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Topic: Factorial function

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Gamma function

Recursion

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	Gamma function - Wikipedia, the free encyclopedia
		In mathematics, the Gamma function extends the factorial function to complex and non-integer numbers (it is already defined on the naturals, and has simple poles at the negative integers).
		The derivatives of the Gamma function are described in terms of the polygamma function.
		The derivative of the logarithm of the Gamma function is called the digamma function; higher derivatives are the polygamma functions.
	en.wikipedia.org /wiki/Gamma_function (872 words)

	Recursion (JScript) (Site not responding. Last check: 2007-10-31)
		To calculate the factorial of any number that is larger than zero, you must calculate the factorial of at least one other number.
		These resources are freed when the recursive function exits, but a function that has too many levels of recursion may use all the available resources.
		Otherwise, a recursive function is used to calculate the factorial.
	msdn.microsoft.com /library/en-us/jscript7/html/recurse.asp?frame=true (680 words)

	Use functional programming techniques to write elegant JavaScript (Site not responding. Last check: 2007-10-31)
		Functional programming describes only the operations to be performed on the inputs to the programs, without use of temporary variables to store intermediate results.
		Functional programming concepts, including anonymous functions, different ways to call functions, and how to pass functions as arguments to other functions.
		Use of functional concepts, with examples of: extending array sort; elegant code for dynamic HTML generation; and application of a sequence of functions.
	www.ibm.com /developerworks/java/library/wa-javascript.html?ca=drs- (1786 words)

	Programming:Haskell Preliminaries - Wikibooks, collection of open-content textbooks
		This refers to the fact that a function and a parameter can always be replaced by the function's return value: for example, 2+2 is (almost) completely indistinguishable from 4.
		The first line says that the factorial of 0 is 1, and the second one says that the factorial of any other number "n" is equal to "n" times the factorial of "n-1".
		A programmer using this function doesn't want to worry about the accumulator argument, so the plain factorial function is used to "wrap up" factorial' by supplying the initial value of the accumulator.
	en.wikibooks.org /wiki/Programming:Haskell_Preliminaries (1175 words)

	CIS Department > Tutorials > Software Design Using C++ > Recursion
		Otherwise, we have the recursive case, where the factorial function calls itself (but with a parameter that is smaller by 1).
		One method is to skip the run-time stack and simply show in boxes the original function call and then the code that it generates, followed by any code that that one generates due to a recursive call, etc. Arrows are used to link each call to the code that it generates.
		function calls itself using the array, 0, and 1 as the parameters, in effect asking that the array section from index 0 to index 1 be sorted.
	cis.stvincent.edu /html/tutorials/swd/recur/recur.html (3217 words)

	[No title]
		Functions can be defined interactively at the command line as well as in independent MSH scripts, and they are invoked the same way that Cmdlets are called.
		The flexibility of proper first class functions is an immense asset for experienced developers, and their availability in MSH substantially increases the elegance of the language.
		Functions can also optionally have these blocks, and you can use the processs block to manipulate content passed into the function with a pipe.
	arstechnica.com /guides/other/msh.ars/9 (1004 words)

	[No title]
		Function main does not know the values of variables C and D in Function1 and variables D and E in Function2.
		No other function will know the value of D or the value of E or the value of A in Function2 or can change them except if they were passed as reference arguments to that function.
		I'll present 4 examples on recursive functions; Factorial, Sum from 1 to N, xN, where N is a positive integer and x is a real number, and a sequence defined recursively.
	www.math.unl.edu /~ijeib/CSIT121_Fall_01/Handout20_Part4.doc (967 words)

	Features
		Once I explained that the factorial of any non-negative integer is defined as the product of all integers from one to the chosen integer, inclusive, I added in passing that I had thought about the factorial function over Thanksgiving dinner.
		I refer to that calculation as the "clink function" and the factorial function is the foundation of it.
		The factorial function is undefined for non-integers and for negative integers.
	www.sas.org /E-Bulletin/2001-12-28/features/features.html (559 words)

	SparkNotes: Discrete Functions: Introduction and Summary
		A function that is defined only for a set of numbers that can be listed, such as the set of whole numbers or the set of integers, is called a discrete function.
		The second section introduces two functions that are derived from the factorial function -- the permutation function and the combination function.
		One particularly interesting function that cannot easily be defined explicitly yields the Fibonacci numbers, which are explored at the end of this section.
	www.sparknotes.com /math/algebra2/discretefunctions/summary.html (293 words)

	All about factorial notation n!
		Because of this relationship, the gamma function is often thought of as a generalization of the factorial function to the domain of complex numbers.
		Uniqueness—The gamma function is the only function which satisfies the mentioned recursive relationship for the domain of complex numbers and is holomorphic and whose restriction to the positive real axis is log-convex.
		Context—The gamma function is generally used in a context similar to that of the factorials (but, of course, where a more general domain is of interest).
	factorielle.free.fr /index_en.html (830 words)

	Halfbakery: Inverse Factorial '?'
		Inverse factorial would only be possible for a very small percentage of numbers, ie those that factorise completely to give consecutive factors right down to 1.
		In trig, the inverse of a function is denoted by a minus one exponent.
		If you can make the leap to accepting factorials of non-natural numbers as a valid concept, then it would follow that should be inverse factorials for numbers that are not factorials of natural numbers.
	www.halfbakery.com /idea/Inverse_20Factorial_20_27_3f_27 (680 words)

	Notes on Recursive Function Calls
		function, it should be fixed within the function.
		Notice that part of this function implements a simple way of finding the last digit of any positive integer.
		This will be true once the function has been called recursively many times.
	andromeda.rutgers.edu /~loftin/cs101spr04/recursive/recursive.html (1062 words)

	Recursive Functions
		Informally, a function defined by recursion is one that computes the result of a call by "calling itself".
		This is a recursive definition of the factorial function, which is ordinarily defined in textbooks by the recursion equations
		There is clearly a close correspondence between the ML notation and the mathematical notation for the factorial function.
	www.cs.cmu.edu /People/rwh/introsml/core/recfns.htm (1265 words)

	Factorial and Gamma Functions (Site not responding. Last check: 2007-10-31)
		function over the complex plane using the algorithm of Kuki, CACM algorithm 421.
		is the general function and may be called with a symbolic or numeric argument.
		function; and for complex numeric arguments, it uses the Kuki algorithm.
	starship.python.net /crew/mike/maxima/html/macref/node24.html (441 words)

	The Factorial Tutorial
		The most important application of the factorial function is a pedagogical one: it is one of the classic exercises in first semester programming.
		The factorial function does not benefit from an optimistic implementation, and so should not be written in E. It can be implemented more efficiently in Java.
		This form of the factorial function is rarely seen in programming texts, because the purpose of the factorial function in general is didactic, not mathematical.
	www.crockford.com /ec/factorial.html (930 words)

	The Factorial Tutorial
		The factorial function is a special case of Pochhammer's polynomial and of the gamma function (aka Euler's integral of the second kind).
		The conventional mathematical notation for the factorial of n is
		The factorial function is used in the study of permutations.
	www.erights.org /history/original-e/factorial.html (790 words)

	A NEW SOLUTION FOR THE PROBABILITY OF COMPLETING SETS IN RANDOM SAMPLING: DEFINITION OF A "TWO-DIMENSIONAL FACTORIAL"
		In developing an improved solution to a classical sampling problem, a new recursive function is obtained which can be termed a "two-dimensional factorial." The sampling problem deals with the probability of completing a subset of unique items when randomly sampled with replacement from a finite population.
		The two-dimensional factorial is partially tabulated, and several of its properties are investigated, including limits for large numbers.
		While the resemblance to the regular factorial function is somewhat superficial, the two-dimensional factorial is still suggested as an appropriate name for the recursive F function introduced here.
	www.jefflindsay.com /2dfactorial.shtml (2435 words)

	Factorial MathProbability Function -- Entisoft Tools 2.0 Object Library (Site not responding. Last check: 2007-10-31)
		Return the factorial of the number X. The factorial of X is the product of every number between 1 and X, except that the factorial of 1 (one) is 1.
		Function returns Null if vX is Null, cannot be interpreted as a number, or is not between 0 (zero) and 170 (the result would be greater than MaxDouble).
		Note: This function will not generate the factorial of a floating point number; it will ignore the fractional part of any floating point numbers by applying the Visual Basic Fix function to the number.
	www.entisoft.com /ESTools/MathProbability_Factorial.HTML (188 words)

	Prolog Tutorial -- 2.2
		Two predicate definitions that calculate the factorial function are in file 2_2.pl, which the reader can view by clicking on the 'Code' link at the bottom of this page.
		A declarative reading of the first (unit) clause says that "the factorial of 0 is 1" and the second clause declares that "the factorial of N is F if N>0 and N1 is N-1 and the factorial of N1 is F1 and F is N*F1".
		All of the arithmetic leaves are true by evaluation (under the intended interpretation), and the lowest link in the tree corresponds to the very first clause of the program for factorial.
	www.csupomona.edu /~jrfisher/www/prolog_tutorial/2_2.html (813 words)

	DeFunctionalization - The Haskell Wiki (Site not responding. Last check: 2007-10-31)
		To perform it, you first need to identify the apply function (it may be inlined, so this isn't necessarily straightforward).
		Or defunctionalization can be thought of as closure conversion followed by moving the functions body out of line.
		The actual representation of the environment is completely unconstrained as the only code that can access it is the body of the function.
	www.haskell.org /hawiki/DeFunctionalization (529 words)

	4GuysFromRolla.com - Recursion, why it's cool.
		The factorial of N is N * the factorial of N-1.
		All recursive functions must have an exit condition, that is a state when it does not recurse upon itself.
		This function is called by passing in the Key of the node that you want to start at.
	www.aspfaqs.com /webtech/060299-2.shtml (1035 words)

	Web Crossing Server-side JavaScript - A Non-Programmer's Guide to Web Crossing
		JavaScript functions defined with the function keyword are hidden from the outside and cannot be called directly from a browser using the URL syntax.
		In the second line our factorial function is called using the local variable n as a parameter.
		The factorial function is explained in the previous section.
	www.seniornet.org /php/images/webximages/docs/Guide/pages/js-03-forms.html (832 words)

	SparkNotes: Discrete Functions: The Factorial Function
		The factorial function F(n) is also represented as "n!
		To find the total number of possibilities when choices are not distinct, divide by the factorial of the number of choices that are the same.
		If 2 choices are the same as each other, and 2 different choices are the same as each other, divide by 2!2!.
	www.sparknotes.com /math/algebra2/discretefunctions/section1.html (331 words)

	Lecture 3 - Recursion As A Programming Technique
		Every invocation of a function call may require space for parameters and local variables, and for an indication of where to return when the function is finished.
		The operations involved in calling a function - allocating, and later releasing, local memory, copying values into the local memory for the parameters, branching to/returning from the function - all contribute to the time overhead.
		There is very little overhead in calling this function, as it has only one word of local memory, for the parameter n.
	www.csi.uottawa.ca /~holte/T26/rec-prog-tech.html (1261 words)

	Writing lisp arithmetic functions
		To take a simple example, let's write a function that returns the slope of the line joining two points.
		Revise the above function to handle the case where a is 0.
		The error function's first argument is a format string, similar to the C language format strings used with printf.
	www.csm.astate.edu /~rossa/cs3543/lispfunctions.html (724 words)

	The Gamma Function: Leonhard Euler's Generalization of the Factorial Function
		The Gamma Function: Leonhard Euler's Generalization of the Factorial Function
		In 1729 Euler proposed a generalization of the factorial function n!=n(n-1)(n-2)··3·2·1.
		The crucial relationship for the gamma function is Γ(x+1)=xΓ(x).
	www.applet-magic.com /gammaf.htm (172 words)

	Scott Bellware [MVP] : Test-Driven Development Example: Factorial
		If I knew that it was important to me to also test a factorial of 3 and of 5, than I would add these to a todo list for testing the function, but I wouldn’t cram them into the same test.
		Since in the first phase of the TDD cycle the test is being calibrating for proper detection of a failure, I want to implement the first iteration of the Factorial method so that it is guaranteed to meaningfully fail the test.
		The requirements state that a factorial of 0 causes an exception to be thrown.
	codebetter.com /blogs/scott.bellware/archive/2005/11/23/135001.aspx (1743 words)

	Unit SpecFun/SpecF87
		The incomplete gamma function is important because of its
		The incomplete beta function is important because of its
		function of the absolute value of gaussian random deviates.
	www.jwrider.com /lab/SPECFUN.htm (720 words)

	Python Journal 2(2) Recursion, by A.J. Chung
		The main feature of a recursive function is that at some point during execution of the function it will make a call to itself.
		This function particularly lends itself to recursion due to the definition of the Fibonacci sequence: The first two numbers are 1, and subsequent numbers are derived by summing the previous two in the sequence.
		The function returns the number of solutions found which must be totalled up during execution.
	pythonjournal.cognizor.com /pyj2.2/RecursionByAJChung.html (2091 words)

	Factorial versus Gamma Function. (Site not responding. Last check: 2007-10-31)
		The normalization of the gamma function to Gamma(n+1) instead of Gamma(n) is due to Legendre and void of any rationality.
		The Gamma function is the Mellin transform of the exponential function.
		Still, there is another way to look at the relation between the factorial function and the Gamma function when we look at the (generalized) factorial powers -- z and w arbitrary complex numbers:
	www.luschny.de /math/factorial/factandgamma.html (359 words)

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