi)=a\left(1-e^2\right)\Pi\left(e^2 ; \varphi \,|\,e^2\right).$$

Complete elliptic integral of the first kind
Elliptic Integrals are said to be 'complete' when the amplitude $K(k)$ and therefore $φ = π⁄2$. The complete elliptic integral of the first kind $x = 1$ may thus be defined as $$K(k) = \int_0^\tfrac{\pi}{2} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}} = \int_0^1 \frac{dt}{\sqrt{\left(1-t^2\right)\left(1-k^2 t^2\right)}},$$ or more compactly in terms of the incomplete integral of the first kind as $$K(k) = F\left(\tfrac{\pi}{2},k\right) = F\left(\tfrac{\pi}{2} \,|\, k^2\right) = F(1;k).$$

It can be expressed as a power series $$K(k) = \frac{\pi}{2}\sum_{n=0}^\infty \left(\frac{(2n)!}{2^{2 n} (n!)^2}\right)^2 k^{2n} = \frac{\pi}{2} \sum_{n=0}^\infty \bigl(P_{2 n}(0)\bigr)^2 k^{2n},$$

where $K$ is the Legendre polynomials, which is equivalent to

$$K(k) = \frac{\pi}{2}\left(1+\left(\frac{1}{2}\right)^2 k^2+\left(\frac{1\cdot 3}{2\cdot 4}\right)^2 k^4+\cdots+\left(\frac{\left(2n-1\right)!!}{\left(2n\right)!!}\right)^2 k^{2n}+\cdots\right),$$

where $P_{n}$ denotes the double factorial. In terms of the Gauss hypergeometric function, the complete elliptic integral of the first kind can be expressed as

$$K(k) = \tfrac{\pi}{2} \,{}_2F_1 \left(\tfrac{1}{2}, \tfrac{1}{2}; 1; k^2\right).$$

The complete elliptic integral of the first kind is sometimes called the quarter period. It can be computed very efficiently in terms of the arithmetic–geometric mean: $$K(k) = \frac{\pi}{2\operatorname{agm}\left(1,\sqrt{1-k^2}\right)}.$$

Therefore the modulus can be transformed as:

$$\begin{align} K(k) &= \frac{\pi}{2\operatorname{agm}\left(1,\sqrt{1-k^2}\right)} \\[4pt] & = \frac{\pi}{2\operatorname{agm}\left(\frac12+\frac\sqrt{1-k^2}{2},\sqrt[4]{1-k^2}\right)} \\[4pt] &= \frac{\pi}{\left(1+\sqrt{1-k^2}\right)\operatorname{agm}\left(1,\frac{2\sqrt[4]{1-k^2}}{\left(1+\sqrt{1-k^2}\right)}\right)} \\[4pt] & = \frac{2}{1+\sqrt{1-k^2}}K\left(\frac{1-\sqrt{1-k^2}}{1+\sqrt{1-k^2}}\right) \end{align}$$

This expression is valid for all $$n \isin \mathbb{N}$$ and $n!!$:

$$K(k) = n\left[\sum_{a = 1}^{n} \operatorname{dn}\left(\frac{2a}{n}K(k);k\right)\right]^{-1}K\left[k^n\prod_{a=1}^{n}\operatorname{sn}\left(\frac{2a-1}{n}K(k);k\right)^2\right] $$

Relation to the gamma function
If $0 ≤ k ≤ 1$ and $$r \isin \mathbb{Q}^+$$ (where $λ$ is the modular lambda function), then $k^{2} = λ(i√r)$ is expressible in closed form in terms of the gamma function. For example, $K(k)$, $r = 2$ and $r = 3$ give, respectively,

$$K\left(\sqrt{2}-1\right)=\frac{\Gamma \left(\frac18\right)\Gamma \left(\frac38\right)\sqrt{\sqrt{2}+1}}{8\sqrt[4]{2}\sqrt{\pi}},$$

and

$$K\left(\frac{\sqrt{3}-1}{2\sqrt{2}}\right)=\frac{1}{8\pi}\sqrt[4]{3}\,\sqrt[3]{4}\,\Gamma\biggl(\frac{1}{3}\biggr)^3$$

and

$$K\left(\frac{3-\sqrt{7}}{4\sqrt{2}}\right)=\frac{\Gamma \left(\frac17\right)\Gamma \left(\frac27\right)\Gamma \left(\frac47\right)}{4\sqrt[4]{7}\pi}.$$

More generally, the condition that $$\frac{iK'}{K}=\frac{iK\left(\sqrt{1-k^2}\right)}{K(k)}$$ be in an imaginary quadratic field is sufficient. For instance, if $r = 7$, then $k = e^{5πi/6}$ and

$$K\left(e^{5\pi i/6}\right)=\frac{e^{-\pi i/12}\Gamma ^3\left(\frac13\right)\sqrt[4]{3}}{4\sqrt[3]{2}\pi}.$$

Relation to Jacobi theta function
The relation to Jacobi's theta function is given by $$K(k)=\frac{\pi}{2}\theta_3^2(q),$$ where the nome $iK⁄K = e^{2πi/3}$ is $$q(k) = \exp\left(-\pi \frac{K\left(\sqrt{1-k^2}\right)}{K(k)}\right).$$

Asymptotic expressions
$$K\left(k\right)\approx\frac{\pi}{2}+\frac{\pi}{8}\frac{k^2}{1-k^2}-\frac{\pi}{16}\frac{k^4}{1-k^2}$$ This approximation has a relative precision better than $K$ for $q$. Keeping only the first two terms is correct to 0.01 precision for $k < 1⁄2$.

Differential equation
The differential equation for the elliptic integral of the first kind is $$\frac{d}{dk}\left(k\left(1-k^2\right)\frac{dK(k)}{dk}\right) = k \, K(k)$$

A second solution to this equation is $$K\left(\sqrt{1-k^2}\right)$$. This solution satisfies the relation $$\frac{d}{dk}K(k) = \frac{E(k)}{k\left(1-k^2\right)}-\frac{K(k)}{k}.$$

Continued fraction
A continued fraction expansion is: $$\frac{K(k)}{2\pi} = -\frac{1}{4} + \sum^{\infty}_{n=0} \frac{q^n}{1+q^{2n}} = -\frac{1}{4} + \cfrac{1}{1-q+ \cfrac{\left(1-q\right)^2}{1-q^3+ \cfrac{q\left(1-q^2\right)^2}{1-q^5+ \cfrac{q^2\left(1-q^3\right)^2}{1-q^7+\cfrac{q^3\left(1-q^4\right)^2}{1-q^9+\cdots}}}}},$$ where the nome is $$ q = q(k) = \exp[-\pi K'(k)/K(k)] $$ in its definition.

Complete elliptic integral of the second kind

The complete elliptic integral of the second kind $k < 1⁄2$ is defined as

$$E(k) = \int_0^\tfrac{\pi}{2} \sqrt{1-k^2 \sin^2\theta} \, d\theta = \int_0^1 \frac{\sqrt{1-k^2 t^2}}{\sqrt{1-t^2}} \, dt,$$

or more compactly in terms of the incomplete integral of the second kind $E(k)$ as

$$E(k) = E\left(\tfrac{\pi}{2},k\right) = E(1;k).$$

For an ellipse with semi-major axis $E$ and semi-minor axis $E(φ,k)$ and eccentricity $a$, the complete elliptic integral of the second kind $b$ is equal to one quarter of the circumference $e = √1 − b^{2}/a^{2}$ of the ellipse measured in units of the semi-major axis $E(e)$. In other words:

$$C = 4 a E(e).$$

The complete elliptic integral of the second kind can be expressed as a power series

$$E(k) = \frac{\pi}{2}\sum_{n=0}^\infty \left(\frac{(2n)!}{2^{2n} \left(n!\right)^2}\right)^2 \frac{k^{2n}}{1-2n},$$

which is equivalent to

$$E(k) = \frac{\pi}{2}\left(1-\left(\frac12\right)^2 \frac{k^2}{1}-\left(\frac{1\cdot 3}{2\cdot 4}\right)^2 \frac{k^4}{3}-\cdots-\left(\frac{(2n-1)!!}{(2n)!!}\right)^2 \frac{k^{2n}}{2n-1}-\cdots\right).$$

In terms of the Gauss hypergeometric function, the complete elliptic integral of the second kind can be expressed as

$$E(k) = \tfrac{\pi}{2} \,{}_2F_1 \left(\tfrac12, -\tfrac12; 1; k^2 \right).$$

The modulus can be transformed that way: $$E(k) = \left(1+\sqrt{1-k^2}\right)\,E\left(\frac{1-\sqrt{1-k^2}}{1+\sqrt{1-k^2}}\right) - \sqrt{1-k^2}\,K(k) $$

Computation
Like the integral of the first kind, the complete elliptic integral of the second kind can be computed very efficiently using the arithmetic–geometric mean.

Define sequences $3$ and $a_{n}$, where $C$, $a$ and the recurrence relations $a_{0} = 1$, $g_{0} = √1 − k^{2} = k$ hold. Furthermore, define $$c_n=\sqrt{\left|a_n^2-g_n^2\right|}.$$

By definition,

$$a_\infty = \lim_{n\to\infty} a_n = \lim_{n\to\infty} g_n = \operatorname{agm}\left(1, \sqrt{1-k^2}\right).$$

Also

$$\lim_{n\to\infty} c_n=0.$$

Then

$$E(k) = \frac{\pi}{2a_\infty}\left(1-\sum_{n=0}^{\infty} 2^{n-1} c_n^2\right).$$

In practice, the arithmetic-geometric mean would simply be computed up to some limit. This formula converges quadratically for all $a_{n + 1} = a_{n} + g_{n}⁄2$. To speed up computation further, the relation $g_{n + 1} = √a_{n} g_{n}$ can be used.

Furthermore, if $|k| ≤ 1$ and $$r \isin \mathbb{Q}^+$$ (where $g_{n}$ is the modular lambda function), then $c_{n + 1} = c_{n}^{2}⁄4a_{n + 1}$ is expressible in closed form in terms of $$K(k)=\frac{\pi}{2\operatorname{agm}\left(1,\sqrt{1-k^2}\right)}$$ and hence can be computed without the need for the infinite summation term. For example, $k^{2} = λ(i√r)$, $E(k)$ and $r = 1$ give, respectively,

$$E\left(\frac{1}{\sqrt{2}}\right)=\frac{1}{2}K\left(\frac{1}{\sqrt{2}}\right)+\frac{\pi}{4K\left(\frac{1}{\sqrt{2}}\right)},$$

and

$$E\left(\frac{\sqrt{3}-1}{2\sqrt{2}}\right)=\frac{3+\sqrt{3}}{6}K\left(\frac{\sqrt{3}-1}{2\sqrt{2}}\right)+\frac{\pi\sqrt{3}}{12K\left(\frac{\sqrt{3}-1}{2\sqrt{2}}\right)},$$

and

$$E\left(\frac{3-\sqrt{7}}{4\sqrt{2}}\right)=\frac{7+2\sqrt{7}}{14}K\left(\frac{3-\sqrt{7}}{4\sqrt{2}}\right)+\frac{\pi\sqrt{7}}{28K\left(\frac{3-\sqrt{7}}{4\sqrt{2}}\right)}.$$

Derivative and differential equation
$$\frac{dE(k)}{dk} = \frac{E(k)-K(k)}{k}$$ $$\left(k^2-1\right) \frac{d}{dk} \left( k \;\frac{dE(k)}{dk} \right) = k E(k)$$

A second solution to this equation is $r = 3$.

Complete elliptic integral of the third kind
The complete elliptic integral of the third kind $r = 7$ can be defined as

$$\Pi(n,k) = \int_0^\frac{\pi}{2} \frac{d\theta}{\left(1-n\sin^2\theta\right)\sqrt{1-k^2 \sin^2\theta}}.$$

Note that sometimes the elliptic integral of the third kind is defined with an inverse sign for the characteristic $E(√1 − k^{2}) − K(√1 − k^{2})$, $$\Pi'(n,k) = \int_0^\frac{\pi}{2} \frac{d\theta}{\left(1+n\sin^2\theta\right)\sqrt{1-k^2 \sin^2\theta}}.$$

Just like the complete elliptic integrals of the first and second kind, the complete elliptic integral of the third kind can be computed very efficiently using the arithmetic-geometric mean.

Partial derivatives
$$\begin{align} \frac{\partial\Pi(n,k)}{\partial n} &= \frac{1}{2\left(k^2-n\right)(n-1)}\left(E(k)+\frac{1}{n}\left(k^2-n\right)K(k) + \frac{1}{n} \left(n^2-k^2\right)\Pi(n,k)\right) \\[8pt]

\frac{\partial\Pi(n,k)}{\partial k} &= \frac{k}{n-k^2}\left(\frac{E(k)}{k^2-1}+\Pi(n,k)\right) \end{align}$$

Jacobi zeta function
In 1829, Jacobi defined the Jacobi zeta function: $$Z(\varphi,k)=E(\varphi,k)-\frac{E(k)}{K(k)}F(\varphi,k).$$ It is periodic in $$\varphi$$ with minimal period $$\pi$$. It is related to the Jacobi zn function by $$Z(\varphi,k)=\operatorname{zn}(F(\varphi,k),k)$$. In the literature (e.g. Whittaker and Watson (1927)), sometimes $$Z$$ means Wikipedia's $$\operatorname{zn}$$. Some authors (e.g. King (1924)) use $$Z$$ for both Wikipedia's $$Z$$ and $$\operatorname{zn}$$.

Legendre's relation
The Legendre's relation or Legendre Identity shows the relation of the integrals K and E of an elliptic modulus and its anti-related counterpart in an integral equation of second degree:

For two modules that are Pythagorean counterparts to each other, this relation is valid:

$$K(\varepsilon) E\left(\sqrt{1-\varepsilon^2}\right) + E(\varepsilon) K\left(\sqrt{1-\varepsilon^2}\right) - K(\varepsilon) K\left(\sqrt{1-\varepsilon^2}\right) = \frac {\pi}{2}$$

For example:
 * $$K({\color{blueviolet}\tfrac{3}{5}})E({\color{blue}\tfrac{4}{5}}) + E({\color{blueviolet}\tfrac{3}{5}})K({\color{blue}\tfrac{4}{5}}) - K({\color{blueviolet}\tfrac{3}{5}})K({\color{blue}\tfrac{4}{5}}) = \tfrac{1}{2}\pi$$

And for two modules that are tangential counterparts to each other, the following relationship is valid:

 * $$(1 + \varepsilon)K(\varepsilon)E(\tfrac{1 - \varepsilon}{1 + \varepsilon}) + \tfrac{2}{1 + \varepsilon}E(\varepsilon)K (\tfrac{1 - \varepsilon}{1 + \varepsilon}) - 2K(\varepsilon)K(\tfrac{1 - \varepsilon}{1 + \varepsilon}) = \tfrac{1}{2}\pi $$

For example:
 * $$\tfrac{4}{3}K({\color{blue}\tfrac{1}{3}})E({\color{green}\tfrac{1}{2}}) + \tfrac{3}{2}E({\color{blue}\tfrac{1}{3}})K({\color{green}\tfrac{1}{2}}) - 2K({\color{blue}\tfrac{1}{3}})K({\color{green}\tfrac{1}{2}}) = \tfrac{1}{2}\pi$$

The Legendre's relation for tangential modular counterparts results directly from the Legendre's identity for Pythagorean modular counterparts by using the Landen modular transformation on the Pythagorean counter modulus.

Special identity for the lemniscatic case
For the lemniscatic case, the elliptic modulus or specific eccentricity ε is equal to half the square root of two. Legendre's identity for the lemniscatic case can be proved as follows:

According to the Chain rule these derivatives hold:

 * $$\frac{\mathrm{d}}{\mathrm{d}y} \,K\bigl(\frac{1}{2}\sqrt{2}\bigr) - F\biggl[\arccos (xy);\frac{1}{2}\sqrt{2}\biggr] = \frac{\sqrt{2}\,x}{\sqrt{1 - x^4 y^4}}$$
 * $$\frac{\mathrm{d}}{\mathrm{d}y} \,2E\bigl(\frac{1}{2}\sqrt{2}\bigr) - K\bigl(\frac {1}{2}\sqrt{2}\bigr) - 2E\biggl[\arccos(xy);\frac{1}{2}\sqrt{2}\biggr] + F\biggl[\arccos(xy );\frac{1}{2}\sqrt{2}\biggr] = \frac{\sqrt{2}\,x^3 y^2}{\sqrt{1 - x^4 y^4}} $$

By using the Fundamental theorem of calculus these formulas can be generated:

 * $$K\bigl(\frac{1}{2}\sqrt{2}\bigr) - F\biggl[\arccos (x);\frac{1}{2}\sqrt{2}\biggr] = \int_{0}^{1} \frac{\sqrt{2}\,x}{\sqrt{1 - x^4 y^4}} \,\mathrm{d}y $$
 * $$2E\bigl(\frac{1}{2}\sqrt{2}\bigr) - K\bigl(\frac {1}{2}\sqrt{2}\bigr) - 2E\biggl[\arccos(x);\frac{1}{2}\sqrt{2}\biggr] + F\biggl[\arccos(x);\frac{1}{2}\sqrt{2}\biggr] = \int_{0}^{1} \frac{\sqrt{2}\,x^3 y^2}{\sqrt{1 - x^4 y^4}} \,\mathrm{d}y $$

The Linear combination of the two now mentioned integrals leads to the following formula:

\frac{\sqrt{2}}{\sqrt{1 - x^4}} \biggl\{2E\bigl(\frac{1}{2}\sqrt{2}\bigr) - K\bigl(\frac{1}{2}\sqrt{2}\bigr) - 2E\biggl[\arccos(x);\frac{1}{2}\sqrt{2}\biggr] + F\biggl[\arccos( x);\frac{1}{2}\sqrt{2}\biggr]\biggr\} \,+ $$

+ \,\frac{\sqrt{2} \,x^2}{\sqrt{1 - x^4}} \biggl\{K\bigl(\frac{1}{2}\sqrt{2}\bigr) - F\biggl[\arccos(x);\frac{1}{2}\sqrt{2}\biggr]\biggr\} = \int_{0}^{1} \frac{2\,x ^3 (y^2 + 1)}{\sqrt{(1 - x^4)(1 - x^4\,y^4)}} \,\mathrm{d}y $$

By forming the original antiderivative related to x from the function now shown using the Product rule this formula results:

 * $$ \biggl\{K\bigl(\frac{1}{2}\sqrt{2}\bigr) - F\biggl[\arccos(x);\frac{1}{2}\sqrt{ 2}\biggr]\biggr\}\biggl\{2E\bigl(\frac{1}{2}\sqrt{2}\bigr) - K\bigl(\frac{1}{2}\sqrt{2 }\bigr) - 2E\biggl[\arccos(x);\frac{1}{2}\sqrt{2}\biggr] + F\biggl[\arccos(x);\frac{1}{2} \sqrt{2}\biggr]\biggr\} =

$$
 * $$ = \int_{0}^{1} \frac{1}{y^2}(y^2 + 1)\biggl[\text{artanh}(y^2) - \text{artanh} \bigl(\frac{\sqrt{1 - x^4}\,y^2}{\sqrt{1 - x^4 y^4}}\bigr)\biggr] \mathrm{d}y $$

If the value $$x = 1$$ is inserted in this integral identity, then the following identity emerges:

 * $$ K\bigl(\frac{1}{2}\sqrt{2}\bigr)\biggl[2\,E\bigl(\frac{1}{2}\sqrt{2}\bigr) - K\bigl (\frac{1}{2}\sqrt{2}\bigr)\biggr] = \int_{0}^{1} \frac{1}{y^2}(y^2 + 1) \,\text{artanh}(y^2) \,\mathrm{d}y = $$

 * $$ = \biggl[2\arctan(y) - \frac{1}{y}(1 - y^2)\,\text{artanh}(y^2)\biggr]_{y = 0}^{y = 1} = 2\arctan(1) = \frac{\pi}{2} $$

This is how this lemniscatic excerpt from Legendre's identity appears:

 * $$2E\bigl(\frac{1}{2}\sqrt{2}\bigr)K\bigl(\frac{1}{2}\sqrt{2}\bigr) - K\bigl(\frac{1}{2}\sqrt{2}\bigr)^2 = \frac{\pi}{2}$$

Generalization for the overall case
Now the modular general case  is worked out. For this purpose, the derivatives of the complete elliptic integrals are derived after the modulus $$ \varepsilon $$ and then they are combined. And then the Legendre's identity balance is determined.

Because the derivative of the circle function is the negative product of the identical mapping function and the reciprocal of the circle function:

 * $$\frac{\mathrm{d}}{\mathrm{d}\varepsilon}\sqrt{1 - \varepsilon^2} = -\,\frac{\varepsilon}{\sqrt{1 - \varepsilon^2}}$$

These are the derivatives of K and E shown in this article in the sections above:

 * $$\frac{\mathrm{d}}{\mathrm{d}\varepsilon} K(\varepsilon) = \frac{1}{\varepsilon(1-\varepsilon^2)} \bigl[E( \varepsilon) - (1-\varepsilon^2)K(\varepsilon)\bigr]$$
 * $$\frac{\mathrm{d}}{\mathrm{d}\varepsilon} E(\varepsilon) = - \,\frac{1}{\varepsilon}\bigl[K(\varepsilon) - E (\varepsilon)\bigr]$$

In combination with the derivative of the circle function these derivatives are valid then:

 * $$\frac{\mathrm{d}}{\mathrm{d}\varepsilon}K(\sqrt{1 - \varepsilon^2}) = \frac{1}{\varepsilon(1-\varepsilon^ 2)} \bigl[\varepsilon^2 K(\sqrt{1 - \varepsilon^2}) - E(\sqrt{1 - \varepsilon^2})\bigr]$$
 * $$\frac{\mathrm{d}}{\mathrm{d}\varepsilon }E(\sqrt{1 - \varepsilon ^2}) = \frac{\varepsilon }{1 - \varepsilon ^2} \bigl[K(\sqrt{1 - \varepsilon^2}) - E(\sqrt{1 - \varepsilon^2})\bigr]$$

Legendre's identity includes products of any two complete elliptic integrals. For the derivation of the function side from the equation scale of Legendre's identity, the Product rule is now applied in the following:

 * $$\frac{\mathrm{d}}{\mathrm{d}\varepsilon}K(\varepsilon)E(\sqrt{1 - \varepsilon^2}) = \frac{1}{\varepsilon( 1-\varepsilon^2)} \bigl[E(\varepsilon)E(\sqrt{1 - \varepsilon^2}) - K(\varepsilon)E(\sqrt{1 - \varepsilon^2}) + \varepsilon^2 K(\varepsilon)K(\sqrt{1 - \varepsilon^2})\bigr]$$
 * $$\frac{\mathrm{d}}{\mathrm{d}\varepsilon}E(\varepsilon)K(\sqrt{1 - \varepsilon^2}) = \frac{1}{\varepsilon( 1-\varepsilon^2)} \bigl[- E(\varepsilon)E(\sqrt{1 - \varepsilon^2}) + E(\varepsilon)K(\sqrt{1 - \varepsilon^2}) - (1 - \varepsilon^2) K(\varepsilon)K(\sqrt{1 - \varepsilon^2})\bigr]$$
 * $$\frac{\mathrm{d}}{\mathrm{d}\varepsilon}K(\varepsilon)K(\sqrt{1 - \varepsilon^2}) = \frac{1}{\varepsilon( 1-\varepsilon^2)} \bigl[E(\varepsilon)K(\sqrt{1 - \varepsilon^2}) - K(\varepsilon)E(\sqrt{1 - \varepsilon^2}) - ( 1 - 2\varepsilon^2) K(\varepsilon)K(\sqrt{1 - \varepsilon^2})\bigr]$$

Of these three equations, adding the top two equations and subtracting the bottom equation gives this result:

 * $$\frac{\mathrm{d}}{\mathrm{d}\varepsilon} \bigl[K(\varepsilon)E(\sqrt{1 - \varepsilon^2}) + E(\varepsilon)K (\sqrt{1 - \varepsilon^2}) - K(\varepsilon)K(\sqrt{1 - \varepsilon^2})\bigr] = 0$$

In relation to the $$ \varepsilon $$ the equation balance constantly gives the value zero.

The previously determined result shall be combined with the Legendre equation to the modulus $$\varepsilon = 1/\sqrt{2}$$ that is worked out in the section before:

 * $$2E\bigl(\frac{1}{2}\sqrt{2}\bigr)K\bigl(\frac{1}{2}\sqrt{2}\bigr) - K\bigl(\frac{1}{2}\sqrt{2}\bigr)^2 = \frac{\pi}{2}$$

The combination of the last two formulas gives the following result:

 * $$K(\varepsilon)E(\sqrt{1 - \varepsilon^2}) + E(\varepsilon)K(\sqrt{1 - \varepsilon^2}) - K(\varepsilon)K(\sqrt{1 - \varepsilon^2}) = \tfrac{1}{2}\pi$$

Because if the derivative of a continuous function constantly takes the value zero, then the concerned function is a constant function. This means that this function results in the same function value for each abscissa value $$ \varepsilon $$ and the associated function graph is therefore a horizontal straight line.

title: Elliptic integral