1.
Calculator
–
An electronic calculator is a small, portable electronic device used to perform operations ranging from basic arithmetic to complex mathematics. The first solid state electronic calculator was created in the 1960s, building on the history of tools such as the abacus. It was developed in parallel with the computers of the day. The pocket sized devices became available in the 1970s, especially after the first microprocessor and they later became used commonly within the petroleum industry. Modern electronic calculators vary, from cheap, give-away, credit-card-sized models to sturdy desktop models with built-in printers and they became popular in the mid-1970s. By the end of decade, calculator prices had reduced to a point where a basic calculator was affordable to most. In addition to general purpose calculators, there are designed for specific markets. For example, there are scientific calculators which include trigonometric and statistical calculations, some calculators even have the ability to do computer algebra. Graphing calculators can be used to graph functions defined on the real line, as of 2016, basic calculators cost little, but the scientific and graphing models tend to cost more. In 1986, calculators still represented an estimated 41% of the worlds general-purpose hardware capacity to compute information, by 2007, this diminished to less than 0. 05%. Modern 2016 electronic calculators contain a keyboard with buttons for digits and arithmetical operations, most basic calculators assign only one digit or operation on each button, however, in more specific calculators, a button can perform multi-function working with key combinations. Large-sized figures and comma separators are used to improve readability. Various symbols for function commands may also be shown on the display, fractions such as 1⁄3 are displayed as decimal approximations, for example rounded to 0.33333333. Also, some fractions can be difficult to recognize in decimal form, as a result, Calculators also have the ability to store numbers into computer memory. Basic types of these only one number at a time. The variables can also be used for constructing formulas, some models have the ability to extend memory capacity to store more numbers, the extended memory address is termed an array index. Power sources of calculators are, batteries, solar cells or mains electricity, some models even have no turn-off button but they provide some way to put off. Crank-powered calculators were also common in the computer era
2.
European Central Bank
–
The European Central Bank is the central bank for the euro and administers monetary policy of the eurozone, which consists of 19 EU member states and is one of the largest currency areas in the world. It is one of the worlds most important central banks and is one of the seven institutions of the European Union listed in the Treaty on European Union, the capital stock of the bank is owned by the central banks of all 28 EU member states. The Treaty of Amsterdam established the bank in 1998, and it is headquartered in Frankfurt, Germany. As of 2015 the President of the ECB is Mario Draghi, former governor of the Bank of Italy, former member of the World Bank, the bank primarily occupied the Eurotower prior to, and during, the construction of the new headquarters. The primary objective of the ECB, mandated in Article 2 of the Statute of the ECB, is to price stability within the Eurozone. The ECB has, under Article 16 of its Statute, the right to authorise the issuance of euro banknotes. Member states can issue euro coins, but the amount must be authorised by the ECB beforehand, the ECB is governed by European law directly, but its set-up resembles that of a corporation in the sense that the ECB has shareholders and stock capital. Its capital is €11 billion held by the central banks of the member states as shareholders. The initial capital allocation key was determined in 1998 on the basis of the population and GDP. Shares in the ECB are not transferable and cannot be used as collateral, the European Central Bank is the de facto successor of the European Monetary Institute. The EMI itself took over from the earlier European Monetary Co-operation Fund, the bank was the final institution needed for EMU, as outlined by the EMU reports of Pierre Werner and President Jacques Delors. It was established on 1 June 1998, the first President of the Bank was Wim Duisenberg, the former president of the Dutch central bank and the European Monetary Institute. The French argued that since the ECB was to be located in Germany and this was opposed by the German, Dutch and Belgian governments who saw Duisenberg as a guarantor of a strong euro. Tensions were abated by an agreement in which Duisenberg would stand down before the end of his mandate. Trichet replaced Duisenberg as President in November 2003, there had also been tension over the ECBs Executive Board, with the United Kingdom demanding a seat even though it had not joined the Single Currency. Under pressure from France, three seats were assigned to the largest members, France, Germany, and Italy, Spain also demanded and obtained a seat. Despite such a system of appointment the board asserted its independence early on in resisting calls for interest rates, when the ECB was created, it covered a Eurozone of eleven members. On 1 December 2009, the Treaty of Lisbon entered into force, ECB according to the article 13 of TEU, on 1 November 2011, Mario Draghi replaced Jean-Claude Trichet as President of the ECB
3.
Logarithm
–
In mathematics, the logarithm is the inverse operation to exponentiation. That means the logarithm of a number is the exponent to which another fixed number, in simple cases the logarithm counts factors in multiplication. For example, the base 10 logarithm of 1000 is 3, the logarithm of x to base b, denoted logb, is the unique real number y such that by = x. For example, log2 =6, as 64 =26, the logarithm to base 10 is called the common logarithm and has many applications in science and engineering. The natural logarithm has the e as its base, its use is widespread in mathematics and physics. The binary logarithm uses base 2 and is used in computer science. Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations and they were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the function in the 18th century. Logarithmic scales reduce wide-ranging quantities to tiny scopes, for example, the decibel is a unit quantifying signal power log-ratios and amplitude log-ratios. In chemistry, pH is a measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and they describe musical intervals, appear in formulas counting prime numbers, inform some models in psychophysics, and can aid in forensic accounting. In the same way as the logarithm reverses exponentiation, the logarithm is the inverse function of the exponential function applied to complex numbers. The discrete logarithm is another variant, it has uses in public-key cryptography, the idea of logarithms is to reverse the operation of exponentiation, that is, raising a number to a power. For example, the power of 2 is 8, because 8 is the product of three factors of 2,23 =2 ×2 ×2 =8. It follows that the logarithm of 8 with respect to base 2 is 3, the third power of some number b is the product of three factors equal to b. More generally, raising b to the power, where n is a natural number, is done by multiplying n factors equal to b. The n-th power of b is written bn, so that b n = b × b × ⋯ × b ⏟ n factors, exponentiation may be extended to by, where b is a positive number and the exponent y is any real number. For example, b−1 is the reciprocal of b, that is, the logarithm of a positive real number x with respect to base b, a positive real number not equal to 1, is the exponent by which b must be raised to yield x
4.
Statistics
–
Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data. In applying statistics to, e. g. a scientific, industrial, or social problem, populations can be diverse topics such as all people living in a country or every atom composing a crystal. Statistics deals with all aspects of data including the planning of data collection in terms of the design of surveys, statistician Sir Arthur Lyon Bowley defines statistics as Numerical statements of facts in any department of inquiry placed in relation to each other. When census data cannot be collected, statisticians collect data by developing specific experiment designs, representative sampling assures that inferences and conclusions can safely extend from the sample to the population as a whole. In contrast, an observational study does not involve experimental manipulation, inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena. A standard statistical procedure involves the test of the relationship between two data sets, or a data set and a synthetic data drawn from idealized model. A hypothesis is proposed for the relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets. Rejecting or disproving the hypothesis is done using statistical tests that quantify the sense in which the null can be proven false. Working from a hypothesis, two basic forms of error are recognized, Type I errors and Type II errors. Multiple problems have come to be associated with this framework, ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis, measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random or systematic, the presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems. Statistics continues to be an area of research, for example on the problem of how to analyze Big data. Statistics is a body of science that pertains to the collection, analysis, interpretation or explanation. Some consider statistics to be a mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is concerned with the use of data in the context of uncertainty, mathematical techniques used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure-theoretic probability theory. In applying statistics to a problem, it is practice to start with a population or process to be studied. Populations can be diverse topics such as all living in a country or every atom composing a crystal. Ideally, statisticians compile data about the entire population and this may be organized by governmental statistical institutes
5.
Floating-point arithmetic
–
In computing, floating-point arithmetic is arithmetic using formulaic representation of real numbers as an approximation so as to support a trade-off between range and precision. A number is, in general, represented approximately to a number of significant digits and scaled using an exponent in some fixed base. For example,1.2345 =12345 ⏟ significand ×10 ⏟ base −4 ⏞ exponent, the term floating point refers to the fact that a numbers radix point can float, that is, it can be placed anywhere relative to the significant digits of the number. This position is indicated as the exponent component, and thus the floating-point representation can be thought of as a kind of scientific notation. The result of dynamic range is that the numbers that can be represented are not uniformly spaced. Over the years, a variety of floating-point representations have been used in computers, however, since the 1990s, the most commonly encountered representation is that defined by the IEEE754 Standard. A floating-point unit is a part of a computer system designed to carry out operations on floating point numbers. A number representation specifies some way of encoding a number, usually as a string of digits, there are several mechanisms by which strings of digits can represent numbers. In common mathematical notation, the string can be of any length. If the radix point is not specified, then the string implicitly represents an integer, in fixed-point systems, a position in the string is specified for the radix point. So a fixed-point scheme might be to use a string of 8 decimal digits with the point in the middle. The scaling factor, as a power of ten, is then indicated separately at the end of the number, floating-point representation is similar in concept to scientific notation. Logically, a floating-point number consists of, A signed digit string of a length in a given base. This digit string is referred to as the significand, mantissa, the length of the significand determines the precision to which numbers can be represented. The radix point position is assumed always to be somewhere within the significand—often just after or just before the most significant digit and this article generally follows the convention that the radix point is set just after the most significant digit. A signed integer exponent, which modifies the magnitude of the number, using base-10 as an example, the number 7005152853504700000♠152853.5047, which has ten decimal digits of precision, is represented as the significand 1528535047 together with 5 as the exponent. In storing such a number, the base need not be stored, since it will be the same for the range of supported numbers. Symbolically, this value is, s b p −1 × b e, where s is the significand, p is the precision, b is the base
6.
Multi-touch
–
In computing, multi-touch is technology that enables a surface to recognize the presence of more than one or more than two points of contact with the surface. The origins of multitouch began at CERN, MIT, University of Toronto, Carnegie Mellon University, the term multi-touch was popularized in 2007 by Apple, though it was in use as early as 1985. This plural-point awareness may be used to implement additional functionality, such as pinch to zoom or to activate certain subroutines attached to predefined gestures, the use of touchscreen technology to control electronic devices pre-dates multi-touch technology and the personal computer. Early synthesizer and electronic instrument builders like Hugh Le Caine and Robert Moog experimented with using touch-sensitive capacitance sensors to control the sounds made by their instruments, early touchscreens only registered one point of touch at a time. On-screen keyboards were thus awkward to use, because key-rollover and holding down a key while typing another were not possible. An exception was a multi-touch reconfigurable touchscreen keyboard/display developed at the Massachusetts Institute of Technology in the early 1970s and this technology was used to develop a new type of human machine interface for the control room of the Super Proton Synchrotron particle accelerator. In a handwritten note dated 11 March 1972, Stumpe presented his proposed solution – a capacitive touch screen with a number of programmable buttons presented on a display. The capacitors were to consist of fine lines etched in copper on a sheet of glass – fine enough, in the final device, a simple lacquer coating prevented the fingers from actually touching the capacitors. In 1976, MIT described a keyboard with variable graphics capable of multi-touch detection, in the early 1980s, The University of Torontos Input Research Group were among the earliest to explore the software side of multi-touch input systems. A1982 system at the University of Toronto used a panel with a camera placed behind the glass. When a finger or several fingers pressed on the glass, the camera would detect the action as one or more spots on an otherwise white background. Since the size of a dot was dependent on pressure, the system was somewhat pressure-sensitive as well, of note, this system was input only and not able to display graphics. In 1983, Bell Labs at Murray Hill published a discussion of touch-screen based interfaces. By 1984, both Bell Labs and Carnegie Mellon University had working multi-touch-screen prototypes – both input and graphics – that could respond interactively in response to multiple finger inputs, the Bell Labs system was based on capacitive coupling of fingers, whereas the CMU system was optical. In 1985, the canonical multitouch pinch-to-zoom gesture was demonstrated, with coordinated graphics, an advance occurred in 1991, when Pierre Wellner published a paper on his multi-touch Digital Desk, which supported multi-finger and pinching motions. Various companies expanded upon these inventions in the beginning of the twenty-first century, the company Fingerworks developed various multi-touch technologies between 1999 and 2005, including Touchstream keyboards and the iGesture Pad. Several studies of technology were published in the early 2000s by Alan Hedge, professor of human factors. Apple acquired Fingerworks and its technology in 2005
7.
Exchange rate
–
In finance, an exchange rate between two currencies is the rate at which one currency will be exchanged for another. It is also regarded as the value of one currency in relation to another currency. For example, an exchange rate of 119 Japanese yen to the United States dollar means that ¥119 will be exchanged for each US$1 or that US$1 will be exchanged for each ¥119. In this case it is said that the price of a dollar in relation to yen is ¥119, trading from 20,15 GMT on Sunday until 22,00 GMT Friday. The spot exchange rate refers to the current exchange rate, the forward exchange rate refers to an exchange rate that is quoted and traded today but for delivery and payment on a specific future date. In the retail currency exchange market, different buying and selling rates will be quoted by money dealers, most trades are to or from the local currency. The buying rate is the rate at which money dealers will buy foreign currency, the quoted rates will incorporate an allowance for a dealers margin in trading, or else the margin may be recovered in the form of a commission or in some other way. Different rates may also be quoted for cash, a form or electronically. The higher rate on documentary transactions has been justified as compensating for the additional time, on the other hand, cash is available for resale immediately, but brings security, storage, and transportation costs, and the cost of tying up capital in a stock of banknotes. Currency for international travel and cross-border payments is predominantly purchased from banks, foreign exchange brokerages and these retail outlets source currency from the inter-bank markets, which are valued by the Bank for International Settlements at 5.3 trillion US dollars per day. The purchase is made at the contract rate. Retail customers will be charged, in the form of commission or otherwise, to cover the providers costs, one form of charge is the use of an exchange rate that is less favourable than the wholesale spot rate. The difference between retail buying and selling prices is referred to as the bid-ask spread, in the foreign exchange market, a currency pair is the quotation of the relative value of a currency unit against the unit of another currency. The quotation EUR/USD1.3225 means that 1 Euro will buy 1.3225 US dollars, in other words, this is the price of a unit of Euro in US dollars. Here, EUR is called the Fixed currency, while USD is called the Variable currency, there is a market convention that determines which is the fixed currency and which is the variable currency. In most parts of the world, the order is, EUR – GBP – AUD – NZD – USD – others. Accordingly, in a conversion from EUR to AUD, EUR is the currency, AUD is the variable currency. Cyprus and Malta, which were quoted as the base to the USD, in some areas of Europe and in the retail market in the United Kingdom, EUR and GBP are reversed so that GBP is quoted as the fixed currency to the euro
8.
Square root
–
In mathematics, a square root of a number a is a number y such that y2 = a, in other words, a number y whose square is a. For example,4 and −4 are square roots of 16 because 42 =2 =16, every nonnegative real number a has a unique nonnegative square root, called the principal square root, which is denoted by √a, where √ is called the radical sign or radix. For example, the square root of 9 is 3, denoted √9 =3. The term whose root is being considered is known as the radicand, the radicand is the number or expression underneath the radical sign, in this example 9. Every positive number a has two roots, √a, which is positive, and −√a, which is negative. Together, these two roots are denoted ± √a, although the principal square root of a positive number is only one of its two square roots, the designation the square root is often used to refer to the principal square root. For positive a, the square root can also be written in exponent notation. Square roots of numbers can be discussed within the framework of complex numbers. In Ancient India, the knowledge of theoretical and applied aspects of square and square root was at least as old as the Sulba Sutras, a method for finding very good approximations to the square roots of 2 and 3 are given in the Baudhayana Sulba Sutra. Aryabhata in the Aryabhatiya, has given a method for finding the root of numbers having many digits. It was known to the ancient Greeks that square roots of positive numbers that are not perfect squares are always irrational numbers, numbers not expressible as a ratio of two integers. This is the theorem Euclid X,9 almost certainly due to Theaetetus dating back to circa 380 BC, the particular case √2 is assumed to date back earlier to the Pythagoreans and is traditionally attributed to Hippasus. Mahāvīra, a 9th-century Indian mathematician, was the first to state that square roots of negative numbers do not exist, a symbol for square roots, written as an elaborate R, was invented by Regiomontanus. An R was also used for Radix to indicate square roots in Gerolamo Cardanos Ars Magna, according to historian of mathematics D. E. Smith, Aryabhatas method for finding the root was first introduced in Europe by Cataneo in 1546. According to Jeffrey A. Oaks, Arabs used the letter jīm/ĝīm, the letter jīm resembles the present square root shape. Its usage goes as far as the end of the century in the works of the Moroccan mathematician Ibn al-Yasamin. The symbol √ for the root was first used in print in 1525 in Christoph Rudolffs Coss
9.
Windows 8.1
–
Windows 8.1 is an upgrade for Windows 8, a computer operating system released by Microsoft. Windows 8.1 is available free of charge for copies of Windows 8. However, unlike previous service packs, Windows 8.1 cannot be acquired via Windows Update, Windows 8.1 also added support for such emerging technologies as high-resolution displays, 3D printing, Wi-Fi Direct, and Miracast streaming, as well as the ReFS file system. Despite these improvements, Windows 8.1 was still criticized for not addressing all digressions of Windows 8, as of March 2017, the market share of Windows 8.1 is 6. 66%. com, and SkyDrive. Lending credibility to the reports, Foley noted that a Microsoft staff member had listed experience with Windows Blue on his LinkedIn profile, a post-RTM build of Windows 8, build 9364, was leaked in March 2013. Shortly afterward on March 26,2013, corporate president of corporate communications Frank X. In early May, press reports announcing the upcoming version in Financial Times, the theme was then echoed and debated in the computer press. Shaw rejected this criticism as extreme, adding that he saw a comparison with Diet Coke as more appropriate, on May 14, Microsoft officially announced that Blue would be named Windows 8.1. Following a keynote presentation focusing on this version, the beta of Windows 8.1 was released on June 26,2013 during Build. Build 9600 of Windows 8.1 was released to OEM hardware partners on August 27,2013, however, after criticism, Microsoft reversed its decision and released the RTM build on MSDN and TechNet on September 9,2013. Prior to the release of Windows 8.1, Microsoft premiered a new television commercial in late-September 2013 that focused on its changes as part of the Windows Everywhere campaign, shortly after its release, Windows RT8. On October 21,2013, Microsoft confirmed that the bug was limited to the original Surface tablet, the company released recovery media and instructions which could be used to repair the device, and restored access to Windows RT8.1 the next day. It was also found changes to screen resolution handling on Windows 8. Users also found the issues to be pronounced when using gaming mice with high resolution and/or polling rates. Microsoft released a patch to fix the bug on certain games in November 2013, on April 8,2014, Microsoft released the Windows 8.1 Update, which included all past updates plus new features. It was unveiled by Microsoft vice president Joe Belfiore at Mobile World Congress on February 23,2014, and detailed in full at Microsofts Build conference on April 2. Belfiore noted that the update would lower the minimum requirements for Windows. Unlike Windows 8.1 itself, this update is distributed through Windows Update
10.
Windows 1.0
–
Windows 1.0 is a graphical personal computer operating environment developed by Microsoft. Windows 1.0 was released on November 20,1985 and it runs as a graphical, 16-bit multi-tasking shell on top of an existing MS-DOS installation. It provides an environment which can run graphical programs designed for Windows and its development was spearheaded by the company founder Bill Gates after he saw a demonstration of a similar software suite known as Visi On at COMDEX. Despite positive responses to its early presentations and support from a number of hardware and software makers, critics felt Windows 1.0 did not meet their expectations. In particular, they felt that Windows 1, despite these criticisms, Windows 1.0 was an important milestone for Microsoft, as it introduced the Microsoft Windows line, and in computer history in general. Windows 1.0 was declared obsolete and Microsoft stopped providing support, the development of Windows began after Microsoft founder Bill Gates saw a demonstration at COMDEX1982 of VisiCorps Visi On, a graphical user interface software suite for IBM PC compatible computers. Microsoft first presented Windows to the public on November 10,1983, requiring two floppy disk drives and 192 KB of RAM, Microsoft described the software as a device driver for MS-DOS2.0. Many manufacturers of MS-DOS computers such as Compaq, Zenith, and DEC promised to support, as did software companies such as Ashton-Tate. From early in Windows history Gates viewed it as Microsofts future, were also saying that only applications that take advantage of Windows will be competitive in the long run. IBM was notably absent from Microsofts announcement, and by late 1984 the press reported a War of the Windows between Windows, IBMs TopView, and Digital Researchs Graphics Environment Manager. Microsoft had promised in November 1983 to ship Windows by April 1984, but subsequently denied that it had announced a release date, and predicted that Windows would ship by June 1985. Deemphasizing multitasking, the stated that Windows purpose, unlike that of TopView, was to turn the computer into a graphics-rich environment while using less memory. Windows 1.0 was officially released on November 20,1985, version 1.01, released in 1985, was the first point-release after Windows 1.00. Version 1.02, released in May 1986, was an international release, version 1.03, released in August 1986, included enhancements that made it consistent with the international release. It included drivers for European keyboards and additional screen and printer drivers, version 1.04, released in April 1987, added support for the new IBM PS/2 computers, although no support for PS/2 mice or new VGA graphics modes was provided. At the same time, Microsoft and IBM announced the introduction of OS/2 and its graphical OS/2 Presentation Manager, in November 1987, Windows 1.0 was succeeded by Windows 2.0. Microsoft supported Windows 1.0 for 16 years, until December 31,2001 – the longest out of all versions of Windows. Windows 1.0 offers limited multitasking of existing MS-DOS programs and concentrates on creating a paradigm, an execution model
11.
Windows 98
–
Windows 98 is a graphical operating system by Microsoft. It is the major release in the Windows 9x line of operating systems. It was released to manufacturing on May 15,1998 and to retail on June 25,1998, like its predecessor, Windows 98 is a hybrid 16-bit and 32-bit monolithic product with an MS-DOS based boot stage. Windows 98 was succeeded by Windows 98 Second Edition on May 5,1999, Microsoft ended mainstream support for Windows 98 and 98 SE on June 30,2002, and extended support on July 11,2006. The famous startup sound for Windows 98 was composed by Microsoft sound engineer Ken Kato, development of Windows 98 began in the 1990s, initially under the development codename Memphis. Many builds were released or leaked, starting with build 1351 on December 15,1996, Windows 98 includes Internet Explorer 4.01 in First Edition and 5.0 in Second Edition. Another feature of this new shell is that dialog boxes now show up in the Alt-Tab sequence, 3D Pinball is included on the CD-ROM but not installed by default. Windows 98 had its own separately purchasable Plus, title bars of windows and dialog boxes now support two-color gradients. Windows menus and tooltips now support slide animation, Windows Explorer includes support for compressed CAB files. The Quick Res and Telephony Location Manager Windows 95 PowerToys are integrated into the operating system. Windows 98 was the first operating system to use the Windows Driver Model, the WDM standard only achieved widespread adoption years later, mostly through Windows 2000 and Windows XP, as they were not compatible with the older VxD standard. Windows Driver Model was introduced largely so that developers would write drivers that were compatible with future versions of Windows. Device driver access in WDM is actually implemented through a VxD device driver, NTKERN creates IRPs and sends them to WDM drivers. Support for WDM audio enables digital mixing, routing and processing of audio streams. WDM Audio allows for software emulation of hardware to support MS-DOS games, DirectSound support. The Windows 95 11-device limitation for MIDI devices is eliminated, a Microsoft GS Wavetable Synthesizer licensed from Roland shipped with Windows 98 for WDM audio drivers. Windows Driver Model also includes Broadcast Driver Architecture, the backbone for TV technologies support in Windows, webTV for Windows utilized BDA to allow viewing television on the computer if a compatible TV tuner card is installed. Windows 98 had more robust USB support than Windows 95 which only had support in OEM versions, Windows 98 supports USB hubs, USB scanners and imaging class devices
12.
Gamma function
–
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers. That is, if n is an integer, Γ =. The gamma function is defined for all numbers except the non-positive integers. The gamma function can be seen as a solution to the interpolation problem. The simple formula for the factorial, x. =1 ×2 × … × x, a good solution to this is the gamma function. There are infinitely many continuous extensions of the factorial to non-integers, the gamma function is the most useful solution in practice, being analytic, and it can be characterized in several ways. The Bohr–Mollerup theorem proves that these properties, together with the assumption that f be logarithmically convex, uniquely determine f for positive, from there, the gamma function can be extended to all real and complex values by using the unique analytic continuation of f. Also see Eulers infinite product definition below where the properties f =1 and f = x f together with the requirement that limn→+∞. nx / f =1 uniquely define the same function. The notation Γ is due to Legendre, if the real part of the complex number z is positive, then the integral Γ = ∫0 ∞ x z −1 e − x d x converges absolutely, and is known as the Euler integral of the second kind. The identity Γ = Γ z can be used to extend the integral formulation for Γ to a meromorphic function defined for all complex numbers z. It is this version that is commonly referred to as the gamma function. When seeking to approximate z. for a number z, it turns out that it is effective to first compute n. for some large integer n. And then use the relation m. = m. backwards n times. Furthermore, this approximation is exact in the limit as n goes to infinity, specifically, for a fixed integer m, it is the case that lim n → + ∞ n. m. =1, and we can ask that the formula is obeyed when the arbitrary integer m is replaced by an arbitrary complex number z lim n → + ∞ n. z. =1. Multiplying both sides by z. gives z. = lim n → + ∞ n. z, Z = lim n → + ∞1 ⋯ n ⋯ z = ∏ n =1 + ∞. Similarly for the function, the definition as an infinite product due to Euler is valid for all complex numbers z except the non-positive integers. By this construction, the function is the unique function that simultaneously satisfies Γ =1, Γ = z Γ for all complex numbers z except the non-positive integers
