how to find closed form of summation

"Closed formula" redirects here. For "closed formula" in the sense of a logic formula with no free variables, see Sentence (mathematical logic).

Mathematical formula built with arithmetic operations and other previously defined functions

In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (east.g., + − × ÷), and functions (e.g., nthursday root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), simply ordinarily no limit, differentiation, or integration. The set of operations and functions may vary with writer and context.

Example: roots of polynomials [edit]

The solutions of any quadratic equation with complex coefficients tin can exist expressed in closed form in terms of add-on, subtraction, multiplication, division, and foursquare root extraction, each of which is an elementary function. For instance, the quadratic equation

${\displaystyle ax^{2}+bx+c=0,}$

is tractable since its solutions can exist expressed as a closed-class expression, i.east. in terms of simple functions:

${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}$

Similarly solutions of cubic and quartic (tertiary and 4th degree) equations tin be expressed using arithmetic, square roots, and nthursday roots. Even so, at that place are quintic equations without such closed-form solutions, for example x ^five −ten + i = 0; this is Abel–Ruffini theorem.

The study of the existence of closed forms for polynomial roots is the initial motivation and one of the main achievements of the expanse of mathematics named Galois theory.

Alternative definitions [edit]

Changing the definition of "well known" to include additional functions can change the set of equations with airtight-grade solutions. Many cumulative distribution functions cannot be expressed in closed grade, unless one considers special functions such equally the error part or gamma office to exist well known. It is possible to solve the quintic equation if general hypergeometric functions are included, although the solution is far besides complicated algebraically to be useful. For many practical computer applications, information technology is entirely reasonable to assume that the gamma function and other special functions are well known since numerical implementations are widely bachelor.

Analytic expression [edit]

An analytic expression (or expression in analytic grade) is a mathematical expression constructed using well-known operations that lend themselves readily to calculation.^{[ vague ] [ citation needed ]} Similar to airtight-course expressions, the prepare of well-known functions allowed tin vary according to context but always includes the basic arithmetics operations (addition, subtraction, multiplication, and partitioning), exponentiation to a existent exponent (which includes extraction of the n th root), logarithms, and trigonometric functions.

However, the class of expressions considered to be analytic expressions tends to be wider than that for airtight-course expressions. In particular, special functions such as the Bessel functions and the gamma function are ordinarily allowed, and oftentimes so are infinite serial and continued fractions. On the other mitt, limits in general, and integrals in detail, are typically excluded.^{[ commendation needed ]}

If an analytic expression involves only the algebraic operations (add-on, subtraction, multiplication, sectionalisation, and exponentiation to a rational exponent) and rational constants and then it is more specifically referred to as an algebraic expression.

Comparison of unlike classes of expressions [edit]

Closed-form expressions are an important sub-class of analytic expressions, which comprise a divisional^{[ commendation needed ]} or an unbounded number of applications of well-known functions. Unlike the broader analytic expressions, the closed-grade expressions do not include infinite series or connected fractions; neither includes integrals or limits. Indeed, by the Stone–Weierstrass theorem, any continuous function on the unit interval can be expressed as a limit of polynomials, then any class of functions containing the polynomials and closed under limits volition necessarily include all continuous functions.

Similarly, an equation or organisation of equations is said to accept a closed-course solution if, and only if, at to the lowest degree ane solution can be expressed as a airtight-grade expression; and it is said to have an analytic solution if and only if at least i solution can exist expressed as an analytic expression. There is a subtle distinction between a "closed-form office" and a "closed-form number" in the give-and-take of a "closed-form solution", discussed in (Grub 1999) and below. A closed-grade or analytic solution is sometimes referred to as an explicit solution.

v t due east	Arithmetic expressions	Polynomial expressions	Algebraic expressions	Closed-form expressions	Analytic expressions	Mathematical expressions
Constant	Yes	Yes	Yes	Yes	Yeah	Yes
Unproblematic arithmetics operation	Yes	Addition, subtraction, and multiplication only	Yes	Yes	Yes	Yes
Finite sum	Yes	Yes	Yep	Aye	Yes	Yes
Finite product	Yeah	Yes	Yes	Yes	Yes	Yes
Finite continued fraction	Yes	No	Yes	Yes	Yes	Yes
Variable	No	Aye	Yes	Yes	Yes	Aye
Integer exponent	No	Yes	Aye	Aye	Yes	Aye
Integer nth root	No	No	Yes	Aye	Yes	Yeah
Rational exponent	No	No	Yes	Yes	Yeah	Yep
Integer factorial	No	No	Yeah	Yes	Yep	Yes
Irrational exponent	No	No	No	Yep	Yes	Yes
Logarithm	No	No	No	Yes	Yep	Yes
Trigonometric function	No	No	No	Aye	Yeah	Yes
Inverse trigonometric office	No	No	No	Yes	Yes	Yep
Hyperbolic function	No	No	No	Yeah	Aye	Yes
Inverse hyperbolic office	No	No	No	Yes	Yes	Yes
Root of a polynomial that is not an algebraic solution	No	No	No	No	Yes	Yes
Gamma role and factorial of a non-integer	No	No	No	No	Yes	Aye
Bessel role	No	No	No	No	Yes	Yes
Special role	No	No	No	No	Yes	Yes
Infinite sum (series) (including power serial)	No	No	No	No	Convergent only	Yes
Infinite production	No	No	No	No	Convergent but	Yep
Infinite connected fraction	No	No	No	No	Convergent only	Aye
Limit	No	No	No	No	No	Yeah
Derivative	No	No	No	No	No	Yes
Integral	No	No	No	No	No	Yes

Dealing with non-closed-form expressions [edit]

Transformation into closed-form expressions [edit]

The expression:

$${\displaystyle f(ten)=\sum _{i=0}^{\infty }{\frac {ten}{2^{i}}}}$$

is not in closed form because the summation entails an infinite number of simple operations. However, past summing a geometric series this expression tin can be expressed in the airtight course:^[i]

$${\displaystyle f(x)=2x.}$$

Differential Galois theory [edit]

The integral of a closed-form expression may or may not itself exist expressible as a closed-form expression. This study is referred to equally differential Galois theory, by analogy with algebraic Galois theory.

The basic theorem of differential Galois theory is due to Joseph Liouville in the 1830s and 1840s and hence referred to as Liouville'south theorem.

A standard case of an elementary role whose antiderivative does not have a closed-form expression is:

$${\displaystyle e^{-10^{ii}},}$$

whose one antiderivative is (upward to a multiplicative abiding) the error role:

$${\displaystyle \operatorname {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt.}$$

Mathematical modelling and computer simulation [edit]

Equations or systems too complex for airtight-form or analytic solutions can frequently be analysed by mathematical modelling and estimator simulation.

Airtight-form number [edit]

Three subfields of the complex numbers C have been suggested as encoding the notion of a "closed-form number"; in increasing order of generality, these are the Liouvillian numbers (non to be dislocated with Liouville numbers in the sense of rational approximation), EL numbers and uncomplicated numbers. The Liouvillian numbers, denoted Fifty , grade the smallest algebraically closed subfield of C closed under exponentiation and logarithm (formally, intersection of all such subfields)—that is, numbers which involve explicit exponentiation and logarithms, but allow explicit and implicit polynomials (roots of polynomials); this is defined in (Ritt 1948, p. sixty). 50 was originally referred to every bit uncomplicated numbers, but this term is now used more broadly to refer to numbers defined explicitly or implicitly in terms of algebraic operations, exponentials, and logarithms. A narrower definition proposed in (Chow 1999, pp. 441–442), denoted East , and referred to as EL numbers, is the smallest subfield of C closed under exponentiation and logarithm—this need non be algebraically airtight, and stand for to explicit algebraic, exponential, and logarithmic operations. "EL" stands both for "exponential–logarithmic" and as an abbreviation for "elementary".

Whether a number is a closed-form number is related to whether a number is transcendental. Formally, Liouvillian numbers and uncomplicated numbers contain the algebraic numbers, and they include some simply not all transcendental numbers. In contrast, EL numbers do not incorporate all algebraic numbers, simply do include some transcendental numbers. Closed-grade numbers tin exist studied via transcendental number theory, in which a major consequence is the Gelfond–Schneider theorem, and a major open question is Schanuel'due south conjecture.

Numerical computations [edit]

For purposes of numeric computations, being in closed form is not in general necessary, every bit many limits and integrals can be efficiently computed.

Conversion from numerical forms [edit]

At that place is software that attempts to find closed-form expressions for numerical values, including RIES,^[2] place in Maple^[three] and SymPy,^[4] Plouffe's Inverter,^[5] and the Inverse Symbolic Computer.^[6]

See also [edit]

• Algebraic solution
• Finitary operation
• Numerical solution
• Computer simulation
• Symbolic regression
• Term (logic)
• Liouvillian function
• Elementary function

References [edit]

1. ^ Holton, Glyn. "Numerical Solution, Airtight-Form Solution". Archived from the original on 4 Feb 2012. Retrieved 31 Dec 2012.
2. ^ Munafo, Robert. "RIES - Notice Algebraic Equations, Given Their Solution". Retrieved 30 April 2012.
3. ^ "identify". Maple Online Help. Maplesoft. Retrieved 30 April 2012.
4. ^ "Number identification". SymPy documentation. Archived from the original on 2018-07-06. Retrieved 2016-12-01 .
5. ^ "Plouffe'south Inverter". Archived from the original on 19 April 2012. Retrieved 30 April 2012.
6. ^ "Inverse Symbolic Calculator". Archived from the original on 29 March 2012. Retrieved 30 April 2012.

Further reading [edit]

• Ritt, J. F. (1948), Integration in finite terms
• Grub, Timothy Y. (May 1999), "What is a Airtight-Form Number?", American Mathematical Monthly, 106 (five): 440–448, arXiv:math/9805045, doi:10.2307/2589148, JSTOR 2589148
• Jonathan M. Borwein and Richard E. Crandall (January 2013), "Closed Forms: What They Are and Why We Care", Notices of the American Mathematical Society, 60 (1): 50–65, doi:ten.1090/noti936

External links [edit]

• Weisstein, Eric W. "Airtight-Grade Solution". MathWorld.

Source: https://en.wikipedia.org/wiki/Closed-form_expression

Posted by: santanafaccons.blogspot.com

Share this post