Citizendia
Your Ad Here

"Series expansion" redirects here. For other notions of the term, see series. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with
As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sinx and Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.
As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sinx and Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.
The exponential function (in blue), and the sum of the first n+1 terms of its Taylor series at 0 (in red).
The exponential function (in blue), and the sum of the first n+1 terms of its Taylor series at 0 (in red). The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x)

In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change It may be regarded as the limit of the Taylor polynomials. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" In Calculus, Taylor's theorem gives a sequence of approximations of a Differentiable function around a given point by Polynomials (the Taylor Taylor series are named in honour of English mathematician Brook Taylor. The English people (from the adjective in Englisc) are a Nation and Ethnic group native to England who predominantly speak English Brook Taylor ('teɪlə(r ( 18 August 1685 &ndash 30 November 1731) was an English mathematician If the series uses the derivatives at zero, the series is also called a Maclaurin series, named after Scottish mathematician Colin Maclaurin. The Scots people ( Scots Gaelic: Albannaich) are a Nation and an Ethnic group indigenous to Scotland. Colin Maclaurin (February 1698 &ndash June 14, 1746) was a Scottish Mathematician.

Contents

Definition

The Taylor series of a real or complex function f(x) that is infinitely differentiable in a neighborhood of a real or complex number a, is the power series

f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+\cdots\,,

which in a more compact form can be written as

\sum_{n=0}^{\infin} \frac{f^{(n)}(a)}{n!} (x-a)^{n}\,,

where n! is the factorial of n and f (n)(a) denotes the nth derivative of f evaluated at the point a; the zeroth derivative of f is defined to be f itself and (x − a)0 and 0! are both defined to be 1. In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + Definition The factorial function is formally defined by n!=\prod_{k=1}^n k In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change

Often f(x) is equal to its Taylor series evaluated at x for all x sufficiently close to a. This is the main reason why Taylor series are important.

In the particular case where a = 0, the series is also called a Maclaurin series.

Examples

The Maclaurin series for any polynomial is the polynomial itself. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations

The Maclaurin series for (1 − x) − 1 is the geometric series

1+x+x^2+x^3+\cdots\!

so the Taylor series for x − 1 at a = 1 is

1-(x-1)+(x-1)^2-(x-1)^3+\cdots\!. In Mathematics, a geometric series is a series with a constant ratio between successive terms.

By integrating the above Maclaurin series we find the Maclaurin series for -\ln(1 - x)\!, where \ln\! denotes the natural logarithm:

x+\frac{x^2}2+\frac{x^3}3+\frac{x^4}4+\cdots\!

and the corresponding Taylor series for \ln(x)\! at a=1\! is

(x-1)-\frac{(x-1)^2}2+\frac{(x-1)^3}3-\frac{(x-1)^4}4+\cdots\!. The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational

The Taylor series for the exponential function ex at a = 0 is

1 + \frac{x^1}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!}+ \cdots \qquad = \qquad 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \frac{x^5}{120} + \cdots\!. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x)

The above expansion holds because the derivative of ex is also ex and e0 equals 1. This leaves the terms (x − 0)n in the numerator and n! in the denominator for each term in the infinite sum.

Convergence

The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full period centered at the origin.
The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full period centered at the origin.
The Taylor polynomials for log(1+x) only provide accurate approximations in the range -1 < x ≤ 1. Note that, for x > 1, the Taylor polynomials of higher degree are worse approximations.
The Taylor polynomials for log(1+x) only provide accurate approximations in the range -1 < x ≤ 1. Note that, for x > 1, the Taylor polynomials of higher degree are worse approximations.

Taylor series need not in general be convergent, but often they are. The limit of a convergent Taylor series of a function f need not in general be equal to the function value f(x), but often it is. If f(x) is equal to its Taylor series in a neighborhood of a, it is said to be analytic in this neighborhood. In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. This article is about both real and complex analytic functions If f(x) is equal to its Taylor series everywhere it is called entire. In Complex analysis, an entire function, also called an integral function is a complex-valued function that is holomorphic everywhere on the The exponential function ex and the trigonometric functions sine and cosine are examples of entire functions. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) Examples of functions that are not entire include the logarithm, the trigonometric function tangent, and its inverse arctan. In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce For these functions the Taylor series do not converge if x is far from a.

A Taylor series can be used to calculate the value of an entire function in every point, if the value of the function, and of all of its derivatives, are known at a single point. Uses of the Taylor series for entire functions include:

  1. The partial sums (the Taylor polynomials) of the series can be used as approximations of the entire function. In Calculus, Taylor's theorem gives a sequence of approximations of a Differentiable function around a given point by Polynomials (the Taylor These approximations are good if sufficiently many terms are included.
  2. The series representation simplifies many mathematical proofs. In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true

Pictured on the right is an accurate approximation of sin(x) around the point a = 0. The pink curve is a polynomial of degree seven:

\sin\left( x \right) \approx x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}\!.

The error in this approximation is no more than \tfrac{|x|^9}{9!}\!. In particular, for |x|<1\!, the error is less than 0. 000003.

In contrast, also shown is a picture of the function log(1+x) and some of its Taylor polynomials around a = 0. The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational In Calculus, Taylor's theorem gives a sequence of approximations of a Differentiable function around a given point by Polynomials (the Taylor These approximations converge to the function only in the region -1 < x ≤ 1; outside of this region the higher-degree Taylor polynomials are worse approximations for the function. This is an example of Runge's phenomenon. In the mathematical field of Numerical analysis, Runge's phenomenon is a problem that occurs when using Polynomial interpolation with polynomials of

Taylor's theorem gives a variety of general bounds on the size of the error in Rn(x) incurred in approximating a function by its nth order Taylor polynomial. In Calculus, Taylor's theorem gives a sequence of approximations of a Differentiable function around a given point by Polynomials (the Taylor

History

The Pythagorean philosopher Zeno considered the problem of summing an infinite series to achieve a finite result, but rejected it as an impossibility: the result was Zeno's paradox. Zeno of Elea (ˈziːnoʊ əv ˈɛliə Greek: Ζήνων ὁ Ἐλεάτης (ca Later, Aristotle proposed a philosophical resolution of the paradox, but the mathematical content was apparently unresolved until taken up by Democritus and then Archimedes. Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. Democritus ( Greek:) was a pre-Socratic Greek Materialist Philosopher (born at Abdera in Thrace ca Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer It was through Archimedes's method of exhaustion that an infinite number of progressive subdivisions could be performed to achieve a finite trigonometric result. The method of exhaustion is a method of finding the Area of a Shape by inscribing inside it a sequence of Polygons whose areas converge to the [1] Liu Hui independently employed a similar method a few centuries later. Liu Hui ( fl 3rd century) was a Chinese Mathematician who lived in the Wei Kingdom. [2]

In the 14th century, the earliest examples of the use of Taylor series and closely-related methods were given by Madhava of Sangamagrama. Mādhava of Sangamagrama (born as Irinjaatappilly Madhavan Namboodiri) (c [3] Though no record of his work survives, writings of later Indian mathematicians suggest that he found a number of special cases of the Taylor series, including those for the trigonometric functions of sine, cosine, tangent, and arctangent. Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. The Kerala school of astronomy and mathematics further expanded his works with various series expansions and rational approximations until the 16th century.

In the 17th century, James Gregory also worked in this area and published several Maclaurin series. As a means of recording the passage of Time, the 17th Century was that Century which lasted from 1601 - 1700 in the Gregorian calendar James Gregory (November 1638 &ndash October 1675 was a Scottish Mathematician and Astronomer. It was not until 1715 however that a general method for constructing these series for all functions for which they exist was finally provided by Brook Taylor[4], after whom the series are now named. Year 1715 ( MDCCXV) was a Common year starting on Tuesday (link will display the full calendar of the Gregorian calendar (or a Brook Taylor ('teɪlə(r ( 18 August 1685 &ndash 30 November 1731) was an English mathematician

The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, who published the special case of the Taylor result in the 18th century. Colin Maclaurin (February 1698 &ndash June 14, 1746) was a Scottish Mathematician.

Properties

The function e−1/x² is not analytic at x = 0: the Taylor series is identically 0, although the function is not.
The function e−1/x² is not analytic at x = 0: the Taylor series is identically 0, although the function is not.

If this series converges for every x in the interval (a − r, a + r) and the sum is equal to f(x), then the function f(x) is said to be analytic in the interval (a − r, a + r). This article is about both real and complex analytic functions If this is true for any r then the function is said to be an entire function. In Complex analysis, an entire function, also called an integral function is a complex-valued function that is holomorphic everywhere on the To check whether the series converges towards f(x), one normally uses estimates for the remainder term of Taylor's theorem. In Calculus, Taylor's theorem gives a sequence of approximations of a Differentiable function around a given point by Polynomials (the Taylor A function is analytic if and only if it can be represented as a power series; the coefficients in that power series are then necessarily the ones given in the above Taylor series formula. ↔ In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 +

The importance of such a power series representation is at least fourfold. First, differentiation and integration of power series can be performed term by term and is hence particularly easy. Second, an analytic function can be uniquely extended to a holomorphic function defined on an open disk in the complex plane, which makes the whole machinery of complex analysis available. This article is about both real and complex analytic functions Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane In Geometry, a disk (also spelled disc) is the region in a plane bounded by a Circle. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex Third, the (truncated) series can be used to compute function values approximately (often by recasting the polynomial into the Chebyshev form and evaluating it with the Clenshaw algorithm). In Mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of Orthogonal polynomials which are related to In the mathematical subfield of Numerical analysis the Clenshaw algorithm (Invented by Charles William Clenshaw) is a recursive method to evaluate

Fourth, algebraic operations can often be done much more readily on the power series representation; for instance the simplest proof of Euler's formula uses the Taylor series expansions for sine, cosine, and exponential functions. This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic This result is of fundamental importance in such fields as harmonic analysis. Harmonic analysis is the branch of Mathematics that studies the representation of functions or signals as the superposition of basic Waves It investigates and generalizes

Note that there are examples of infinitely differentiable functions f(x) whose Taylor series converge, but are not equal to f(x). In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability For instance, the function defined pointwise by f(x) = e−1/x² if x ≠ 0 and f(0) = 0 is an example of a non-analytic smooth function. In Mathematics, Smooth functions (also called infinitely differentiable functions and Analytic functions are two very important types of functions. All its derivatives at x = 0 are zero, so the Taylor series of f(x) at 0 is zero everywhere, even though the function is nonzero for every x ≠ 0. This particular pathology does not afflict Taylor series in complex analysis. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex There, the area of convergence of a Taylor series is always a disk in the complex plane (possibly with radius 0), and where the Taylor series converges, it converges to the function value. In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis Notice that e−1/z² does not approach 0 as z approaches 0 along the imaginary axis, hence this function is not continuous as a function on the complex plane.

Since every sequence of real or complex numbers can appear as coefficients in the Taylor series of an infinitely differentiable function defined on the real line, the radius of convergence of a Taylor series can be zero. [5] There are even infinitely differentiable functions defined on the real line whose Taylor series have a radius of convergence 0 everywhere. [6]

Some functions cannot be written as Taylor series because they have a singularity; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable x; see Laurent series. In Mathematics, a singularity is in general a point at which a given mathematical object is not defined or a point of an exceptional set where it fails to be In Mathematics, the Laurent series of a complex function f ( z) is a representation of that function as a Power series which includes terms For example, f(x) = e^{-1/x^2}\! can be written as a Laurent series.

The Parker-Sochacki method is a recent advance in finding Taylor series which are solutions to differential equations. In Mathematics, the Parker-Sochacki method is an Algorithm for solving systems of Differential equations which has been developed by G A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the This algorithm is an extension of the Picard iteration. In Numerical analysis, fixed point iteration is a method of computing fixed points of Iterated functions More specifically given a function f

List of Taylor series of some common functions

The cosine function in the complex plane.
The cosine function in the complex plane. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted
An 8th degree approximation of the cosine function in the complex plane.
An 8th degree approximation of the cosine function in the complex plane. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted
The two above curves put together.
The two above curves put together.

Several important Maclaurin series expansions follow. All these expansions are valid for complex arguments x\!.

Exponential function:

\mathrm{e}^{x} = \sum^{\infin}_{n=0} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\quad\mbox{ for all } x\!

Natural logarithm:

\ln(1-x) = -\sum^{\infin}_{n=1} \frac{x^n}n\quad\mbox{ for } |x| < 1\!

Finite geometric series:

\frac{1-x^{m + 1}}{1-x} = \sum^{m}_{n=0} x^n\quad\mbox{ for } x \not= 1\mbox{ and } m\in\mathbb{N}_0\!

Infinite geometric series:

\frac{1}{1-x} = \sum^{\infin}_{n=0} x^n\quad\mbox{ for } |x| < 1\!

Variants of the infinite geometric series:

\frac{x^m}{1-x} = \sum^{\infin}_{n=m} x^n\quad\mbox{ for } |x| < 1 \mbox{ and } m\in\mathbb{N}_0\!
\frac{x}{(1-x)^2} = \sum^{\infin}_{n=1}n x^n\quad\mbox{ for } |x| < 1\!

Square root:

\sqrt{1+x} = \sum_{n=0}^\infty \frac{(-1)^n(2n)!}{(1-2n)n!^24^n}x^n \quad\mbox{ for } |x|<1\!

Binomial series (includes the square root for α = 1/2 and the infinite geometric series for α = −1):

(1+x)^\alpha = \sum_{n=0}^\infty {\alpha \choose n} x^n\quad\mbox{ for all } |x| < 1 \mbox{ and all complex } \alpha\!
with generalized binomial coefficients
{\alpha\choose n} = \prod_{k=1}^n \frac{\alpha-k+1}k = \frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}\!

Trigonometric functions:

\sin x = \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1}\quad = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\mbox{ for all } x\!
\cos x = \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n}\quad = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\mbox{ for all } x\!
\tan x = \sum^{\infin}_{n=1} \frac{B_{2n} (-4)^n (1-4^n)}{(2n)!} x^{2n-1}\quad = x + \frac{x^3}{3} + \frac{2 x^5}{15} + \cdots\mbox{ for } |x| < \frac{\pi}{2}\!
where the Bs are Bernoulli numbers. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational In Mathematics, a geometric series is a series with a constant ratio between successive terms. In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose In Mathematics, the binomial series generalizes the purely algebraic formula of the Binomial theorem to complex values of α In Mathematics, the binomial coefficient \tbinom nk is the Coefficient of the x   k term in the Polynomial In Mathematics, the Bernoulli numbers are a Sequence of Rational numbers with deep connections to Number theory.
\sec x = \sum^{\infin}_{n=0} \frac{(-1)^n E_{2n}}{(2n)!} x^{2n}\quad\mbox{ for } |x| < \frac{\pi}{2}\!
\arcsin x = \sum^{\infin}_{n=0} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}\quad\mbox{ for } |x| < 1\!
\arctan x = \sum^{\infin}_{n=0} \frac{(-1)^n}{2n+1} x^{2n+1}\quad\mbox{ for } |x| \le 1\!

Hyperbolic functions:

\sinh x = \sum^{\infin}_{n=0} \frac{x^{2n+1}}{(2n+1)!} \quad\mbox{ for all } x\!
\cosh x = \sum^{\infin}_{n=0} \frac{x^{2n}}{(2n)!} \quad\mbox{ for all } x\!
\tanh x = \sum^{\infin}_{n=1} \frac{B_{2n} 4^n (4^n-1)}{(2n)!} x^{2n-1}\quad\mbox{ for } |x| < \frac{\pi}{2}\!
\mathrm{arsinh} (x) = \sum^{\infin}_{n=0} \frac{(-1)^n (2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}\quad\mbox{ for } |x| < 1\!
\mathrm{artanh} (x) = \sum^{\infin}_{n=0} \frac{x^{2n+1}}{2n+1} \quad\mbox{ for } |x| < 1\!

Lambert's W function:

W_0(x) = \sum^{\infin}_{n=1} \frac{(-n)^{n-1}}{n!} x^n\quad\mbox{ for } |x| < \frac{1}{\mathrm{e}}\!

The numbers B_k\! appearing in the summation expansions of tan(x) and tanh(x) are the Bernoulli numbers. In Mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular functions In Mathematics, The Lambert W function, named after Johann Heinrich Lambert, also called the Omega function or product log, is the In Mathematics, the Bernoulli numbers are a Sequence of Rational numbers with deep connections to Number theory. The E_k\! in the expansion of sec(x) are Euler numbers. For other uses see Euler number (topology and Eulerian number.

Calculation of Taylor series

Several methods exist for the calculation of Taylor series of a large number of functions. One can attempt to use the Taylor series as-is and generalize the form of the coefficients, or one can use manipulations such as substitution, multiplication or division, addition or subtraction of standard Taylor series to construct the Taylor series of a function, by virtue of Taylor series being power series. In some cases, one can also derive the Taylor series by repeatedly applying integration by parts. In Calculus, and more generally in Mathematical analysis, integration by parts is a rule that transforms the Integral of products of functions into other Particularly convenient is the use of computer algebra systems to calculate Taylor series. A computer algebra system ( CAS) is a software program that facilitates Symbolic mathematics.

First example

Compute the 7th degree Maclaurin polynomial for the function

f(x)=\ln\cos x, \quad x\in(-\pi/2, \pi/2)\!.

First, rewrite the function as

f(x)=\ln(1+(\cos x-1))\!.

We have for the natural logarithm (by using the big O notation)

\ln(1+x) = x - \frac{x^2}2 + \frac{x^3}3 + \mathcal{O}(x^4)\!

and for the cosine function

\cos x - 1 = -\frac{x^2}2 + \frac{x^4}{24} - \frac{x^6}{720} + \mathcal{O}(x^8)\!

The latter series expansion has a zero constant term, which enables us to substitute the second series into the first one and to easily omit terms of higher order than the 7th degree by using the big O notation:

\begin{align}f(x)&=\ln(1+(\cos x-1))\\ &=\bigl(\cos x-1\bigr) - \frac12\bigl(\cos x-1\bigr)^2 + \frac13\bigl(\cos x-1\bigr)^3+ \mathcal{O}\bigl((\cos x-1)^4\bigr)\\&=\biggl(-\frac{x^2}2 + \frac{x^4}{24} - \frac{x^6}{720} +\mathcal{O}(x^8)\biggr)-\frac12\biggl(-\frac{x^2}2+\frac{x^4}{24}+\mathcal{O}(x^6)\biggr)^2+\frac13\biggl(-\frac{x^2}2+\mathcal{O}(x^4)\biggr)^3 + \mathcal{O}(x^8)\\ & =-\frac{x^2}2 + \frac{x^4}{24}-\frac{x^6}{720} - \frac{x^4}8 + \frac{x^6}{48} - \frac{x^6}{24} +\mathcal{O}(x^8)\\ & =- \frac{x^2}2 - \frac{x^4}{12} - \frac{x^6}{45}+\mathcal{O}(x^8). \end{align}\!

Since the cosine is an even function, the coefficients for all the odd powers x, x3, x5, x7, . In mathematics big O notation (so called because it uses the symbol O) describes the limiting behavior of a function for very small or very large arguments In Mathematics, the constant term of a Polynomial is the term of degree 0 In Mathematics, even functions and odd functions are functions which satisfy particular Symmetry relations with respect to taking Additive  .  . have to be zero.

Second example

Suppose we want the Taylor series at 0 of the function

g(x)=\frac{e^x}{\cos x}\!.

We have for the exponential function

e^x = \sum^\infty_{n=0} {x^n\over n!} =1 + x + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!}+\cdots\!

and, as in the first example,

\cos x = 1 - {x^2 \over 2!} + {x^4 \over 4!} - \cdots\!

Assume the power series is

{e^x \over \cos x} = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots\!

Then multiplication with the denominator and substitution of the series of the cosine yields

\begin{align} e^x &= (c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots)\cos x\\ &=\left(c_0 + c_1 x + c_2 x^2 + c_3 x^3 + c_4x^4 + \cdots\right)\left(1 - {x^2 \over 2!} + {x^4 \over 4!} - \cdots\right)\\&=c_0 - {c_0 \over 2}x^2 + {c_0 \over 4!}x^4 + c_1x - {c_1 \over 2}x^3 + {c_1 \over 4!}x^5 + c_2x^2 - {c_2 \over 2}x^4 + {c_2 \over 4!}x^6 + c_3x^3 - {c_3 \over 2}x^5 + {c_3 \over 4!}x^7 +\cdots \end{align}\!

Collecting the terms up to fourth order yields

=c_0 + c_1x + \left(c_2 - {c_0 \over 2}\right)x^2 + \left(c_3 - {c_1 \over 2}\right)x^3+\left(c_4+{c_0 \over 4!}-{c_2\over 2}\right)x^4 + \cdots\!

Comparing coefficients with the above series of the exponential function yields the desired Taylor series

\frac{e^x}{\cos x}=1 + x + x^2 + {2x^3 \over 3} + {x^4 \over 2} + \cdots\!.

Taylor series as definitions

Classically, algebraic functions are defined by an algebraic equation, and transcendental functions (including those discussed above) are defined by some property that holds for them, such as a differential equation. In Mathematics, an algebraic function is informally a function which satisfies a Polynomial equation whose coefficients are themselves polynomials A transcendental function is a function that does not satisfy a Polynomial equation whose Coefficients are themselves polynomials in contrast to an A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the For example the exponential function is the function which is everywhere equal to its own derivative, and assumes the value 1 at the origin. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) However, one may equally well define an analytic function by its Taylor series. This article is about both real and complex analytic functions

Taylor series are used to define functions in diverse areas of mathematics. In particular, this is true in areas where the classical definitions of functions break down. For example, using Taylor series, one may define analytical functions of matrices and operators, such as the matrix exponential or matrix logarithm. In Mathematics, the matrix exponential is a Matrix function on square matrices analogous to the ordinary Exponential function. In Mathematics, the logarithm of a matrix is a Matrix function which generalizes the scalar Logarithm to matrices.

In other areas, such as formal analysis, it is more convenient to work directly with the power series themselves. In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + Thus one may define a solution of a differential equation as a power series which, one hopes to prove, is the Taylor series of the desired solution.

Taylor series in several variables

The Taylor series may also be generalized to functions of more than one variable with

T(x_1,\cdots,x_d) =
=\sum_{n_1=0}^{\infin} \cdots \sum_{n_d=0}^{\infin} \frac{\partial^{n_1}}{\partial x_1^{n_1}} \cdots \frac{\partial^{n_d}}{\partial x_d^{n_d}} \frac{f(a_1,\cdots,a_d)}{n_1!\cdots n_d!} (x_1-a_1)^{n_1}\cdots (x_d-a_d)^{n_d}\!.

For example, for a function that depends on two variables, x and y, the Taylor series to second order about the point (a, b) is:

f(x,y)\!
\approx f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b) \!
+ \frac{1}{2!}\left[ f_{xx}(a,b)(x-a)^2 + 2f_{xy}(a,b)(x-a)(y-b) + f_{yy}(a,b)(y-b)^2 \right]\!

where the subscripts denote the respective partial derivatives. In Mathematics, a partial derivative of a function of several variables is its Derivative with respect to one of those variables with the others held constant

A second-order Taylor series expansion of a scalar-valued function of more than one variable can be compactly written as

T(\mathbf{x}) = f(\mathbf{a}) + \mathrm{D} f(\mathbf{a})^T (\mathbf{x} - \mathbf{a}) + \frac{1}{2!} (\mathbf{x} - \mathbf{a})^T \mathrm{D}^2 f(\mathbf{a}) (\mathbf{x} - \mathbf{a}) + \cdots\!

where D f(\mathbf{a})\! is the gradient and D^2 f(\mathbf{a})\! is the Hessian matrix. In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar In Mathematics, the Hessian matrix is the Square matrix of second-order Partial derivatives of a function. Applying the multi-index notation the Taylor series for several variables becomes

T(\mathbf{x}) = \sum_{|\alpha| \ge 0}^{}{\frac{\mathrm{D}^{\alpha}f(\mathbf{a})}{\alpha !}(\mathbf{x}-\mathbf{a})^{\alpha}}\!

in full analogy to the single variable case. The Mathematical notation of multi-indices simplifies formulae used in Multivariable calculus, Partial differential equations and the theory of distributions

See also

Notes

  1. ^ Kline, M. In Calculus, Taylor's theorem gives a sequence of approximations of a Differentiable function around a given point by Polynomials (the Taylor In Mathematics, the Laurent series of a complex function f ( z) is a representation of that function as a Power series which includes terms In Complex analysis, a field of Mathematics, a complex -valued function f of a complex variable z is holomorphic In the Mathematical field of Numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is the interpolation Polynomial Mādhava of Sangamagrama (born as Irinjaatappilly Madhavan Namboodiri) (c The Difference Engine was an automatic mechanical calculator designed to tabulate polynomial functions. (1990) Mathematical Thought from Ancient to Modern Times. Oxford University Press. pp. 35-37.
  2. ^ Boyer, C. and Merzbach, U. (1991) A History of Mathematics. John Wiley and Sons. pp. 202-203.
  3. ^ Neither Newton nor Leibniz - The Pre-History of Calculus and Celestial Mechanics in Medieval Kerala. MAT 314. Canisius College. Retrieved on 2006-07-09. Year 2006 ( MMVI) was a Common year starting on Sunday of the Gregorian calendar. Events 455 - Roman military commander Avitus is proclaimed Emperor of the Western Roman Empire.
  4. ^ Taylor, Brook, Methodus Incrementorum Directa et Inversa [Direct and Reverse Methods of Incrementation] (London, 1715), pages 21-23 (Proposition VII, Theorem 3, Corollary 2). Translated into English in D. J. Struik, A Source Book in Mathematics 1200-1800 (Cambridge, Massachusetts: Harvard University Press, 1969), pages 329-332.
  5. ^ Exercise 12 on page 418 in Walter Rudin, Real and Complex Analysis. McGraw-Hill, New Dehli 1980, ISBN 0-07-099557-5
  6. ^ Exercise 13, same book

References

External links

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
 Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic