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Encyclopedia > Taylor polynomial
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As the degree of the Taylor series rises, it approaches the correct function. This image shows sin(x) 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 sin(x) and Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.

In mathematics, the Taylor series of an infinitely often differentiable real (or complex) function f defined on an open interval (ar, a + r) is the power series Download high resolution version (800x800, 33 KB)sin(x) and Taylor approximations of sin(x) File links The following pages link to this file: Taylor series Wikipedia:List of images/Mathematics Categories: GFDL images ... Download high resolution version (800x800, 33 KB)sin(x) and Taylor approximations of sin(x) File links The following pages link to this file: Taylor series Wikipedia:List of images/Mathematics Categories: GFDL images ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... 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. ...

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

Here, n! is the factorial of n and f (n)(a) denotes the nth derivative of f at the point a. If a = 0, the series is also called a Maclaurin series. In mathematics, the factorial of a natural number n is the product of all positive integers less than and equal to n. ... Jump to: navigation, search In mathematics, the derivative is one of the two central concepts of calculus. ...

Contents


History

Limited sets of series expansions for particular functions were known in India by Madhava in the fourteenth century. In the 17th century, James Gregory also worked in this area and published several Maclaurin series. It wasn't until 1715 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. Madhava (माधव) of Sangamagrama (1350-1425) was a major mathematician of the Kerala school who is considered the father of mathematical analysis for having taken the decisive step from the finite procedures of ancient mathematics to treat their limit-passage to infinity, which is the kernel of modern classical analysis. ... Jump to: navigation, search This 14th-century statue from south India depicts the gods Shiva (on the left) and Uma (on the right}. It is housed in the Smithsonian Institution in Washington, D.C. As a means of recording the passage of time, the 14th century was that century which... James Gregory (November 1638 – October 1675), was a Scottish mathematician and astronomer. ... // Events September 1 - King Louis XIV of France dies after a reign of 72 years, leaving the throne of his exhausted and indebted country to his great-grandson Louis XV. Regent for the new, five years old monarch is Philippe dOrléans, nephew of Louis XIV. September - First of... Brook Taylor (August 18, 1685 – December 29, 1731) was an English mathematician. ...


Properties

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 called analytic. 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. 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 whose coefficients depend only on the derivatives of the function at that point. ... 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 in the polynomial into the Chebyshev form and evaluating it with the Clenshaw algorithm). 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. 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. ... Jump to: navigation, search In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... Jump to: navigation, search This article may be too technical for most readers to understand. ... In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev (Пафнутий Чебышёв), are special polynomials. ... In the mathematical subfield of numerical analysis the Clenshaw algorithm is a recursive method to evaluate polynomials in Chebyshev form. ... Eulers formula, named after Leonhard Euler, is a mathematical formula in the subfield of complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ...

The function e-1/x² is not analytic: the Taylor series is 0, although the function is not.
The function e-1/x² is not analytic: the Taylor series is 0, although the function is not.

Note that there are examples of infinitely often differentiable functions f(x) whose Taylor series converge, but are not equal to f(x). For instance, for the function defined piecewise by saying that f(x) = exp(−1/x²) if x ≠ 0 and f(0) = 0, all the derivatives are zero at x = 0, so the Taylor series of f(x) is zero, and its radius of convergence is infinite, even though the function most definitely is not zero. This particular pathology does not afflict complex-valued functions of a complex variable. Notice that exp(−1/z²) does not approach 0 as z approaches 0 along the imaginary axis. e^(-1/x²) (made by me) File links The following pages link to this file: Taylor series Non-perturbative Categories: GFDL images ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... In mathematics, the radius of convergence of a power series where the center a and the coefficients cn are complex numbers (which may, in particular, be real numbers) is the nonnegative quantity r (which may be a real number or ∞) such that the series converges if and diverges if In... Jump to: navigation, search In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...


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) = exp(−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 Parker-Sochacki theorem is a recent advance in finding Taylor series which are solutions to differential equations. This theorem is an expansion on the Picard iteration. In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...


Taylor series for several variables

The Taylor series may also be generalised 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}

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

where nabla f(mathbf{a}) is the gradient and nabla^2 f(mathbf{a}) is the Hessian matrix. Applying the multi-index notation the Taylor series for several variables becomes In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows. ... In mathematics, the Hessian matrix is the square matrix of second partial derivatives of a scalar-valued function. ... The notion of multi-indices simplifies formula used in the calculus of several variables, 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.


List of Taylor series of some common functions

Several important Taylor/Maclaurin series expansions follow. All these expansions are also valid for complex arguments x.


Exponential function and natural logarithm: Jump to: navigation, search The exponential function is one of the most important functions in mathematics. ... The natural logarithm, invented by John Napier, is the logarithm to the base e, where e is equal to 2. ...

mathrm{e}^{x} = sum^{infin}_{n=0} frac{x^n}{n!}quadmbox{ for all } x
ln(1+x) = sum^{infin}_{n=0} frac{(-1)^n}{n+1} x^{n+1}quadmbox{ for } left| x right| < 1

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{x^m}{1-x} = sum^{infin}_{n=m} x^nquadmbox{ for } left| x right| < 1

Binomial theorem: Jump to: navigation, search // Edahls Theorem This is another way to solve binomials that may be easier for some people. ...

(1+x)^alpha = sum^{infin}_{n=0} {alpha choose n} x^nquadmbox{ for all } left| x right| < 1quadmbox{ and all complex } alpha

Trigonometric functions: Jump to: navigation, search In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

sin x = sum^{infin}_{n=0} frac{(-1)^n}{(2n+1)!} x^{2n+1}quadmbox{ for all } x
cos x = sum^{infin}_{n=0} frac{(-1)^n}{(2n)!} x^{2n}quadmbox{ for all } x
tan x = sum^{infin}_{n=1} frac{B_{2n} (-4)^n (1-4^n)}{(2n)!} x^{2n-1}quadmbox{ for } left| x right| < frac{pi}{2}
sec x = sum^{infin}_{n=0} frac{(-1)^n E_{2n}}{(2n)!} x^{2n}quadmbox{ for } left| x right| < frac{pi}{2}
arcsin x = sum^{infin}_{n=0} frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}quadmbox{ for } left| x right| < 1
arctan x = sum^{infin}_{n=0} frac{(-1)^n}{2n+1} x^{2n+1}quadmbox{ for } left| x right| < 1

Hyperbolic functions: Jump to: navigation, search In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ...

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

Lambert's W function: Jump to: navigation, search In mathematics, Lamberts W function, named after Johann Heinrich Lambert, also called the Omega function, 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 } left| x right| < frac{1}{mathrm{e}}

The numbers Bk appearing in the expansions of tan(x) and tanh(x) are the Bernoulli numbers. The binomial expansion uses binomial coefficients. 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. ... In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is defined to be the natural number and (Here, for a natural number m, m! denotes the factorial of m. ... 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. In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...


For example, consider the function

f(x)=ln{(1+cos{x})} ,

for which we want a Taylor series about 0.


We have:

ln(1+x) = sum^{infin}_{n=1} frac{(-1)^{n+1}}{n} x^n = x - {x^2over 2}+{x^3 over 3} - {x^4 over 4} + cdots quadmbox{ for } left| x right| < 1
cos x = sum^{infin}_{n=0} frac{(-1)^n}{(2n)!} x^{2n} = 1 -{x^2over 2!}+{x^4over 4!}mp cdots quadmbox{ for all } x

We can simply substitute the second series into the first. Doing so,

(1 -{x^2over 2!}+{x^4over 4!}+cdots )-{1over 2}(1 -{x^2over 2!}+{x^4over 4!}+cdots )^2+{1over 3}(1 -{x^2over 2!}+{x^4over 4!}+cdots)^3-{1over 4}(1 -{x^2over 2!}+{x^4over 4!}+cdots)^4+cdots

Expanding by using multinomial coefficients gives the requisite Taylor series. In mathematics, the multinomial formula is an expression of a power of a sum in terms of powers of the addends. ...


Or, for example, consider

We have

mathrm{e}^x = 1 + x + {x^2 over 2!} + {x^3 over 3!} + cdots
sin{x} = x - {x^3 over 3!} + {x^5 over 5!} mp cdots

Then,

Assume the power series is

c_0 + c_1 x + c_2 x^2 + c_3 x^3 + cdots = {1 + x + {x^2 over 2!} + {x^3 over 3!} + cdots over x - {x^3 over 3!} + {x^5 over 5!} + cdots}

Then

=left(c_0 + c_1 x + c_2 x^2 + c_3 x^3 + cdotsright)left(x - {x^3 over 3!} + {x^5 over 5!} -+ cdotsright) = 1 + x + {x^2 over 2!} + {x^3 over 3!} + cdots
=c_0x - {c_0 over 3!}x^3 + {c_0over 5!}x^5 + c_1x^2 - {c_1 over 3!}x^4 + {c_1over 5!}x^6 + c_2 x^3 - {c_2 over 3!} x^5 + {c_2 over 5!} x^7 + c_3x^4-{c_3over 3!}x^6 + cdots
= 1 + x + {x^2 over 2!} + {x^3 over 3!} + cdots
=c_0x + c_1x^2 + c_2 x^3 - {c_0 over 3!}x^3 + c_3x^4- {c_1 over 3!}x^4 + cdots
=c_0x + c_1x^2 + (c_2 - {c_0 over 3!})x^3 + (c_3-{c_1 over 3!})x^4 + cdots = 1 + x + {x^2 over 2!} + {x^3 over 3!} + cdots

Comparing coefficients yields the Taylor series for the function.


See also

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 whose coefficients depend only on the derivatives of the function at that point. ... In the mathematical subfield 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. ... Madhava (माधव) of Sangamagrama (1350-1425) was a major mathematician of the Kerala school who is considered the father of mathematical analysis for having taken the decisive step from the finite procedures of ancient mathematics to treat their limit-passage to infinity, which is the kernel of modern classical analysis. ...

External links


  Results from FactBites:
 
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