Taylor series

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"Series expansion" redirects here. For other notions of the term, see series.

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 is a representation or approximation of a function as a sum of terms calculated from the values of its derivatives at a single point.

Specifically, the Taylor series of an infinitely differentiable real (or complex) function f, defined on an open interval (a − r, a + r), is the power series

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

Here, n! is the factorial of n and f ⁽ⁿ⁾(a) denotes the nth derivative of f at the point a; the zeroth derivative of f is defined to be the function itself.

The series is named in honor of English mathematician Brook Taylor. If a = 0, the series is also called a Maclaurin series, named after Scottish mathematician Colin Maclaurin.

Functions that are equal to their Taylor series around any point a in their domain are called analytic functions. The trigonometric functions sine and cosine are examples of such functions. A Taylor series can be used to produce all the values of an analytic function, if the value of the function, and of all of its derivatives, is known at a single point. Uses of the Taylor series include:

The partial sums of the series can be used as a good approximation of the values of the function.
Replacing the function with the series is a tool used in many mathematical proofs.

Pictured in the right are increasingly accurate approximations of sin(x) around the point a = 0. The yellow curve is a polynomial of degree seven:

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

1 History
2 Properties
3 List of Taylor series of some common functions
4 Calculation of Taylor series
5 Taylor series as definitions
6 See also
7 References
8 External links

[edit] History

The Taylor series, power series, and infinite series expansions of functions may have been first discovered in India by Madhava in the 14th century. Though no record of his work survives, writings of later Indian mathematicians suggest that he found a number of special cases of the Taylor series, including the Taylor series for the trigonometric functions of sine, cosine, tangent and arctangent, and the second-order Taylor series approximations of the sine and cosine functions, which he extended to the third-order Taylor series approximation of the sine function. He is also thought to have discovered the power series of the radius, diameter, circumference, angle θ, π and π/4, along with rational approximations of π, and infinite continued fractions. His students and followers in the Kerala School further expanded his works with various series expansions and rational approximations until the 16th century.

In the 17th century, James Gregory also worked in this area and published several Maclaurin series. It was not until 1715 however that a general method for constructing these series for all functions for which they exist was finally provided by Brook Taylor, after whom the series are now named.

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.

[edit] Properties

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

If this series converges for every x in the interval (a − r, a + r) and the sum is equal to f(x), then the function f(x) is said to be analytic in the interval (a − r, a + r). If this is true for any r then the function is said to be an entire function. To check whether the series converges towards f(x), one normally uses estimates for the remainder term of Taylor's theorem. A function is analytic if and only if it can be represented as a power series; the coefficients in that power series are then necessarily the ones given in the above Taylor series formula.

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.

Note that there are examples of infinitely differentiable functions f(x) whose Taylor series converge, but are not equal to f(x). For instance, for the function defined piecewise by saying that f(x) = e^−1/x² if x ≠ 0 and f(0) = 0, all the derivatives are zero at x = 0, so the Taylor series of f(x) 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 e^−1/z² does not approach 0 as z approaches 0 along the imaginary axis.

Some functions cannot be written as Taylor series because they have a singularity; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable x; see Laurent series. For example, f(x) = e^−1/x² can be written as a Laurent series.

The Parker-Sochacki method is a recent advance in finding Taylor series which are solutions to differential equations. This algorithm is an extension of the Picard iteration.

[edit] List of Taylor series of some common functions

The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full cycle centered on the origin.

The cosine function.

An 8th degree approximation of the cosine function in the complex plane.

The two above curves put together.

An animation of the approximation.

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

Exponential function and natural logarithm:

$\mathrm{e}^{x} = \sum^{\infin}_{n=0} \frac{x^n}{n!}\quad\mbox{ for all } x$

$\ln(1+x) = \sum^{\infin}_{n=0} \frac{(-1)^n}{n+1} x^{n+1}\quad\mbox{ for } \left| x \right| < 1$

Geometric series:

$\frac{x^m}{1-x} = \sum^{\infin}_{n=m} x^n\quad\mbox{ for } \left| x \right| < 1$

Binomial theorem:

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

Trigonometric functions:

$\sin x = \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1}\quad\mbox{ for all } x$

$\cos x = \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n}\quad\mbox{ for all } x$

$\tan x = \sum^{\infin}_{n=1} \frac{B_{2n} (-4)^n (1-4^n)}{(2n)!} x^{2n-1}\quad = x + \frac{x^3}{3} + \frac{2 x^5}{15} + .. \mbox{ for } \left| x \right| < \frac{\pi}{2}$

where the Bs are Bernoulli numbers.

$\sec x = \sum^{\infin}_{n=0} \frac{(-1)^n E_{2n}}{(2n)!} x^{2n}\quad\mbox{ for } \left| x \right| < \frac{\pi}{2}$

$\arcsin x = \sum^{\infin}_{n=0} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}\quad\mbox{ for } \left| x \right| < 1$

$\arctan x = \sum^{\infin}_{n=0} \frac{(-1)^n}{2n+1} x^{2n+1}\quad\mbox{ for } \left| x \right| \leq 1$

Hyperbolic functions:

$\sinh \left(x\right) = \sum^{\infin}_{n=0} \frac{1}{(2n+1)!} x^{2n+1}\quad\mbox{ for all } x$

$\cosh \left(x\right) = \sum^{\infin}_{n=0} \frac{1}{(2n)!} x^{2n}\quad\mbox{ for all } x$

$\tanh\left(x\right) = \sum^{\infin}_{n=1} \frac{B_{2n} 4^n (4^n-1)}{(2n)!} x^{2n-1}\quad\mbox{ for } \left|x\right| < \frac{\pi}{2}$

$\mathrm{arsinh} \left(x\right) = \sum^{\infin}_{n=0} \frac{(-1)^n (2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}\quad\mbox{ for } \left| x \right| < 1$

$\mathrm{artanh} \left(x\right) = \sum^{\infin}_{n=0} \frac{1}{2n+1} x^{2n+1}\quad\mbox{ for } \left| x \right| < 1$

Lambert's W function:

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

The numbers B_k appearing in the summation expansions of tan(x) and tanh(x) are the Bernoulli numbers. The binomial expansion uses binomial coefficients. The E_k in the expansion of sec(x) are Euler numbers.

[edit] 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.

For example, consider the function

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

for which we want a Taylor series about 0.

We have:

$\ln(1+x) = \sum^{\infin}_{n=1} \frac{(-1)^{n+1}}{n} x^n = x - {x^2\over 2}+{x^3 \over 3} - {x^4 \over 4} + \cdots \quad\mbox{ for } \left| x \right| < 1$

$\cos x = \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n} = 1 -{x^2\over 2!}+{x^4\over 4!}- \cdots \quad\mbox{ for all } x$

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

$\left(1 -{x^2\over 2!}+{x^4\over 4!}-\cdots\right)-{1\over 2}\left(1 -{x^2\over 2!}+{x^4\over 4!}-\cdots\right)^2 +{1\over 3}\left(1 -{x^2\over 2!}+{x^4\over 4!}-\cdots\right)^3-\cdots$

Expanding by using multinomial coefficients gives the requisite Taylor series.

Or, for example, consider

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

We have

$e^x = 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!} +\cdots$

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

Then,

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

Assume the power series is

$c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots = \frac{1 + x + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!} + \cdots}{1 - {x^2 \over 2!} + {x^4 \over 4!} - \cdots}$

Then

$\left(c_0 + c_1 x + c_2 x^2 + c_3 x^3 + c_4x^4 + \cdots\right)\left(1 - {x^2 \over 2!} + {x^4 \over 4!} - \cdots\right) = 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!} + \cdots$

$=c_0 - {c_0 \over 2}x^2 + {c_0 \over 4!}x^4 + c_1x - {c_1 \over 2}x^3 + {c_1 \over 4!}x^5 + c_2x^2 - {c_2 \over 2}x^4 + {c_2 \over 4!}x^6 + c_3x^3 - {c_3 \over 2}x^5 + {c_3 \over 4!}x^7 +\cdots$

Collecting the terms up to fourth order yields

$=c_0 + c_1x + \left(c_2 - {c_0 \over 2}\right)x^2 + \left(c_3 - {c_1 \over 2}\right)x^3+\left(c_4+{c_0 \over 4!}-{c_2\over 2}\right)x^4 + \cdots = 1 + x + {x^2 \over 2!} + {x^3 \over 3!} +{x^4 \over 4!} + \cdots$

Comparing coefficients finally yields the Taylor series for the function

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

[edit] Taylor series as definitions

Classically the above functions are defined by some property that holds for them, for example the exp function is defined as the function that is equal to its own derivative. However in computable analysis functions must be defined by algorithms rather than properties, so the above Taylor expansions are used as primary definitions rather than derived results. This is also likely to be the case in software implementations of the functions.

[edit] See also

Laurent series
Taylor's theorem
Holomorphic functions are analytic — a proof that a holomorphic function can be expressed as a Taylor power series
Newton's divided difference interpolation
Madhava of Sangamagrama (credited with the first use of "Taylor" series)
Difference engine

[edit] References

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