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| {{for|other notions of series expansion|Series (mathematics)}}
| | Hi there. My title is Garland even though it is not the title on my birth certificate. Interviewing is what she does in her working day job but quickly her husband and her will start their own company. To play croquet is the pastime I will by no means quit doing. Her husband and her selected to reside in Alabama.<br><br>Check out my blog post: [http://www.johnwilkie.co.uk/users.php?mode=profile&uid=58653 www.johnwilkie.co.uk] |
| {{good article}}
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| [[File:sintay_SVG.svg|thumb|300 px|As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sin(x) and its Taylor approximations, polynomials of degree <span style="color:red;">1</span>, <span style="color:orange;">3</span>, <span style="color:yellow;">5</span>, <span style="color:green;">7</span>, <span style="color:blue;">9</span>, <span style="color:indigo;">11</span> and <span style="color:violet;">13</span>.]]
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| [[File:Exp series.gif|right|thumb|The [[exponential function]] ''e''<sup>''x''</sup> (in blue), and the sum of the first ''n''+1 terms of its Taylor series at 0 (in red).]]
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| {{Calculus |Series}}
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| In [[mathematics]], a '''Taylor series''' is a representation of a [[function (mathematics)|function]] as an [[Series (mathematics)|infinite sum]] of terms that are calculated from the values of the function's [[derivative]]s at a single point.
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| The concept of a Taylor series was discovered by the Scottish mathematician [[James Gregory (mathematician)|James Gregory]] and formally introduced by the English mathematician [[Brook Taylor]] in 1715. If the Taylor series is centered at zero, then that series is also called a '''Maclaurin series''', named after the Scottish mathematician [[Colin Maclaurin]], who made extensive use of this special case of Taylor series in the 18th century.
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| It is common practice to approximate a function by using a finite number of terms of its Taylor series. [[Taylor's theorem]] gives quantitative estimates on the error in this approximation. Any finite number of initial terms of the Taylor series of a function is called a [[Taylor polynomial]]. The Taylor series of a function is the [[limit of a sequence|limit]] of that function's Taylor polynomials, provided that the limit exists. A function may not be equal to its Taylor series, even if its Taylor series converges at every point. A function that is equal to its Taylor series in an [[open interval]] (or a disc in the [[complex plane]]) is known as an [[analytic function]].
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| ==Definition==
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| The Taylor series of a [[real-valued function|real]] or [[complex-valued function]] ''ƒ''(''x'') that is [[infinitely differentiable function|infinitely differentiable]] at a [[real number|real]] or [[complex number]] ''a'' is the [[power series]]
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| <!--
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| As stated below, the Taylor series need not equal the function. So please don't write f(x)=... here. In other words,
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| DO NOT CHANGE ANYTHING ABOUT THIS FORMULA-->:<math>f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f''(a)}{2!} (x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+ \cdots. </math><!---->
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| which can be written in the more compact [[Summation#Capital-sigma_notation|sigma notation]] as
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| :<math> \sum_{n=0} ^ {\infty} \frac {f^{(n)}(a)}{n!} \, (x-a)^{n}</math>
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| where ''n''! denotes the [[factorial]] of ''n'' and ''ƒ''<sup> (''n'')</sup>(''a'') denotes the ''n''th [[derivative]] of ''ƒ'' evaluated at the point ''a''. The derivative of order zero ''ƒ'' is defined to be ''ƒ'' itself and {{nowrap|(''x'' − ''a'')<sup>0</sup>}} and 0! are both defined to be 1. When {{nowrap|''a'' {{=}} 0}}, the series is also called a Maclaurin series.
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| ==Examples==
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| The Maclaurin series for any [[polynomial]] is the polynomial itself.
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| The Maclaurin series for {{nowrap|(1 − ''x'')<sup>−1</sup>}} at ''a'' = 0 is the [[geometric series]]
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| :<math>1+x+x^2+x^3+\cdots\!</math>
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| so the Taylor series for ''x''<sup>−1</sup> at {{nowrap|''a'' {{=}} 1}} is
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| :<math>1-(x-1)+(x-1)^2-(x-1)^3+\cdots.\!</math>
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| By integrating the above Maclaurin series we find the Maclaurin series for {{nowrap|log(1 − ''x'')}}, where log denotes the [[natural logarithm]]:
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| :<math>-x-\frac{1}{2}x^2-\frac{1}{3}x^3-\frac{1}{4}x^4-\cdots\!</math>
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| and the corresponding Taylor series for log(''x'') at {{nowrap|''a'' {{=}} 1}} is
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| :<math>(x-1)-\frac{1}{2}(x-1)^2+\frac{1}{3}(x-1)^3-\frac{1}{4}(x-1)^4+\cdots,\!</math>
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| and more generally, the corresponding Taylor series for log(''x'') at some ''a'' = ''x''<sub>0</sub> is:
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| : <math> \log ( x_0 ) + \frac{1}{x_0} ( x - x_0 ) - \frac{1}{x_0^2}\frac{( x - x_0 )^2}{2} + \cdots.</math>
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| The Taylor series for the [[exponential function]] e<sup>''x''</sup> at ''a'' = 0 is
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| :<math>1 + \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\! = \sum_{n=0}^\infty \frac{x^n}{n!}.</math>
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| The above expansion holds because the derivative of e<sup>''x''</sup> with respect to x is also e<sup>''x''</sup> and e<sup>0</sup> equals 1. This leaves the terms {{nowrap|(''x'' − 0)<sup>''n''</sup>}} in the numerator and ''n''<nowiki>!</nowiki> in the denominator for each term in the infinite sum.
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| ==History==
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| The Greek philosopher [[Zeno of Elea|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 result.<ref>{{cite book |last=Kline |first=M. |year=1990 |title=Mathematical Thought from Ancient to Modern Times |location=New York |publisher=Oxford University Press |pages=35–37 |isbn=0-19-506135-7 }}</ref> [[Liu Hui]] independently employed a similar method a few centuries later.<ref>{{cite book |last=Boyer |first=C. |last2=Merzbach |first2=U. |year=1991 |title=A History of Mathematics |location= |publisher=John Wiley and Sons |edition=Second revised |pages=202–203 |isbn=0-471-09763-2 }}<!--I'm sure there are better refs than this. Hui gave fairly "rigorous" bounds on the convergence, if I recall. But it isn't addressed here.--></ref>
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| In the 14th century, the earliest examples of the use of Taylor series and closely related methods were given by [[Madhava of Sangamagrama]].<ref name="MAT 314">{{cite web| publisher=Canisius College| work=MAT 314|url=http://www.canisius.edu/topos/rajeev.asp| title=Neither Newton nor Leibniz – The Pre-History of Calculus and Celestial Mechanics in Medieval Kerala| accessdate=2006-07-09}}</ref><ref name="dani">{{cite journal| title=Ancient Indian Mathematics – A Conspectus| author= S. G. Dani|journal= Resonance |volume=17|issue=3|year=2012|pages=236–246}}</ref> Though no record of his work survives, writings of later [[Indian mathematics|Indian mathematicians]] suggest that he found a number of special cases of the Taylor series, including those for the [[trigonometric function]]s of [[sine]], [[cosine]], [[tangent (trigonometric function)|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.
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| In the 17th century, [[James Gregory (astronomer and mathematician)|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]],<ref>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.</ref> after whom the series are now named.
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| 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.
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| ==Analytic functions==
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| [[Image:Exp neg inverse square.svg|200px|thumb|right|The function <strong style="color:#803300">e<sup>−1/x²</sup></strong> is not analytic at ''x'' = 0: the Taylor series is identically 0, although the function is not.]]
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| If ''f''(''x'') is given by a convergent power series in an open disc (or interval in the real line) centered at ''b'' in the complex plane, it is said to be [[analytic function|analytic]] in this disc. Thus for ''x'' in this disc, ''f'' is given by a convergent power series
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| :<math>f(x) = \sum_{n=0}^\infty a_n(x-b)^n.</math>
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| Differentiating by ''x'' the above formula ''n'' times, then setting ''x''=''b'' gives:
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| :<math>\frac{f^{(n)}(b)}{n!} = a_n</math>
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| and so the power series expansion agrees with the Taylor series. Thus a function is analytic in an open disc centered at ''b'' if and only if its Taylor series converges to the value of the function at each point of the disc.
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| If ''f''(''x'') is equal to its Taylor series for all ''x'' in the complex plane, it is called [[entire function|entire]]. The polynomials and the [[exponential function]] ''e''<sup>''x''</sup> and the [[trigonometric function]]s 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 [[convergent series|converge]] if ''x'' is far from ''b''. 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.
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| Uses of the Taylor series for analytic functions include:
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| # The partial sums (the [[Taylor polynomial]]s) of the series can be used as approximations of the entire function. These approximations are good if sufficiently many terms are included.
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| #Differentiation and integration of power series can be performed term by term and is hence particularly easy.
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| #An [[analytic function]] is uniquely extended to a [[holomorphic function]] on an [[open disk]] in the [[complex number|complex plane]]. This makes the machinery of [[complex analysis]] available.
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| #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]]).
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| #Algebraic operations can be done readily on the power series representation; for instance the [[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]].
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| #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.
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| ==Approximation and convergence==
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| [[Image:Taylorsine.svg|300px|thumb|right|The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full period centered at the origin.]]
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| [[Image:LogTay.svg|300px|thumb|right|The Taylor polynomials for log(''1+x'') only provide accurate approximations in the range {{nowrap|−1 < ''x'' ≤ 1}}. Note that, for {{nowrap|''x'' > 1}}, the Taylor polynomials of higher degree are '''''worse''''' approximations.]]
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| [[Image:Logarithm GIF.gif|300px|thumb|right|The Taylor approximations for log(''1+x'') (black). For ''x > 1'', any Taylor approximation is invalid.]]
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| Pictured on the right is an accurate approximation of sin(''x'') around the point ''x'' = 0. The pink curve is a polynomial of degree seven:
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| :<math>\sin\left( x \right) \approx x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}.\!</math>
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| The error in this approximation is no more than |''x''|<sup>9</sup>/9<nowiki>!</nowiki>. In particular, for {{nowrap|−1 < ''x'' < 1}}, the error is less than 0.000003.
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| In contrast, also shown is a picture of the natural logarithm function {{nowrap|[[Natural logarithm|log]](1 + ''x'')}} and some of its [[Taylor polynomial]]s around ''a'' = 0. 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 similar to [[Runge's phenomenon]].
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| The '''error''' incurred in approximating a function by its ''n''th-degree Taylor polynomial is called the '''remainder''' or ''[[Residual (numerical analysis)|residual]]'' and is denoted by the function ''R<sub>n</sub>(x)''.
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| [[Taylor's theorem]] can be used to obtain a bound on the size of the remainder.
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| In general, Taylor series need not be [[convergent series|convergent]] at all. And in fact the set of functions with a convergent Taylor series is a [[meager set]] in the [[Fréchet space]] of [[smooth functions]]. And even if the Taylor series of a function ''f'' does converge, its limit need not in general be equal to the value of the function ''f''(''x''). For example, the function
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| :<math>
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| f(x) = \begin{cases}
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| e^{-1/x^2}&\mathrm{if}\ x\not=0\\
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| 0&\mathrm{if}\ x=0
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| \end{cases}
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| </math>
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| is [[infinitely differentiable]] at {{nowrap|''x'' {{=}} 0}}, and has all derivatives zero there. Consequently, the Taylor series of ''f''(''x'') about {{nowrap|''x'' {{=}} 0}} is identically zero. However, ''f''(''x'') is not equal to the zero function, and so it is not equal to its Taylor series around the origin.
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| In [[real analysis]], this example shows that there are [[infinitely differentiable function]]s ''f''(''x'') whose Taylor series are ''not'' equal to ''f''(''x'') even if they converge. By contrast, the [[holomorphic function]]s studied in [[complex analysis]] always possess a convergent Taylor series, and even the Taylor series of [[meromorphic function]]s, which might have singularities, never converge to a value different from the function itself. The complex function e<sup>−''z''<sup>−2</sup></sup>, however, does not approach 0 when ''z'' approaches 0 along the imaginary axis, so it is not [[Continuous function|continuous]] in the complex plane and its Taylor series is undefined at 0.
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| More generally, every sequence of real or complex numbers can appear as [[coefficient]]s in the Taylor series of an infinitely differentiable function defined on the real line, a consequence of [[Borel's lemma]] (see also [[Non-analytic smooth function#Application to Taylor series|Non-analytic smooth function]]). 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.<ref>{{Citation | last = Rudin | first = Walter | author-link = Walter Rudin| title = Real and Complex Analysis | place = New Dehli | publisher = McGraw-Hill | year = 1980 | page = 418, Exercise 13 | isbn = 0-07-099557-5 | postscript = <!--none-->}}</ref>
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| Some functions cannot be written as Taylor series because they have a [[singularity (mathematics)|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''<sup>−''x''<sup>−2</sup></sup> can be written as a Laurent series.
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| ===Generalization===
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| There is, however, a generalization<ref>{{citation|first=William|last=Feller|authorlink=William Feller|title=An introduction to probability theory and its applications, Volume 2|edition=3rd|publisher=Wiley|year=1971|pages=230–232}}.</ref><ref>{{citation|first1=Einar|last1=Hille|authorlink1=Einar Hille|first2=Ralph S.|last2=Phillips|authorlink2=Ralph S. Phillips|title=Functional analysis and semi-groups|publisher=American Mathematical Society|series=AMS Colloquium Publications|volume=31|year=1957|page=300–327}}.</ref> of the Taylor series that does converge to the value of the function itself for any [[bounded function|bounded]] [[continuous function]] on (0,∞), using the calculus of [[finite differences]]. Specifically, one has the following theorem, due to [[Einar Hille]], that for any ''t'' > 0,
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| :<math>\lim_{h\to 0^+}\sum_{n=0}^\infty \frac{t^n}{n!}\frac{\Delta_h^nf(a)}{h^n} = f(a+t).</math>
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| Here Δ{{su|p=''n''|b=''h''}} is the ''n''-th finite difference operator with step size ''h''. 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 ''f'' is analytic at ''a'', the terms in the series converge to the terms of the Taylor series, and in this sense generalizes the usual Taylor series.
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| In general, for any infinite sequence ''a''<sub>''i''</sub>, the following power series identity holds:
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| :<math>\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>
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| So in particular,
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| :<math>f(a+t) = \lim_{h\to 0^+} e^{-t/h}\sum_{j=0}^\infty f(a+jh) \frac{(t/h)^j}{j!}.</math>
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| The series on the right is the [[expectation value]] of ''f''(a + ''X''), where ''X'' is a [[Poisson distribution|Poisson distributed]] [[random variable]] that takes the value ''jh'' with probability ''e''<sup>−''t''/''h''</sup>(''t''/''h'')<sup>''j''</sup>/''j''!. Hence,
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| :<math>f(a+t) = \lim_{h\to 0^+} \int\limits_{-\infty}^\infty f(a+x)dP_{t/h,h}(x).</math>
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| The [[law of large numbers]] implies that the identity holds.
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| ==List of Maclaurin series of some common functions==
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| :''See also [[List of mathematical series]]''
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| [[Image:TaylorCosCosEnhanced.svg|150px|thumb|right|The real part of the cosine function in the [[complex plane]]]]
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| [[Image:TaylorCosPolSVG.svg|150px|thumb|right|An 8th-degree approximation of the cosine function in the [[complex plane]]]]
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| [[Image:TaylorCosAllSVG.svg|150px|thumb|right|The two above curves put together]]
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| <!-- [[Image:TaylorCosAnim.gif|280px|thumb|right|An animation of the approximation.]] -->
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| Several important Maclaurin series expansions follow.<ref>Most of these can be found in {{harv|Abramowitz|Stegun|1970}}.</ref> All these expansions are valid for complex arguments ''x''.
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| [[Exponential function]]
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| :<math>e^{x} = \sum^{\infty}_{n=0} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\quad\text{ for all } x\!</math>
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| [[Natural logarithm]]:
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| :<math>\log(1-x) = - \sum^{\infty}_{n=1} \frac{x^n}n\quad\text{ for } |x| < 1</math>
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| :<math>\log(1+x) = \sum^\infty_{n=1} (-1)^{n+1}\frac{x^n}n\quad\text{ for } |x| < 1</math>
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| [[Geometric series]] (see article for variants):
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| :<math>\frac{1}{1-x} = \sum^\infty_{n=0} x^n\quad\text{ for }|x| < 1\!</math>
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| [[Binomial series]] (includes the square root for ''α'' = 1/2 and the infinite geometric series for ''α'' = −1):
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| :<math>(1+x)^\alpha = \sum_{n=0}^\infty {\alpha \choose n} x^n\quad\text{ for all }|x| < 1 \text{ and all complex } \alpha\!</math>
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| which, with the first several terms written out explicitly for the common [[square root]] cases, is:
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| :<math>(1+x)^{0.5} = \textstyle 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</math>
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| :<math>(1+x)^{-0.5} = \textstyle 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</math>
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| with generalized [[binomial coefficient]]s
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| : <math>{\alpha\choose n} = \prod_{k=1}^n \frac{\alpha-k+1}k = \frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}</math>
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| [[Trigonometric function]]s:
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| :<math>\sin x = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\quad\text{ for all } x\!</math>
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| :<math>\cos x = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\quad\text{ for all } x\!</math>
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| :<math>\tan x = \sum^{\infty}_{n=1} \frac{B_{2n} (-4)^n (1-4^n)}{(2n)!} x^{2n-1} = x + \frac{x^3}{3} + \frac{2 x^5}{15} + \cdots\quad\text{ for }|x| < \frac{\pi}{2}\!</math>
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| :<math>\sec x = \sum^{\infty}_{n=0} \frac{(-1)^n E_{2n}}{(2n)!} x^{2n}\quad\text{ for }|x| < \frac{\pi}{2}\!</math>
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| :<math>\arcsin x = \sum^{\infty}_{n=0} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}\quad\text{ for }|x| \le 1\!</math>
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| :<math>\arccos x ={\pi\over 2}-\arcsin x={\pi\over 2}- \sum^{\infty}_{n=0} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}\quad\text{ for }|x| \le 1\!</math>
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| :<math>\arctan x = \sum^{\infty}_{n=0} \frac{(-1)^n}{2n+1} x^{2n+1}\quad\text{ for }|x| \le 1, x\not=\pm i\!</math>
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| [[Hyperbolic function]]s:
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| :<math>\sinh x = \sum^{\infty}_{n=0} \frac{x^{2n+1}}{(2n+1)!} = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots\quad\text{ for all } x\!</math>
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| :<math>\cosh x = \sum^{\infty}_{n=0} \frac{x^{2n}}{(2n)!} = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots\quad\text{ for all } x\!</math>
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| :<math>\tanh x = \sum^{\infty}_{n=1} \frac{B_{2n} 4^n (4^n-1)}{(2n)!} x^{2n-1} = x-\frac{1}{3}x^3+\frac{2}{15}x^5-\frac{17}{315}x^7+\cdots \quad\text{ for }|x| < \frac{\pi}{2}\!</math>
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| :<math>\mathrm{arcsinh} (x) = \sum^{\infty}_{n=0} \frac{(-1)^n (2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}\quad\text{ for }|x| \le 1\!</math>
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| :<math>\mathrm{arctanh} (x) = \sum^{\infty}_{n=0} \frac{x^{2n+1}}{2n+1} \quad\text{ for }|x| \le 1, x\not=\pm 1\!</math>
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| The numbers ''B''<sub>''k''</sub> appearing in the ''summation'' expansions of tan(''x'') and tanh(''x'') are the [[Bernoulli numbers]]. The ''E''<sub>''k''</sub> in the expansion of sec(''x'') are [[Euler number]]s.
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| ==Calculation of Taylor series==
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| 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 system]]s to calculate Taylor series.
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| ===First example===
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| Compute the 7th degree Maclaurin polynomial for the function
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| :<math>f(x)=\log\cos x, \quad x\in(-\pi/2, \pi/2)\!</math>.
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| First, rewrite the function as
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| :<math>f(x)=\log(1+(\cos x-1))\!</math>.
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| We have for the natural logarithm (by using the [[big O notation]])
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| :<math>\log(1+x) = x - \frac{x^2}2 + \frac{x^3}3 + {O}(x^4)\!</math>
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| and for the cosine function
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| :<math>\cos x - 1 = -\frac{x^2}2 + \frac{x^4}{24} - \frac{x^6}{720} + {O}(x^8)\!</math>
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| 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:
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| | |
| :<math>\begin{align}f(x)&=\log(1+(\cos x-1))\\
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| &=\bigl(\cos x-1\bigr) - \frac12\bigl(\cos x-1\bigr)^2 + \frac13\bigl(\cos x-1\bigr)^3+ {O}\bigl((\cos x-1)^4\bigr)\\&=\biggl(-\frac{x^2}2 + \frac{x^4}{24} - \frac{x^6}{720} +{O}(x^8)\biggr)-\frac12\biggl(-\frac{x^2}2+\frac{x^4}{24}+{O}(x^6)\biggr)^2+\frac13\biggl(-\frac{x^2}2+O(x^4)\biggr)^3 + {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} +O(x^8)\\
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| & =- \frac{x^2}2 - \frac{x^4}{12} - \frac{x^6}{45}+O(x^8). \end{align}\!</math>
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| Since the cosine is an [[even function]], the coefficients for all the odd powers ''x'', ''x''<sup>3</sub>, ''x''<sup>5</sub>, ''x''<sup>7</sub>, {{nowrap|.}}.. have to be zero.
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| | |
| ===Second example===
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| Suppose we want the Taylor series at 0 of the function
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| : <math>g(x)=\frac{e^x}{\cos x}.\!</math>
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| We have for the exponential function
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| : <math>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\!</math>
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| and, as in the first example,
| |
| : <math>\cos x = 1 - {x^2 \over 2!} + {x^4 \over 4!} - \cdots\!</math>
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| Assume the power series is
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| : <math>{e^x \over \cos x} = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots\!</math>
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| Then multiplication with the denominator and substitution of the series of the cosine yields
| |
| : <math>\begin{align} e^x &= (c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots)\cos x\\
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| &=\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}\!</math>
| |
| Collecting the terms up to fourth order yields
| |
| : <math>=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\!</math>
| |
| Comparing coefficients with the above series of the exponential function yields the desired Taylor series
| |
| : <math>\frac{e^x}{\cos x}=1 + x + x^2 + {2x^3 \over 3} + {x^4 \over 2} + \cdots.\!</math>
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| | |
| ===Third example===
| |
| Here we use a method called "Indirect Expansion" to expand the given function.
| |
| This method uses the known function of Taylor series for expansion.
| |
| | |
| Q: Expand the following function as a power series of x
| |
| : <math>(1 + x)e^x</math>.
| |
| We know the Taylor series of function <math>e^x</math> is:
| |
| : <math>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\!, -\infty<x<+\infty</math>
| |
| Thus,
| |
| : <math>\begin{align}(1+x)e^x &= e^x + xe^x = \sum^\infty_{n=0} {x^n\over n!} + \sum^\infty_{n=0} {x^{n+1}\over n!} = 1 + \sum^\infty_{n=1} {x^n\over n!} + \sum^\infty_{n=0} {x^{n+1}\over n!} \\ &= 1 + \sum^\infty_{n=1} {x^n\over n!} + \sum^\infty_{n=1} {x^{n}\over (n-1)!} =1 + \sum^\infty_{n=1}\left({1\over n!} + {1\over (n-1)!}\right)x^n \\ &= 1 + \sum^\infty_{n=1}{n+1\over n!}x^n, -\infty<x<+\infty \\ &= \sum^\infty_{n=0}{n+1\over n!}x^n\end{align}</math>
| |
| | |
| ==Taylor series as definitions==
| |
| Classically, [[algebraic function]]s are defined by an algebraic equation, and [[transcendental function]]s (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 "[[operator (mathematics)|operator]]s" 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 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 variables==
| |
| The Taylor series may also be generalized to functions of more than one variable with
| |
| :<math>
| |
| \begin{align}
| |
| T(x_1,\dots,x_d) &= \sum_{n_1=0}^\infty \sum_{n_2=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,\dots,a_d) \\
| |
| &= f(a_1, \dots,a_d) + \sum_{j=1}^d \frac{\partial f(a_1, \dots,a_d)}{\partial x_j} (x_j - a_j) \\
| |
| &\quad {} + \frac{1}{2!} \sum_{j=1}^d \sum_{k=1}^d \frac{\partial^2 f(a_1, \dots,a_d)}{\partial x_j \partial x_k} (x_j - a_j)(x_k - a_k) \\
| |
| &\quad {} + \frac{1}{3!} \sum_{j=1}^d\sum_{k=1}^d\sum_{l=1}^d \frac{\partial^3 f(a_1, \dots,a_d)}{\partial x_j \partial x_k \partial x_l} (x_j - a_j)(x_k - a_k)(x_l - a_l) + \dots
| |
| \end{align}
| |
| </math>
| |
| | |
| | |
| 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:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| f(x,y) &\approx f(a,b) +(x-a)\, f_x(a,b) +(y-b)\, f_y(a,b) \\
| |
| &\quad {} + \frac{1}{2!}\left[ (x-a)^2\,f_{xx}(a,b) + 2(x-a)(y-b)\,f_{xy}(a,b) +(y-b)^2\, f_{yy}(a,b) \right],
| |
| \end{align}
| |
| </math>
| |
| | |
| where the subscripts denote the respective [[partial derivative]]s.
| |
| | |
| A second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as
| |
| :<math>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\!
| |
| \,,</math>
| |
| | |
| where <math>D f(\mathbf{a})\!</math> is the [[gradient]] of <math>\,f</math> evaluated at <math>\mathbf{x} = \mathbf{a}</math> and <math>D^2 f(\mathbf{a})\!</math> is the [[Hessian matrix]]. Applying the [[multi-index notation]] the Taylor series for several variables becomes
| |
| | |
| :<math>T(\mathbf{x}) = \sum_{|\alpha| \ge 0}^{}\frac{(\mathbf{x}-\mathbf{a})^{\alpha}}{\alpha !}\,({\mathrm{\partial}^{\alpha}}\,f)(\mathbf{a})\,,</math>
| |
| | |
| which is to be understood as a still more abbreviated [[multi-index]] version of the first equation of this paragraph, again in full analogy to the single variable case.
| |
| | |
| === Example ===
| |
| [[Image:Second Order Taylor.svg|200px|thumb|right|Second-order Taylor series approximation (in orange) of a function <math>f(x,y) = e^x\log{(1+y)}</math> around origin.]]
| |
| Compute a second-order Taylor series expansion around point (''a'', ''b'') = (0, 0) of a function
| |
| :<math>f(x,y)=e^x\log(1+y).\,</math>
| |
| Firstly, we compute all partial derivatives we need
| |
| :<math>f_x(a,b)=e^x\log(1+y)\bigg|_{(x,y)=(0,0)}=0\,,</math>
| |
| | |
| :<math>f_y(a,b)=\frac{e^x}{1+y}\bigg|_{(x,y)=(0,0)}=1\,,</math>
| |
| | |
| :<math>f_{xx}(a,b)=e^x\log(1+y)\bigg|_{(x,y)=(0,0)}=0\,,</math>
| |
| | |
| :<math>f_{yy}(a,b)=-\frac{e^x}{(1+y)^2}\bigg|_{(x,y)=(0,0)}=-1\,,</math>
| |
| | |
| :<math>f_{xy}(a,b)=f_{yx}(a,b)=\frac{e^x}{1+y}\bigg|_{(x,y)=(0,0)}=1.</math>
| |
| | |
| The Taylor series is
| |
| | |
| :<math>\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)^2\,f_{xx}(a,b) + 2(x-a)(y-b)\,f_{xy}(a,b) +(y-b)^2\, f_{yy}(a,b) \right]+
| |
| \cdots\,,\end{align}</math>
| |
| | |
| which in this case becomes
| |
| | |
| :<math>\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 - \frac{y^2}{2} + \cdots. \end{align}
| |
| </math>
| |
| | |
| Since {{nowrap|log(1 + ''y'')}} is analytic in |''y''| < 1, we have
| |
| :<math>e^x\log(1+y)= y + xy - \frac{y^2}{2} + \cdots</math>
| |
| for |''y''| < 1.
| |
| | |
| ==Fractional Taylor series==
| |
| | |
| With the emergence of [[fractional calculus]], a natural question arises about what the Taylor Series expansion would be. Odibat and Shawagfeh answered this in 2007.<ref>{{cite doi|10.1016/j.amc.2006.07.102}}</ref> By using the Caputo fractional derivative, <math>0<\alpha<1\,\!</math>, and <math>x+\,\!</math> indicating the limit as we approach <math>x\,\!</math> from the right, the fractional Taylor series can be written as
| |
| | |
| :<math>f(x+\Delta x) = f(x) + {D_x}^\alpha f(x+)\frac{(\Delta x)^\alpha}{\Gamma(\alpha+1)} + {D_x}^\alpha {D_x}^\alpha f(x+)\frac{(\Delta x)^{2\alpha}}{\Gamma(2\alpha+1)} + \cdots.</math>
| |
| | |
| == Comparison with Fourier series ==
| |
| {{main|Fourier series}}
| |
| The trigonometric [[Fourier series]] allows one to express a [[periodic function]] (or a function defined on a compact interval) as an infinite sum of [[trigonometric function]]s ([[sine]]s and [[cosine]]s). In this sense, the Fourier series is analogous to Taylor series, since the latter allows to express a function as an infinite sum of [[power function|powers]]. Nevertheless both types of series differ in several relevant issues:
| |
| * The computation of Taylor series requires the knowledge of the function on an arbitrary small [[Neighbourhood_(mathematics)|neighbourhood]] of a point, whereas the computation of the Fourier series requires knowing the function on its whole domain [[interval (mathematics)|interval]]. In a certain sense one could say that the Taylor series is "local" and the Fourier series is "global."
| |
| * The computation of the Taylor series requires the function to be of class C<sup>∞</sup>, whereas that of Fourier series only requires the function to be [[integrable function|integrable]] (and thus may not even be continuous).
| |
| * 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 convergence|pointwise]] to the function, and [[uniform convergence|uniformly]] on every compact set. Concerning the Fourier series, if the function is [[Square-integrable function|square-integrable]] then the series converges in [[Convergence in quadratic mean|quadratic mean]], but additional requirements are needed to ensure the pointwise or uniform convergence (for instance, if the function is periodic and of class C<sup>1</sup> then the convergence is uniform).
| |
| * Finally, in practice one wants to approximate the function with a finite number of terms, let's 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 also==
| |
| * [[Laurent series]]
| |
| * [[Madhava series]]
| |
| * [[Newton polynomial|Newton's divided difference interpolation]]
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| *{{Citation| last1=Abramowitz| first1=Milton| author1-link=Milton Abramowitz| last2=Stegun| first2=Irene A.| author2-link=Irene Stegun| title=[[Abramowitz and Stegun|Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables]]| publisher=[[Dover Publications]]| location=New York| id=Ninth printing| year=1970| postscript=<!--none-->}}
| |
| *{{citation| last1=Thomas|first1=George B. Jr.|last2=Finney|first2=Ross L.| title=Calculus and Analytic Geometry (9th ed.)| publisher=Addison Wesley| year=1996| isbn=0-201-53174-7| postscript=<!--none-->}}
| |
| *{{citation| last=Greenberg|first=Michael| title=Advanced Engineering Mathematics (2nd ed.)| publisher=Prentice Hall| year=1998| isbn=0-13-321431-1| postscript=<!--none-->}}
| |
| | |
| ==External links==
| |
| * {{springer|title=Taylor series|id=p/t092320}}
| |
| * {{MathWorld| urlname= TaylorSeries| title= Taylor Series}}
| |
| * [http://www-groups.dcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch9_3.html Madhava of Sangamagramma ]
| |
| * [http://math.fullerton.edu/mathews/c2003/TaylorSeriesMod.html Taylor Series Representation Module by John H. Mathews]
| |
| * "[http://csma31.csm.jmu.edu/physics/rudmin/ParkerSochacki.htm Discussion of the Parker-Sochacki Method]"
| |
| * [http://stud3.tuwien.ac.at/~e0004876/taylor/Taylor_en.html Another Taylor visualisation] — where you can choose the point of the approximation and the number of derivatives
| |
| * [http://numericalmethods.eng.usf.edu/topics/taylor_series.html Taylor series revisited for numerical methods] at [http://numericalmethods.eng.usf.edu Numerical Methods for the STEM Undergraduate]
| |
| * [http://cinderella.de/files/HTMLDemos/2C02_Taylor.html Cinderella 2: Taylor expansion]
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| * [http://www.sosmath.com/calculus/tayser/tayser01/tayser01.html Taylor series]
| |
| * [http://www.efunda.com/math/taylor_series/inverse_trig.cfm Inverse trigonometric functions Taylor series]
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| [[Category:Real analysis]]
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| [[Category:Complex analysis]]
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| [[Category:Mathematical series]]
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| {{Link GA|ca}}
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