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In mathematical logic, a formal calculation, or formal operation, is a calculation that is systematic but without a rigorous justification. It involves manipulating symbols in an expression using a generic substitution without proving that the necessary conditions hold. Essentially, it involves the form of an expression without considering its underlying meaning. This reasoning can either serve as positive evidence that some statement is true when it is difficult or unnecessary to provide proof or as an inspiration for the creation of new (completely rigorous) definitions.
However, this interpretation of the term formal is not universally accepted, and some consider it to mean quite the opposite: a completely rigorous argument, as in formal mathematical logic.
Formal calculations can lead to results that are wrong in one context, but correct in another context. The equation
holds if q has an absolute value less than 1. Ignoring this restriction, and substituting q = 2 to leads to
Substituting q=2 into the proof of the first equation, yields a formal calculation that produces the last equation. But it is wrong over the real numbers, since the series does not converge. However, in other contexts (e.g. working with 2-adic numbers, or with integers modulo a power of 2), the series does converge. The formal calculation implies that the last equation must be valid in those contexts.
Another example is obtained by substituting q=-1. The resulting series 1-1+1-1+... is divergent (over the real and the p-adic numbers) but a value can be assigned to it with an alternative method of summation, such as Cesàro summation. The resulting value, 1/2, is the same as that obtained by the formal computation.
Formal power series is a concept that adopts the form of power series from real analysis. The word "formal" indicates that the series need not converge. In mathematics, and especially in algebra, a formal series is an infinite sum that is considered independently from any notion of convergence and can be manipulated with algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.).
A formal power series is a special kind of formal series, which may be viewed as a generalization of a polynomial, where the number of terms is allowed to be infinite, with no requirements of convergence. Thus, the series may no longer represent a function of its variable, merely a formal sequence of coefficients, in contrast to a power series, which defines a function by taking numerical values for the variable within a radius of convergence. In a formal power series, the powers of the variable are used only as position-holders for the coefficients, so that the coefficient of is the fifth term in the sequence. In combinatorics, the method of generating functions uses formal power series to represent numerical sequences and multisets, for instance allowing concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved. More generally, formal power series can include series with any finite (or countable) number of variables, and with coefficients in an arbitrary ring.
Rings of formal power series are complete local rings, which supports calculus-like methods in the purely algebraic framework of algebraic geometry and commutative algebra. They are analogous to p-adic integers, which can be defined as formal series of the powers of p.
To solve the differential equation
these symbols can be treated as ordinary algebraic symbols, and without giving any justification regarding the validity of this step, we take reciprocals of both sides:
A simple antiderivative:
Because this is a formal calculation, it is acceptable to let and obtain another solution:
The final solutions can be checked to confirm that they solve the equation.
The cross product can be expressed as the following determinant:
where is a positively oriented orthonormal basis of a three-dimensional oriented Euclidean vector space, while are scalars such that , and similar for .
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century.
In probability theory, the central limit theorem (CLT) establishes that, in many situations, for independent and identically distributed random variables, the sampling distribution of the standardized sample mean tends towards the standard normal distribution even if the original variables themselves are not normally distributed.
In number theory, given a prime number p, the p-adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; p-adic numbers can be written in a form similar to decimals, but with digits based on a prime number p rather than ten, and extending to the left rather than to the right. Formally, given a prime number p, a p-adic number can be defined as a series
A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Common forms of the Fourier series below.
In mathematics, a power series is an infinite series of the form
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series.
In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function.
In mathematics, a system of linear equations is a collection of one or more linear equations involving the same variables.
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-sides of the equations. It is named after Gabriel Cramer (1704–1752), who published the rule for an arbitrary number of unknowns in 1750, although Colin Maclaurin also published special cases of the rule in 1748.
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region.
In linear algebra, an n-by-n square matrix A is called invertible, if there exists an n-by-n square matrix B such that
In mathematics, a quadratic form is a polynomial with terms all of degree two. For example,
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function.
In mathematics, an algebraic equation or polynomial equation is an equation of the form
In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable in the input dataset and the output of the (linear) function of the independent variable.
In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series
In mathematics, and more specifically in analysis, a holonomic function is a smooth function of several variables that is a solution of a system of linear homogeneous differential equations with polynomial coefficients and satisfies a suitable dimension condition in terms of D-modules theory. More precisely, a holonomic function is an element of a holonomic module of smooth functions. Holonomic functions can also be described as differentiably finite functions, also known as D-finite functions. When a power series in the variables is the Taylor expansion of a holonomic function, the sequence of its coefficients, in one or several indices, is also called holonomic. Holonomic sequences are also called P-recursive sequences: they are defined recursively by multivariate recurrences satisfied by the whole sequence and by suitable specializations of it. The situation simplifies in the univariate case: any univariate sequence that satisfies a linear homogeneous recurrence relation with polynomial coefficients, or equivalently a linear homogeneous difference equation with polynomial coefficients, is holonomic.
In mathematics, 1 + 2 + 4 + 8 + ⋯ is the infinite series whose terms are the successive powers of two. As a geometric series, it is characterized by its first term, 1, and its common ratio, 2. As a series of real numbers it diverges to infinity, so the sum of this series is infinity.
Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.
Tau functions are an important ingredient in the modern mathematical theory of integrable systems, and have numerous applications in a variety of other domains. They were originally introduced by Ryogo Hirota in his direct method approach to soliton equations, based on expressing them in an equivalent bilinear form.