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Encyclopedia > Taylor series
As the degree of the Taylor series 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 series 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 polynomial at 0 (in red).

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. It may be regarded as the limit of the Taylor polynomials. Taylor series are named in honour of English mathematician Brook Taylor. If the series uses the derivatives at zero, the series is also called a Maclaurin series, named after Scottish mathematician Colin Maclaurin. In mathematics, a series is often represented as the sum of a sequence of terms. ... Image File history File links Sintay. ... Image File history File links Sintay. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... The exponential function is one of the most important functions in mathematics. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... Addition is one of the basic operations of arithmetic. ... This article is about derivatives and differentiation in mathematical calculus. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... As the degree of the Taylor series rises, it approaches the correct function. ... This article is about the English as an ethnic group and nation. ... Brook Taylor (August 18, 1685 – November 30, 1731) was an English mathematician. ... This article is about the Scottish as an ethnic group. ... Colin Maclaurin Colin Maclaurin (February, 1698 - 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 In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... 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 as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...

 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 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. For factorial rings in mathematics, see unique factorisation domain. ... This article is about derivatives and differentiation in mathematical calculus. ...


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 that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...


The Maclaurin series for (1 − x) − 1 is the geometric series In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ...

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.

By integrating the above Maclaurin series we find the Maclaurin series for − ln(1 − x), where ln denotes the natural logarithm: The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ...

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 Maclaurin series for the exponential function ex at a = 0 is The exponential function is one of the most important functions in mathematics. ...

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 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 Taylor series need not in general be a convergent series, but often it is. The limit of a convergent Taylor series 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. If f(x) is equal to its Taylor series everywhere it is called entire. The exponential function ex and the trigonometric functions sine and cosine are examples of such functions. Examples of functions that are not entire include the logarithm, the trigonometric function tangent, and its inverse arctan. For these functions the Taylor series do not even converge if x is far from a. In mathematics, a series is the sum of the terms of a sequence of numbers. ... In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ... In mathematics, an analytic function is a function that is locally given by a convergent power series. ... In complex analysis, an entire function is a function that is holomorphic everywhere (ie complex-differentiable at every point) on the whole complex plane. ... The exponential function is one of the most important functions in mathematics. ... Sine redirects here. ... Logarithms to various bases: is to base e, is to base , and is to base . ... Sine redirects here. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, a series is the sum of the terms of a sequence of numbers. ...

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.

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, is known at a single point. Uses of the Taylor series for entire functions include: Image File history File links Taylorsine. ... Image File history File links Taylorsine. ...

  1. The partial sums (the Taylor polynomials) of the series can be used as approximations of the entire function. These approximations are good if sufficiently many terms are included.
  2. The series representation simplifies many mathematical proofs.

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: As the degree of the Taylor series rises, it approaches the correct function. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ...

sinleft( 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.


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. Later, Aristotle proposed a philosophical resolution of the paradox, but the mathematical content was apparently unresolved until taken up by Democritus and then Archimedes. It was through Archimedes's method of exhaustion that an infinite number of progressive subdivisions could be performed to achieve a finite trigonometric result.[1] Liu Hui independently employed a similar method several centuries later.[2] The Pythagoreans were an Hellenic organization of astronomers, musicians, mathematicians, and philosophers; who believed that all things are, essentially, numeric. ... Zeno of Elea (IPA:zɛnoʊ, ɛlɛɑː)(circa 490 BC? – circa 430 BC?) was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides. ... Zenos paradoxes are a set of paradoxes devised by Zeno of Elea to support Parmenides doctrine that all is one and that contrary to the evidence of our senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion. ... For other uses, see Aristotle (disambiguation). ... ‎ Democritus (Greek: ) was a pre-Socratic Greek materialist philosopher (born at Abdera in Thrace ca. ... For other uses, see Archimedes (disambiguation). ... 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 area of the containing shape. ... A possible likeness of Liu Hui on japanpostage stamp This is a Chinese name; the family name is 劉 (Liu) Liu Hui 劉徽 was a Chinese mathematician who lived in the 200s in the Wei Kingdom. ...


In the 14th century, the earliest examples of the use of Taylor series and closely-related methods were given by Madhava of Sangamagrama.[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. The Kerala school of astronomy and mathematics further expanded his works with various series expansions and rational approximations until the 16th century. This 14th-century statue from south India depicts the gods Shiva (on the left) and Uma (on the right). ... Madhavan (മാധവന്) of Sangamagramam (1350–1425) was a prominent mathematician-astronomer from Kerala, India. ... This article is under construction. ... Sine redirects here. ... For other uses, see tangent (disambiguation). ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... The Kerala School was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, South India which included as its prominent members Parameshvara, Nilakantha Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ... (15th century - 16th century - 17th century - more centuries) As a means of recording the passage of time, the 16th century was that century which lasted from 1501 to 1600. ...


In the 17th century, James Gregory also worked in this area and published several Maclaurin series. 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, after whom the series are now named. (16th century - 17th century - 18th century - more centuries) As a means of recording the passage of time, the 17th century was that century which lasted from 1601-1700. ... James Gregory For other people with the same name, see James Gregory. ... Year 1715 (MDCCXV) was a common year starting on Tuesday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Friday of the 11-day slower Julian calendar). ... Brook Taylor (August 18, 1685 – November 30, 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 Colin Maclaurin (February, 1698 - 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 (ar, a + r) and the sum is equal to f(x), then the function f(x) is said to be analytic in the interval (ar, a + r). If this is true for any r then the function is said to be an entire function. To check whether the series converges towards f(x), one normally uses estimates for the remainder term of Taylor's theorem. 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. Image File history File links Expinvsq. ... In mathematics, an analytic function is a function that is locally given by a convergent power series. ... In complex analysis, an entire function is a function that is holomorphic everywhere (ie complex-differentiable at every point) on the whole complex plane. ... The exponential function (continuous red line) and the corresponding Taylors polynomial about a = 0 of degree four (dashed green line) In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial... ↔ ⇔ ≡ logical symbols representing iff. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...


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. 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, an analytic function is a function that is locally given by a convergent power series. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... A disk is the inside of a circle. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ... In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev (&#1055;&#1072;&#1092;&#1085;&#1091;&#1090;&#1080;&#1081; &#1063;&#1077;&#1073;&#1099;&#1096;&#1105;&#1074;), are special polynomials. ... In the mathematical subfield of numerical analysis the Clenshaw algorithm (Invented by Charles William Clenshaw) is a recursive method to evaluate polynomials in Chebyshev form. ...


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 result is of fundamental importance in such fields as harmonic analysis. Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ... Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. ...


Note that there are examples of infinitely differentiable functions f(x) whose Taylor series converge, but are not equal to f(x). For instance, for the function defined pointwise by saying that f(x) = e−1/x² if x ≠ 0 and f(0) = 0, all the derivatives are zero at x = 0, 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. 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. 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. In mathematics, a smooth function is one that is infinitely differentiable, i. ... Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...


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 (Exercise 12 on page 418 in Walter Rudin, Real and Complex Analysis. McGraw-Hill, New Dehli 1980, ISBN 0-07-099557-5). There are even infinitely differentiable functions defined on the real line whose Taylor series have a radius of convergence 0 everywhere (Exercise 13, same book).


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. For example, f(x) = e−1/x² can be written as a 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 well-behaved in some particular way, such as differentiability. ... A Laurent series is defined with respect to a particular point c and a path of integration γ. The path of integration must lie in an annulus (shown here in red) inside of which f(z) is holomorphic. ...


The Parker-Sochacki method is a recent advance in finding Taylor series which are solutions to differential equations. This algorithm is an extension of the Picard iteration. In mathematics, the Parker-Sochacki method is an algorithm for solving systems of differential equations, which has been developed by G. Edgar Parker and James Sochacki, of the James Madison University Mathematics Department. ... Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ... In mathematics, the Picard–Lindelöf theorem on existence and uniqueness of solutions of differential equations (Picard 1890, Lindelöf 1894) states that an initial value problem has exactly one solution if f is Lipschitz continuous in , continuous in as long as stays bounded. ...


List of Taylor series of some common functions

The cosine function.
The cosine function.
An 8th degree approximation of the cosine function in the complex plane.
An 8th degree approximation of the cosine function in the complex plane.
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. Image File history File links Download high-resolution version (1249x941, 448 KB) Graph of z = Re(cos(x+iy)), selfmade with MuPad I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version... Image File history File links Download high-resolution version (1249x941, 448 KB) Graph of z = Re(cos(x+iy)), selfmade with MuPad I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version... Image File history File links Download high-resolution version (1250x957, 471 KB) Graph of z = Re(1-(x+I*y)^2/2! + (x+I*y)^4/4! - (x+I*y)^6/6! +(x+I*y)^8/8!), selfmade with MuPad I, the creator of this work, hereby grant the permission... Image File history File links Download high-resolution version (1250x957, 471 KB) Graph of z = Re(1-(x+I*y)^2/2! + (x+I*y)^4/4! - (x+I*y)^6/6! +(x+I*y)^8/8!), selfmade with MuPad I, the creator of this work, hereby grant the permission... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... Image File history File links Download high-resolution version (1250x959, 705 KB) Graphs of z = Re(cos(x+iy)) and an eight-degree taylor series approximation, selfmade with MuPad. ... Image File history File links Download high-resolution version (1250x959, 705 KB) Graphs of z = Re(cos(x+iy)) and an eight-degree taylor series approximation, selfmade with MuPad. ...


Exponential function: The exponential function is one of the most important functions in mathematics. ...

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

Natural logarithm: The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ...

ln(1-x) = -sum^{infin}_{n=1} frac{x^n}nquadmbox{ for } |x| < 1

Finite geometric series: In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ...

frac{1-x^{m + 1}}{1-x} = sum^{m}_{n=0} x^nquadmbox{ for } x not= 1mbox{ and } minmathbb{N}_0

Infinite geometric series:

frac{1}{1-x} = sum^{infin}_{n=0} x^nquadmbox{ for } |x| < 1

Variants of the infinite geometric series:

frac{x^m}{1-x} = sum^{infin}_{n=m} x^nquadmbox{ for } |x| < 1 mbox{ and } minmathbb{N}_0
frac{x}{(1-x)^2} = sum^{infin}_{n=1}n x^nquadmbox{ for } |x| < 1

Square root: In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ...

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

Binomial series (includes the square root for α = 1/2 and the infinite geometric series for α = −1): In mathematics, the binomial series generalizes the purely algebraic binomial theorem. ...

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

Trigonometric functions: In mathematics, particularly in combinatorics, a binomial coefficient is a coefficient of any of the terms in the expansion of the binomial (x+1)n. ... Sine redirects here. ...

sin x = sum^{infin}_{n=0} frac{(-1)^n}{(2n+1)!} x^{2n+1}quad = x - frac{x^3}{3!} + frac{x^5}{5!} - cdotsmbox{ 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!} - cdotsmbox{ 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.
sec x = sum^{infin}_{n=0} frac{(-1)^n E_{2n}}{(2n)!} x^{2n}quadmbox{ for } |x| < frac{pi}{2}
arcsin x = sum^{infin}_{n=0} frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}quadmbox{ for } |x| < 1
arctan x = sum^{infin}_{n=0} frac{(-1)^n}{2n+1} x^{2n+1}quadmbox{ for } |x| le 1

Hyperbolic functions: In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. ... A ray through the origin intercepts the hyperbola in the point , where is the area between the ray, its mirror image with respect to the -axis, and the hyperbola (see animated version with comparison with the trigonometric (circular) functions). ...

sinh x = sum^{infin}_{n=0} frac{x^{2n+1}}{(2n+1)!} quadmbox{ for all } x
cosh x = sum^{infin}_{n=0} frac{x^{2n}}{(2n)!} quadmbox{ for all } x
tanh x = sum^{infin}_{n=1} frac{B_{2n} 4^n (4^n-1)}{(2n)!} x^{2n-1}quadmbox{ for } |x| < frac{pi}{2}
mathrm{arcsinh} (x) = sum^{infin}_{n=0} frac{(-1)^n (2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}quadmbox{ for } |x| < 1
mathrm{arctanh} (x) = sum^{infin}_{n=0} frac{x^{2n+1}}{2n+1} quadmbox{ for } |x| < 1

Lambert's W function: In mathematics, Lamberts W function, named after Johann Heinrich Lambert, also called the Omega function or product log, is the inverse function of where ew is the exponential function and w is any complex number. ...

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

The numbers Bk appearing in the summation expansions of tan(x) and tanh(x) are the Bernoulli numbers. The Ek in the expansion of sec(x) are Euler numbers. In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. ... The Euler numbers are a sequence En of integers defined by the following Taylor series expansion: (Note that e, the base of the natural logarithm, is also occasionally called Eulers number, as is the Euler characteristic. ...

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. Particularly convenient is the use of computer algebra systems to calculate Taylor series. In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, hopefully simpler, integrals. ... 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)=lncos x, quad xin(-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) For other uses, see Big O. In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithms usage of computational resources (usually running time or memory). ...

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 expansion has no constant term on the right-hand side, 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-1bigr) - frac12bigl(cos x-1bigr)^2 + frac13bigl(cos x-1bigr)^3+ mathcal{O}bigl((cos x-1)^4bigr) &=biggl(-frac{x^2}2 + frac{x^4}{24} - frac{x^6}{720} +mathcal{O}(x^8)biggr) - frac12biggl(-frac{x^2}2+frac{x^4}{24}+mathcal{O}(x^6)biggr)^2 +frac13biggl(-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, . . . have to be zero. In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking negatives. ...


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^nover 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 + cdotsright)left(1 - {x^2 over 2!} + {x^4 over 4!} - cdotsright) &=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_2over 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, the above functions are defined by some property that holds for them. For example, the exponential function is defined as the function that is equal to its own derivative. However, in computable analysis, functions must be defined by algorithms rather than properties, so the above Taylor expansions are used as primary definitions rather than derived results. This is also likely to be the case in software implementations of the functions. The exponential function is one of the most important functions in mathematics. ...


Using Taylor series, one may define analytical functions of matrices and operators, such as 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 generalization of the scalar logarithm to matrices. ...


Taylor series for 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 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].

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}) + nabla f(mathbf{a})^T (mathbf{x} - mathbf{a}) + frac{1}{2!} (mathbf{x} - mathbf{a})^T nabla^2 f(mathbf{a}) (mathbf{x} - mathbf{a}) + cdots

where nabla f(mathbf{a}) is the gradient and nabla^2 f(mathbf{a}) is the Hessian matrix (not to be confused with the Laplacian, which sometimes has the same notation). Applying the multi-index notation the Taylor series for several variables becomes For other uses, see Gradient (disambiguation). ... In mathematics, the Hessian matrix is the square matrix of second order partial derivatives of a function. ... In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ... The notion of multi-indices simplifies formulae used in the multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an array of indices. ...

 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.


See also

The exponential function (continuous red line) and the corresponding Taylors polynomial about a = 0 of degree four (dashed green line) In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial... A Laurent series is defined with respect to a particular point c and a path of integration γ. The path of integration must lie in an annulus (shown here in red) inside of which f(z) is holomorphic. ... In complex analysis, a complex-valued function f of a complex variable is holomorphic at a point a iff it is differentiable at every point within some open disk centered at a, and is analytic at a if in some open disk centered at a it can be expanded as... In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is the interpolation polynomial for a given set of data points in the Newton form. ... Madhavan (മാധവന്) of Sangamagramam (1350–1425) was a prominent mathematician-astronomer from Kerala, India. ... Part of Babbages Difference engine, assembled after his death by Babbages son, using parts found in his laboratory. ...

Notes

  1. ^ Kline, M. (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. ... is the 190th day of the year (191st in leap years) in the Gregorian calendar. ...

References

  • Thomas, George B. Jr.; Finney, Ross L. (1996). Calculus and Analytic Geometry (9th ed.). Addison Wesley. ISBN 0-201-53174-7. 
  • Greenberg, Michael (1998). Advanced Engineering Mathematics (2nd ed.). Prentice Hall. ISBN 0-13-321431-1. 

External links


  Results from FactBites:
 
Taylor series - Wikipedia, the free encyclopedia (1202 words)
The Taylor series, power series, and infinite series expansions of functions may have been first discovered in India by Madhava in the 14th century.
He is also thought to have discovered the power series of the radius, diameter, circumference, angle θ, π and π/4, along with rational approximations of π, and infinite continued fractions.
The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, who published the special case of the Taylor result in the 17th century.
PlanetMath: Taylor series (745 words)
In contrast with the complex case, it turns out that all holomorphic functions are infinitely differentiable and have Taylor series that converge to them.
Taylor series and polynomials can be generalized to Banach spaces: for details, see Taylor's formula in Banach spaces.
This is version 19 of Taylor series, born on 2001-11-08, modified 2005-10-14.
  More results at FactBites »

 
 

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