Taylor series
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In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.
The partial sum formed by the first Template:Math terms of a Taylor series is a polynomial of degree Template:Mvar that is called the Template:Mvarth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate as Template:Mvar increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point Template:Mvar if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing Template:Mvar. This implies that the function is analytic at every point of the interval (or disk).
DefinitionEdit
The Taylor series of a real or complex-valued function Template:Math, that is infinitely differentiable at a real or complex number Template:Math, is the power series <math display="block"> f(a) + \frac {f'(a)}{1!}(x-a) + \frac{f(a)}{2!} (x-a)^2+ \cdots = \sum_{n=0} ^ {\infty} \frac {f^{(n)}(a)}{n!} (x-a)^{n}. </math> Here, Template:Math denotes the factorial of Template:Mvar. The function Template:Math denotes the Template:Mvarth derivative of Template:Mvar evaluated at the point Template:Mvar. The derivative of order zero of Template:Mvar is defined to be Template:Mvar itself and Template:Math and Template:Math are both defined to be 1. This series can be written by using sigma notation, as in the right side formula.Template:Sfn With Template:Math, the Maclaurin series takes the form:Template:Sfn <math display="block"> f(0)+\frac {f'(0)}{1!} x+ \frac{f(0)}{2!} x^2+ \cdots = \sum_{n=0} ^ {\infty} \frac {f^{(n)}(0)}{n!} x^{n}. </math>
ExamplesEdit
The Taylor series of any polynomial is the polynomial itself.
The Maclaurin series of Template:Math is the geometric series
<math display="block">1 + x + x^2 + x^3 + \cdots.</math>
So, by substituting Template:Mvar for Template:Math, the Taylor series of Template:Math at Template:Math is
<math display="block">1 - (x-1) + (x-1)^2 - (x-1)^3 + \cdots.</math>
By integrating the above Maclaurin series, we find the Maclaurin series of Template:Math, where Template:Math denotes the natural logarithm:
<math display="block">-x - \tfrac{1}{2}x^2 - \tfrac{1}{3}x^3 - \tfrac{1}{4}x^4 - \cdots.</math>
The corresponding Taylor series of Template:Math at Template:Math is
<math display="block">(x-1) - \tfrac{1}{2}(x-1)^2 + \tfrac{1}{3}(x-1)^3 - \tfrac{1}{4}(x-1)^4 + \cdots,</math>
and more generally, the corresponding Taylor series of Template:Math at an arbitrary nonzero point Template:Mvar is:
<math display="block">\ln a + \frac{1}{a} (x - a) - \frac{1}{a^2}\frac{\left(x - a\right)^2}{2} + \cdots.</math>
The Maclaurin series of the exponential function Template:Math is
<math display="block">\begin{align}
\sum_{n=0}^\infty \frac{x^n}{n!} &= \frac{x^0}{0!} + \frac{x^1}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!}+ \cdots \\ &= 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \frac{x^5}{120} + \cdots. \end{align}</math>
The above expansion holds because the derivative of Template:Math with respect to Template:Mvar is also Template:Math, and Template:Math equals 1. This leaves the terms Template:Math in the numerator and Template:Math in the denominator of each term in the infinite sum.
HistoryEdit
The ancient Greek philosopher Zeno of Elea considered the problem of summing an infinite series to achieve a finite result, but rejected it as an impossibility;Template:Sfn 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 Archimedes, as it had been prior to Aristotle by the Presocratic Atomist Democritus. It was through Archimedes's method of exhaustion that an infinite number of progressive subdivisions could be performed to achieve a finite result.Template:Sfn Liu Hui independently employed a similar method a few centuries later.Template:Sfn
In the 14th century, the earliest examples of specific Taylor series (but not the general method) were given by Indian mathematician Madhava of Sangamagrama.Template:Sfn Though no record of his work survives, writings of his followers in the Kerala school of astronomy and mathematics suggest that he found the Taylor series for the trigonometric functions of sine, cosine, and arctangent (see Madhava series). During the following two centuries his followers developed further series expansions and rational approximations.
In late 1670, James Gregory was shown in a letter from John Collins several Maclaurin series Template:Nobr <math display=inline>\cos x,</math> <math display=inline>\arcsin x,</math> and Template:Nobr derived by Isaac Newton, and told that Newton had developed a general method for expanding functions in series. Newton had in fact used a cumbersome method involving long division of series and term-by-term integration, but Gregory did not know it and set out to discover a general method for himself. In early 1671 Gregory discovered something like the general Maclaurin series and sent a letter to Collins including series for <math display=inline>\arctan x,</math> <math display=inline>\tan x,</math> <math display=inline>\sec x,</math> <math display=inline>\ln\, \sec x</math> (the integral of Template:Nobr <math display=inline>\ln\, \tan\tfrac12{\bigl(\tfrac12\pi + x\bigr)}</math> (the [[integral of the secant function|integral of Template:Math]], the inverse Gudermannian function), <math display=inline>\arcsec \bigl(\sqrt2 e^x\bigr),</math> and <math display=inline>2 \arctan e^x - \tfrac12\pi</math> (the Gudermannian function). However, thinking that he had merely redeveloped a method by Newton, Gregory never described how he obtained these series, and it can only be inferred that he understood the general method by examining scratch work he had scribbled on the back of another letter from 1671.<ref>Template:Multiref</ref>
In 1691–1692, Isaac Newton wrote down an explicit statement of the Taylor and Maclaurin series in an unpublished version of his work De Quadratura Curvarum. However, this work was never completed and the relevant sections were omitted from the portions published in 1704 under the title Tractatus de Quadratura Curvarum.
It was not until 1715 that a general method for constructing these series for all functions for which they exist was finally published by Brook Taylor,<ref>Template:Multiref</ref> after whom the series are now named.
The Maclaurin series was named after Colin Maclaurin, a Scottish mathematician, who published a special case of the Taylor result in the mid-18th century.
Analytic functionsEdit
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If Template:Math is given by a convergent power series in an open disk centred at Template:Mvar in the complex plane (or an interval in the real line), it is said to be analytic in this region. Thus for Template:Mvar in this region, Template:Mvar is given by a convergent power series
<math display="block">f(x) = \sum_{n=0}^\infty a_n(x-b)^n.</math>
Differentiating by Template:Mvar the above formula Template:Mvar times, then setting Template:Math gives:
<math display="block">\frac{f^{(n)}(b)}{n!} = a_n</math>
and so the power series expansion agrees with the Taylor series. Thus a function is analytic in an open disk centered at Template:Mvar if and only if its Taylor series converges to the value of the function at each point of the disk.
If Template:Math is equal to the sum of its Taylor series for all Template:Mvar in the complex plane, it is called entire. The polynomials, exponential function Template:Math, and the trigonometric functions sine and cosine, are examples of entire functions. Examples of functions that are not entire include the square root, the logarithm, the trigonometric function tangent, and its inverse, arctan. For these functions the Taylor series do not converge if Template:Mvar is far from Template:Mvar. That is, the Taylor series diverges at Template:Mvar if the distance between Template:Mvar and Template:Mvar is larger than the radius of convergence. The Taylor series can be used to calculate the value of an entire function at 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 analytic functions include:
- The partial sums (the Taylor polynomials) of the series can be used as approximations of the function. These approximations are good if sufficiently many terms are included.
- Differentiation and integration of power series can be performed term by term and is hence particularly easy.
- An analytic function is uniquely extended to a holomorphic function on an open disk in the complex plane. This makes the machinery of complex analysis available.
- The (truncated) series can be used to compute function values numerically, (often by recasting the polynomial into the Chebyshev form and evaluating it with the Clenshaw algorithm).
- Algebraic operations can be done readily on the power series representation; for instance, Euler's formula follows from Taylor series expansions for trigonometric and exponential functions. This result is of fundamental importance in such fields as harmonic analysis.
- Approximations using the first few terms of a Taylor series can make otherwise unsolvable problems possible for a restricted domain; this approach is often used in physics.
Approximation error and convergenceEdit
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Pictured is an accurate approximation of Template:Math around the point Template:Math. The pink curve is a polynomial of degree seven:
<math display="block">\sin{x} \approx x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}.\!</math>
The error in this approximation is no more than Template:Math. For a full cycle centered at the origin (Template:Math) the error is less than 0.08215. In particular, for Template:Math, the error is less than 0.000003.
In contrast, also shown is a picture of the natural logarithm function Template:Math and some of its Taylor polynomials around Template:Math. These approximations converge to the function only in the region Template:Math; outside of this region the higher-degree Taylor polynomials are worse approximations for the function.
The error incurred in approximating a function by its Template:Mvarth-degree Taylor polynomial is called the remainder or residual and is denoted by the function Template:Math. Taylor's theorem can be used to obtain a bound on the size of the remainder.
In general, Taylor series need not be convergent at all. In fact, the set of functions with a convergent Taylor series is a meager set in the Fréchet space of smooth functions. Even if the Taylor series of a function Template:Mvar does converge, its limit need not be equal to the value of the function Template:Math. For example, the function
<math display="block"> f(x) = \begin{cases}
e^{-1/x^2} & \text{if } x \neq 0 \\[3mu] 0 & \text{if } x = 0
\end{cases} </math>
is infinitely differentiable at Template:Math, and has all derivatives zero there. Consequently, the Taylor series of Template:Math about Template:Math is identically zero. However, Template:Math is not the zero function, so does not equal its Taylor series around the origin. Thus, Template:Math is an example of a non-analytic smooth function.
In real analysis, this example shows that there are infinitely differentiable functions Template:Math whose Taylor series are not equal to Template:Math even if they converge. By contrast, the holomorphic functions studied in complex analysis always possess a convergent Taylor series, and even the Taylor series of meromorphic functions, which might have singularities, never converge to a value different from the function itself. The complex function Template:Math, however, does not approach 0 when Template:Mvar approaches 0 along the imaginary axis, so it is not continuous in the complex plane and its Taylor series is undefined at 0.
More generally, 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, a consequence of Borel's lemma. As a result, the radius of convergence of a Taylor series can be zero. There are even infinitely differentiable functions defined on the real line whose Taylor series have a radius of convergence 0 everywhere.Template:Sfn
A function cannot be written as a Taylor series centred at a singularity; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable Template:Mvar; see Laurent series. For example, Template:Math can be written as a Laurent series.
GeneralizationEdit
The generalization of the Taylor series does converge to the value of the function itself for any bounded continuous function on Template:Math, and this can be done by using the calculus of finite differences. Specifically, the following theorem, due to Einar Hille, that for any Template:Math,<ref>Template:Multiref</ref>
<math display="block" >\lim_{h\to 0^+}\sum_{n=0}^\infty \frac{t^n}{n!}\frac{\Delta_h^nf(a)}{h^n} = f(a+t).</math>
Here Template:Math is the Template:Mvarth finite difference operator with step size Template:Mvar. The series is precisely the Taylor series, except that divided differences appear in place of differentiation: the series is formally similar to the Newton series. When the function Template:Mvar is analytic at Template:Mvar, the terms in the series converge to the terms of the Taylor series, and in this sense generalizes the usual Taylor series.
In general, for any infinite sequence Template:Math, the following power series identity holds:
<math display="block">\sum_{n=0}^\infty\frac{u^n}{n!}\Delta^na_i = e^{-u}\sum_{j=0}^\infty\frac{u^j}{j!}a_{i+j}.</math>
So in particular,
<math display="block">f(a+t) = \lim_{h\to 0^+} e^{-t/h}\sum_{j=0}^\infty f(a+jh) \frac{(t/h)^j}{j!}.</math>
The series on the right is the expected value of Template:Math, where Template:Mvar is a Poisson-distributed random variable that takes the value Template:Math with probability Template:Math. Hence,
<math display="block">f(a+t) = \lim_{h\to 0^+} \int_{-\infty}^\infty f(a+x)dP_{t/h,h}(x).</math>
The law of large numbers implies that the identity holds.Template:Sfn
List of Maclaurin series of some common functionsEdit
Template:See also Several important Maclaurin series expansions follow. All these expansions are valid for complex arguments Template:Mvar.
Exponential functionEdit
The exponential function <math>e^x</math> (with base Template:Mvar) has Maclaurin seriesTemplate:Sfn
<math display="block"> e^{x} = \sum^{\infty}_{n=0} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots. </math> It converges for all Template:Mvar.
The exponential generating function of the Bell numbers is the exponential function of the predecessor of the exponential function:
<math display="block">\exp(\exp{x}-1) = \sum_{n=0}^{\infty} \frac{B_n}{n!}x^{n}</math>
Natural logarithmEdit
The natural logarithm (with base Template:Mvar) has Maclaurin series<ref name="bileodeau-abramowitz">Template:Multiref</ref>
<math display="block"> \begin{align} \ln(1-x) &= - \sum^{\infty}_{n=1} \frac{x^n}n = -x - \frac{x^2}2 - \frac{x^3}3 - \cdots , \\ \ln(1+x) &= \sum^\infty_{n=1} (-1)^{n+1}\frac{x^n}n = x - \frac{x^2}2 + \frac{x^3}3 - \cdots . \end{align}</math>
The last series is known as Mercator series, named after Nicholas Mercator (since it was published in his 1668 treatise Logarithmotechnia).Template:Sfn Both of these series converge for <math>|x| < 1</math>. (In addition, the series for Template:Math converges for Template:Math, and the series for Template:Math converges for Template:Math.)<ref name="bileodeau-abramowitz" />
Geometric seriesEdit
The geometric series and its derivatives have Maclaurin series
<math display="block">\begin{align} \frac{1}{1-x} &= \sum^\infty_{n=0} x^n \\ \frac{1}{(1-x)^2} &= \sum^\infty_{n=1} nx^{n-1} \\ \frac{1}{(1-x)^3} &= \sum^\infty_{n=2} \frac{(n-1)n}{2} x^{n-2}. \end{align}</math>
All are convergent for <math>|x| < 1</math>. These are special cases of the binomial series given in the next section.
Binomial seriesEdit
The binomial series is the power series
<math display="block">(1+x)^\alpha = \sum_{n=0}^\infty \binom{\alpha}{n} x^n</math>
whose coefficients are the generalized binomial coefficientsTemplate:Sfn
<math display="block">\binom{\alpha}{n} = \prod_{k=1}^n \frac{\alpha-k+1}k = \frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}.</math>
(If Template:Math, this product is an empty product and has value 1.) It converges for <math>|x| < 1</math> for any real or complex number Template:Mvar.
When Template:Math, this is essentially the infinite geometric series mentioned in the previous section. The special cases Template:Math and Template:Math give the square root function and its inverse:Template:Sfn
<math display="block">\begin{align} (1+x)^\frac{1}{2} &= 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - \frac{5}{128}x^4 + \frac{7}{256}x^5 - \cdots &= \sum^{\infty}_{n=0} \frac{(-1)^{n-1}(2n)!}{4^n (n!)^2 (2n-1)} x^n, \\ (1+x)^{-\frac{1}{2}} &= 1 -\frac{1}{2}x + \frac{3}{8}x^2 - \frac{5}{16}x^3 + \frac{35}{128}x^4 - \frac{63}{256}x^5 + \cdots &= \sum^{\infty}_{n=0} \frac{(-1)^n(2n)!}{4^n (n!)^2} x^n. \end{align} </math>
When only the linear term is retained, this simplifies to the binomial approximation.
Trigonometric functionsEdit
The usual trigonometric functions and their inverses have the following Maclaurin series:Template:Sfn
<math display="block">\begin{align} \sin x &= \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1} &&= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots && \text{for all } x\\[6pt] \cos x &= \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n} &&= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots && \text{for all } x\\[6pt] \tan x &= \sum^{\infty}_{n=1} \frac{B_{2n} (-4)^n \left(1-4^n\right)}{(2n)!} x^{2n-1} &&= x + \frac{x^3}{3} + \frac{2 x^5}{15} + \cdots && \text{for }|x| < \frac{\pi}{2}\\[6pt] \sec x &= \sum^{\infty}_{n=0} \frac{(-1)^n E_{2n}}{(2n)!} x^{2n} &&=1+\frac{x^2}{2}+\frac{5x^4}{24}+\cdots && \text{for }|x| < \frac{\pi}{2}\\[6pt] \arcsin x &= \sum^{\infty}_{n=0} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1} &&=x+\frac{x^3}{6}+\frac{3x^5}{40}+\cdots && \text{for }|x| \le 1\\[6pt] \arccos x &=\frac{\pi}{2}-\arcsin x\\&=\frac{\pi}{2}- \sum^{\infty}_{n=0} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}&&=\frac{\pi}{2}-x-\frac{x^3}{6}-\frac{3x^5}{40}-\cdots&& \text{for }|x| \le 1\\[6pt] \arctan x &= \sum^{\infty}_{n=0} \frac{(-1)^n}{2n+1} x^{2n+1} &&=x-\frac{x^3}{3} + \frac{x^5}{5}-\cdots && \text{for }|x| \le 1,\ x\neq\pm i \end{align}</math>
All angles are expressed in radians. The numbers Template:Math appearing in the expansions of Template:Math are the Bernoulli numbers. The Template:Math in the expansion of Template:Math are Euler numbers.Template:Sfn
Hyperbolic functionsEdit
The hyperbolic functions have Maclaurin series closely related to the series for the corresponding trigonometric functions:Template:Sfn
<math display="block">\begin{align} \sinh x &= \sum^{\infty}_{n=0} \frac{x^{2n+1}}{(2n+1)!} &&= x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots && \text{for all } x\\[6pt] \cosh x &= \sum^{\infty}_{n=0} \frac{x^{2n}}{(2n)!} &&= 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots && \text{for all } x\\[6pt] \tanh x &= \sum^{\infty}_{n=1} \frac{B_{2n} 4^n \left(4^n-1\right)}{(2n)!} x^{2n-1} &&= x-\frac{x^3}{3}+\frac{2x^5}{15}-\frac{17x^7}{315}+\cdots && \text{for }|x| < \frac{\pi}{2}\\[6pt] \operatorname{arsinh} x &= \sum^{\infty}_{n=0} \frac{(-1)^n (2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1} &&=x - \frac{x^3}{6} + \frac{3x^5}{40} - \cdots && \text{for }|x| \le 1\\[6pt] \operatorname{artanh} x &= \sum^{\infty}_{n=0} \frac{x^{2n+1}}{2n+1} &&=x + \frac{x^3}{3} + \frac{x^5}{5} +\cdots && \text{for }|x| \le 1,\ x\neq\pm 1 \end{align}</math>
The numbers Template:Math appearing in the series for Template:Math are the Bernoulli numbers.Template:Sfn
Polylogarithmic functionsEdit
The polylogarithms have these defining identities:
<math display="block">\begin{align} \text{Li}_{2}(x) &= \sum_{n = 1}^{\infty} \frac{1}{n^2} x^{n} \\\text{Li}_{3}(x) &= \sum_{n = 1}^{\infty} \frac{1}{n^3} x^{n} \end{align}</math>
The Legendre chi functions are defined as follows:
<math display="block">\begin{align} \chi_{2}(x) &= \sum_{n = 0}^{\infty} \frac{1}{(2n + 1)^2} x^{2n + 1} \\ \chi_{3}(x) &= \sum_{n = 0}^{\infty} \frac{1}{(2n + 1)^3} x^{2n + 1} \end{align}</math>
And the formulas presented below are called inverse tangent integrals:
<math display="block">\begin{align} \text{Ti}_{2}(x) &= \sum_{n = 0}^{\infty} \frac{(-1)^{n}}{(2n + 1)^2} x^{2n + 1} \\ \text{Ti}_{3}(x) &= \sum_{n = 0}^{\infty} \frac{(-1)^{n}}{(2n + 1)^3} x^{2n + 1} \end{align}</math>
In statistical thermodynamics these formulas are of great importance.
Elliptic functionsEdit
The complete elliptic integrals of first kind K and of second kind E can be defined as follows:
<math display="block">\begin{align} \frac{2}{\pi}K(x) &= \sum_{n = 0}^{\infty} \frac{[(2n)!]^2}{16^{n}(n!)^4}x^{2n} \\ \frac{2}{\pi}E(x) &= \sum_{n = 0}^{\infty} \frac{[(2n)!]^2}{(1 - 2n)16^{n}(n!)^4}x^{2n} \end{align}</math>
The Jacobi theta functions describe the world of the elliptic modular functions and they have these Taylor series:
<math display="block">\begin{align} \vartheta_{00}(x) &= 1 + 2\sum_{n = 1}^{\infty} x^{n^2} \\ \vartheta_{01}(x) &= 1 + 2\sum_{n = 1}^{\infty} (-1)^{n} x^{n^2} \end{align}</math>
The regular partition number sequence P(n) has this generating function:
<math display="block">\vartheta_{00}(x)^{-1/6}\vartheta_{01}(x)^{-2/3}\biggl[\frac{\vartheta_{00}(x)^4 - \vartheta_{01}(x)^4}{16\,x}\biggr]^{-1/24} = \sum_{n=0}^{\infty} P(n)x^n = \prod_{k = 1}^{\infty} \frac{1}{1 - x^{k}}</math>
The strict partition number sequence Q(n) has that generating function:
<math display="block">\vartheta_{00}(x)^{1/6}\vartheta_{01}(x)^{-1/3}\biggl[\frac{\vartheta_{00}(x)^4 - \vartheta_{01}(x)^4}{16\,x}\biggr]^{1/24} = \sum_{n=0}^{\infty} Q(n)x^n = \prod_{k = 1}^{\infty} \frac{1}{1 - x^{2k - 1}}</math>
Calculation of Taylor seriesEdit
Several methods exist for the calculation of Taylor series of a large number of functions. One can attempt to use the definition of the Taylor series, though this often requires generalizing the form of the coefficients according to a readily apparent pattern. Alternatively, 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.
First exampleEdit
In order to compute the 7th degree Maclaurin polynomial for the function
<math display="block">f(x)=\ln(\cos x),\quad x\in\bigl({-\tfrac\pi2}, \tfrac\pi2\bigr),</math>
one may first rewrite the function as
<math display="block">f(x)={\ln}\bigl(1+(\cos x-1)\bigr),</math>
the composition of two functions <math>x \mapsto \ln(1 + x)</math> and <math>x \mapsto \cos x - 1.</math> The Taylor series for the natural logarithm is (using big O notation)
<math display="block">\ln(1+x) = x - \frac{x^2}2 + \frac{x^3}3 + O{\left(x^4\right)}</math>
and for the cosine function
<math display="block">\cos x - 1 = -\frac{x^2}2 + \frac{x^4}{24} - \frac{x^6}{720} + O{\left(x^8\right)}.</math>
The first several terms from the second series can be substituted into each term of the first series. Because the first term in the second series has degree 2, three terms of the first series suffice to give a 7th-degree polynomial:
<math display="block">\begin{align}f(x) &= \ln\bigl(1+(\cos x-1)\bigr) \\ &= (\cos x-1) - \tfrac12(\cos x-1)^2 + \tfrac13(\cos x-1)^3+ O{\left((\cos x-1)^4\right)} \\ &= - \frac{x^2}2 - \frac{x^4}{12} - \frac{x^6}{45}+O{\left(x^8\right)}. \end{align}\!</math>
Since the cosine is an even function, the coefficients for all the odd powers are zero.
Second exampleEdit
Suppose we want the Taylor series at 0 of the function
<math display="block">g(x)=\frac{e^x}{\cos x}.\!</math>
The Taylor series for the exponential function is
<math display="block">e^x =1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!}+\cdots,</math>
and the series for cosine is
<math display="block">\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots.</math>
Assume the series for their quotient is
<math display="block">\frac{e^x}{\cos x} = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + c_4x^4 + \cdots</math>
Multiplying both sides by the denominator <math>\cos x</math> and then expanding it as a series yields
<math display="block">\begin{align} e^x &= \left(c_0 + c_1 x + c_2 x^2 + c_3 x^3 + c_4x^4 + \cdots\right)\left(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\right) \\[5mu] &= c_0 + c_1x + \left(c_2 - \frac{c_0}{2}\right)x^2 + \left(c_3 - \frac{c_1}{2}\right)x^3+\left(c_4-\frac{c_2}{2}+\frac{c_0}{4!}\right)x^4 + \cdots \end{align}</math>
Comparing the coefficients of <math>g(x)\cos x</math> with the coefficients of <math>e^x,</math>
<math display="block"> c_0 = 1,\ \ c_1 = 1,\ \ c_2 - \tfrac12 c_0 = \tfrac12,\ \ c_3 - \tfrac12 c_1 = \tfrac16,\ \ c_4 - \tfrac12 c_2 + \tfrac1{24} c_0 = \tfrac1{24},\ \ldots. </math>
The coefficients <math>c_i</math> of the series for <math>g(x)</math> can thus be computed one at a time, amounting to long division of the series for <math>e^x</math> and Template:Nobr
<math display="block">\frac{e^x}{\cos x}=1 + x + x^2 + \tfrac23 x^3 + \tfrac12 x^4 + \cdots.</math>
Third exampleEdit
Here we employ a method called "indirect expansion" to expand the given function. This method uses the known Taylor expansion of the exponential function. In order to expand Template:Math as a Taylor series in Template:Mvar, we use the known Taylor series of function Template:Math:
<math display="block">e^x = \sum^\infty_{n=0} \frac{x^n}{n!} =1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!}+\cdots.</math>
Thus,
<math display="block">\begin{align}(1+x)e^x &= e^x + xe^x = \sum^\infty_{n=0} \frac{x^n}{n!} + \sum^\infty_{n=0} \frac{x^{n+1}}{n!} = 1 + \sum^\infty_{n=1} \frac{x^n}{n!} + \sum^\infty_{n=0} \frac{x^{n+1}}{n!} \\ &= 1 + \sum^\infty_{n=1} \frac{x^n}{n!} + \sum^\infty_{n=1} \frac{x^n}{(n-1)!} =1 + \sum^\infty_{n=1}\left(\frac{1}{n!} + \frac{1}{(n-1)!}\right)x^n \\ &= 1 + \sum^\infty_{n=1}\frac{n+1}{n!}x^n\\ &= \sum^\infty_{n=0}\frac{n+1}{n!}x^n.\end{align}</math>
Taylor series as definitionsEdit
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. For example, the exponential function is the function which is equal to its own derivative everywhere, and assumes the value 1 at the origin. However, one may equally well define an analytic function by its Taylor series.
Taylor series are used to define functions and "operators" 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 extend analytic functions to sets of matrices and operators, such as the matrix exponential or matrix logarithm.
In other areas, such as formal analysis, it is more convenient to work directly with the power series themselves. 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 variablesTemplate:AnchorEdit
The Taylor series may also be generalized to functions of more than one variable with<ref>Template:Multiref</ref>
<math display="block">\begin{align} T(x_1,\ldots,x_d) &= \sum_{n_1=0}^\infty \cdots \sum_{n_d = 0}^\infty \frac{(x_1-a_1)^{n_1}\cdots (x_d-a_d)^{n_d}}{n_1!\cdots n_d!}\,\left(\frac{\partial^{n_1 + \cdots + n_d}f}{\partial x_1^{n_1}\cdots \partial x_d^{n_d}}\right)(a_1,\ldots,a_d) \\ &= f(a_1, \ldots,a_d) + \sum_{j=1}^d \frac{\partial f(a_1, \ldots,a_d)}{\partial x_j} (x_j - a_j) + \frac{1}{2!} \sum_{j=1}^d \sum_{k=1}^d \frac{\partial^2 f(a_1, \ldots,a_d)}{\partial x_j \partial x_k} (x_j - a_j)(x_k - a_k) \\ & \qquad \qquad + \frac{1}{3!} \sum_{j=1}^d\sum_{k=1}^d\sum_{l=1}^d \frac{\partial^3 f(a_1, \ldots,a_d)}{\partial x_j \partial x_k \partial x_l} (x_j - a_j)(x_k - a_k)(x_l - a_l) + \cdots \end{align}</math>
For example, for a function <math>f(x,y)</math> that depends on two variables, Template:Mvar and Template:Mvar, the Taylor series to second order about the point Template:Math is
<math display="block">f(a,b) +(x-a) f_x(a,b) +(y-b) f_y(a,b) + \frac{1}{2!}\Big( (x-a)^2 f_{xx}(a,b) + 2(x-a)(y-b) f_{xy}(a,b) +(y-b)^2 f_{yy}(a,b) \Big)</math>
where the subscripts denote the respective partial derivatives.
Second-order Taylor series in several variablesEdit
A second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as
<math display="block">T(\mathbf{x}) = f(\mathbf{a}) + (\mathbf{x} - \mathbf{a})^\mathsf{T} D f(\mathbf{a}) + \frac{1}{2!} (\mathbf{x} - \mathbf{a})^\mathsf{T} \left \{D^2 f(\mathbf{a}) \right \} (\mathbf{x} - \mathbf{a}) + \cdots,</math>
where Template:Math is the gradient of Template:Mvar evaluated at Template:Math and Template:Math is the Hessian matrix. Applying the multi-index notation the Taylor series for several variables becomes
<math display="block">T(\mathbf{x}) = \sum_{|\alpha| \geq 0}\frac{(\mathbf{x}-\mathbf{a})^\alpha}{\alpha !} \left({\mathrm{\partial}^{\alpha}}f\right)(\mathbf{a}),</math>
which is to be understood as a still more abbreviated multi-index version of the first equation of this paragraph, with a full analogy to the single variable case.
ExampleEdit
In order to compute a second-order Taylor series expansion around point Template:Math of the function <math display="block">f(x,y)=e^x\ln(1+y),</math>
one first computes all the necessary partial derivatives:
<math display="block">\begin{align} f_x &= e^x\ln(1+y) \\[6pt] f_y &= \frac{e^x}{1+y} \\[6pt] f_{xx} &= e^x\ln(1+y) \\[6pt] f_{yy} &= - \frac{e^x}{(1+y)^2} \\[6pt] f_{xy} &=f_{yx} = \frac{e^x}{1+y} . \end{align}</math>
Evaluating these derivatives at the origin gives the Taylor coefficients
<math display="block">\begin{align} f_x(0,0) &= 0 \\ f_y(0,0) &=1 \\ f_{xx}(0,0) &=0 \\ f_{yy}(0,0) &=-1 \\ f_{xy}(0,0) &=f_{yx}(0,0)=1. \end{align}</math>
Substituting these values in to the general formula
<math display="block">\begin{align} T(x,y) = &f(a,b) +(x-a) f_x(a,b) +(y-b) f_y(a,b) \\ &{}+\frac{1}{2!}\left( (x-a)^2f_{xx}(a,b) + 2(x-a)(y-b)f_{xy}(a,b) +(y-b)^2 f_{yy}(a,b) \right)+ \cdots \end{align}</math>
produces
<math display="block">\begin{align} T(x,y) &= 0 + 0(x-0) + 1(y-0) + \frac{1}{2}\big( 0(x-0)^2 + 2(x-0)(y-0) + (-1)(y-0)^2 \big) + \cdots \\ &= y + xy - \tfrac12 y^2 + \cdots \end{align}</math>
Since Template:Math is analytic in Template:Math, we have
<math display="block">e^x\ln(1+y)= y + xy - \tfrac12 y^2 + \cdots, \qquad |y| < 1.</math>
Comparison with Fourier seriesEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The trigonometric Fourier series enables one to express a periodic function (or a function defined on a closed interval Template:Math) as an infinite sum of trigonometric functions (sines and cosines). In this sense, the Fourier series is analogous to Taylor series, since the latter allows one to express a function as an infinite sum of powers. Nevertheless, the two series differ from each other in several relevant issues:
- The finite truncations of the Taylor series of Template:Math about the point Template:Math are all exactly equal to Template:Math at Template:Math. In contrast, the Fourier series is computed by integrating over an entire interval, so there is generally no such point where all the finite truncations of the series are exact.
- The computation of Taylor series requires the knowledge of the function on an arbitrary small neighbourhood of a point, whereas the computation of the Fourier series requires knowing the function on its whole domain interval. In a certain sense one could say that the Taylor series is "local" and the Fourier series is "global".
- The Taylor series is defined for a function which has infinitely many derivatives at a single point, whereas the Fourier series is defined for any integrable function. In particular, the function could be nowhere differentiable. (For example, Template:Math could be a Weierstrass function.)
- The convergence of both series has very different properties. Even if the Taylor series has positive convergence radius, the resulting series may not coincide with the function; but if the function is analytic then the series converges pointwise to the function, and uniformly on every compact subset of the convergence interval. Concerning the Fourier series, if the function is square-integrable then the series converges in quadratic mean, but additional requirements are needed to ensure the pointwise or uniform convergence (for instance, if the function is periodic and of class C1 then the convergence is uniform).
- Finally, in practice one wants to approximate the function with a finite number of terms, say with a Taylor polynomial or a partial sum of the trigonometric series, respectively. In the case of the Taylor series the error is very small in a neighbourhood of the point where it is computed, while it may be very large at a distant point. In the case of the Fourier series the error is distributed along the domain of the function.
See alsoEdit
- Asymptotic expansion
- Newton polynomial
- Padé approximant – best approximation by a rational function
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- Approximation theory
- Function approximation
NotesEdit
ReferencesEdit
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External linksEdit
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- {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web
|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:TaylorSeries%7CTaylorSeries.html}} |title = Taylor Series |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }} Template:Series (mathematics) Template:Authority control