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Stirling's Approximation
In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less precise result was first stated by Abraham de Moivre. One way of stating the approximation involves the logarithm of the factorial: \ln(n!) = n\ln n - n +O(\ln n), where the big O notation means that, for all sufficiently large values of n, the difference between \ln(n!) and n\ln n-n will be at most proportional to the logarithm of n. In computer science applications such as the worst-case lower bound for comparison sorting, it is convenient to instead use the binary logarithm, giving the equivalent form \log_2 (n!) = n\log_2 n - n\log_2 e +O(\log_2 n). The error term in either base can be expressed more precisely as \tfrac12\log(2\pi n)+O(\tfrac1n), corresponding to an approximate formula for the factorial itself, n! \sim \sqr ...
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Wallis Product
The Wallis product is the infinite product representation of : :\begin \frac & = \prod_^ \frac = \prod_^ \left(\frac \cdot \frac\right) \\ pt& = \Big(\frac \cdot \frac\Big) \cdot \Big(\frac \cdot \frac\Big) \cdot \Big(\frac \cdot \frac\Big) \cdot \Big(\frac \cdot \frac\Big) \cdot \; \cdots \\ \end It was published in 1656 by John Wallis. Proof using integration Wallis derived this infinite product using interpolation, though his method is not regarded as rigorous. A modern derivation can be found by examining \int_0^\pi \sin^n x\,dx for even and odd values of n, and noting that for large n, increasing n by 1 results in a change that becomes ever smaller as n increases. Let :I(n) = \int_0^\pi \sin^n x\,dx. (This is a form of Wallis' integrals.) Integrate by parts: :\begin u &= \sin^x \\ \Rightarrow du &= (n-1) \sin^x \cos x\,dx \\ dv &= \sin x\,dx \\ \Rightarrow v &= -\cos x \end :\begin \Rightarrow I(n) &= \int_0^\pi \sin^n x\,dx \\ ...
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Ratio Test
In mathematics, the ratio test is a convergence tests, test (or "criterion") for the convergent series, convergence of a series (mathematics), series :\sum_^\infty a_n, where each term is a real number, real or complex number and is nonzero when is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test. The test The usual form of the test makes use of the limit (mathematics), limit The ratio test states that: * if ''L'' 1 then the series divergent series, diverges; * if ''L'' = 1 or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case. It is possible to make the ratio test applicable to certain cases where the limit ''L'' fails to exist, if limit superior and limit inferior are used. The test criteria can also be refined so that the test is sometimes conclusive even when ''L'' = 1. More specifically, let ...
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Approximation Error
The approximation error in a given data value represents the significant discrepancy that arises when an exact, true value is compared against some approximation derived for it. This inherent error in approximation can be quantified and expressed in two principal ways: as an absolute error, which denotes the direct numerical magnitude of this discrepancy irrespective of the true value's scale, or as a relative error, which provides a scaled measure of the error by considering the absolute error in proportion to the exact data value, thus offering a context-dependent assessment of the error's significance. An approximation error can manifest due to a multitude of diverse reasons. Prominent among these are limitations related to computing machine precision, where digital systems cannot represent all real numbers with perfect accuracy, leading to unavoidable truncation or rounding. Another common source is inherent measurement error, stemming from the practical limitations of inst ...
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On-Line Encyclopedia Of Integer Sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the OEIS Foundation in 2009, and is its chairman. OEIS records information on integer sequences of interest to both professional and amateur mathematicians, and is widely cited. , it contains over 370,000 sequences, and is growing by approximately 30 entries per day. Each entry contains the leading terms of the sequence, keywords, mathematical motivations, literature links, and more, including the option to generate a graph or play a musical representation of the sequence. The database is searchable by keyword, by subsequence, or by any of 16 fields. There is also an advanced search function called SuperSeeker which runs a large number of different algorithms to identify sequences related to the input. History Neil Sloane started col ...
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Stirling Series Relative Error
Stirling (; ; ) is a city in central Scotland, northeast of Glasgow and north-west of Edinburgh. The market town, surrounded by rich farmland, grew up connecting the royal citadel, the medieval old town with its merchants and tradesmen, the Old Bridge and the port. Located on the River Forth, Stirling is the administrative centre for the Stirling council area, and is traditionally the county town and historic county of Stirlingshire. Stirling's key position as the lowest bridging point of the River Forth before it broadens towards the Firth of Forth made it a focal point for travel north or south. It has been said that "Stirling, like a huge brooch clasps Highlands and Lowlands together". The city's status as "Gateway to the Highlands" also historically lent it great strategic importance—the credo "he who holds Stirling, holds Scotland" is sometimes attributed to Robert the Bruce. When Stirling was temporarily under Anglo-Saxon sway, according to a 9th-century legend, it ...
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Probability Density Function
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would be equal to that sample. Probability density is the probability per unit length, in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. More precisely, the PDF is used to specify the probability of the random variable falling ''within ...
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Central Limit Theorem
In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distribution, standard normal distribution. This holds even if the original variables themselves are not Normal distribution, normally distributed. There are several versions of the CLT, each applying in the context of different conditions. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions. This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern form it was only precisely stated as late as 1920. In statistics, the CLT can be stated as: let X_1, X_2, \dots, X_n denote a Sampling ...
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Normal Distribution
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac e^\,. The parameter is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma^2 is the variance. The standard deviation of the distribution is (sigma). A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution c ...
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Poisson Distribution
In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event. It can also be used for the number of events in other types of intervals than time, and in dimension greater than 1 (e.g., number of events in a given area or volume). The Poisson distribution is named after French mathematician Siméon Denis Poisson. It plays an important role for discrete-stable distributions. Under a Poisson distribution with the expectation of ''λ'' events in a given interval, the probability of ''k'' events in the same interval is: :\frac . For instance, consider a call center which receives an average of ''λ ='' 3 calls per minute at all times of day. If the calls are independent, receiving one does not change the probability of when the next on ...
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Method Of Steepest Descent
In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase. The saddle-point approximation is used with integrals in the complex plane, whereas Laplace’s method is used with real integrals. The integral to be estimated is often of the form :\int_Cf(z)e^\,dz, where ''C'' is a contour, and λ is large. One version of the method of steepest descent deforms the contour of integration ''C'' into a new path integration ''C′'' so that the following conditions hold: # ''C′'' passes through one or more zeros of the derivative ''g''′(''z''), # the imaginary part of ''g''(''z'') is constant on ''C′''. The method of steepest descent was first published by , who used it to estimate Bessel functions and pointed out that it occurred in the ...
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Cauchy's Integral Formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under uniform limits – a result that does not hold in real analysis. Theorem Let be an open subset of the complex plane , and suppose the closed disk defined as D = \bigl\ is completely contained in . Let be a holomorphic function, and let be the circle, oriented counterclockwise, forming the boundary of . Then for every in the interior of , f(a) = \frac \oint_\gamma \frac\,dz.\, The proof of this statement uses the Cauchy integral theorem and like that theorem, it only requires to ...
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