Encyclopedia
In
mathematics, the
Taylor series of an
infinitely
differentiable real function
f, defined on an open interval , is the power series shown below. The series is named in honor of
English mathematician
Brook Taylor.
Here,
n! is the
factorial of
n and
f denotes the
nth
derivative of
f at the point
a.
If
a = 0, the series is also called a
Maclaurin series, named after
Scottish mathematician
Colin Maclaurin.
Functions that involve rational operations such as addition, subtraction, multiplication and division are relatively easy to evaluate. Many other functions aren't so easy to evaluate, like those that involve
logarithms or
trigonometric functions such as cos. These and many other functions are approximately equal to their
Taylor series within a certain range and so the partial sums of this series can be used as a good approximation.
Pictured in the right are increasingly accurate approximations of sin around the point
a = 0. The yellow curve is a polynomial of degree seven:
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 ?,
p and p/4, along with rational approximations of p, 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.
Properties
If this series converges for every
x in the interval and the sum is equal to
f, then the function
f is said to be
analytic in the interval . If this is true for any
r then the function is said to be
analytic. To check whether the series converges towards
f, 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 series can be used to compute function values approximately . 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 whose Taylor series converge, but are
not equal to
f. For instance, for the function defined piecewise by saying that
f = e
−1/x² if
x ? 0 and
f = 0, all the derivatives are zero at
x = 0, so the Taylor series of
f 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 = 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.
List of Taylor series of some common functions
Several important Taylor/Maclaurin series expansions follow. All these expansions are also valid for complex arguments
x.
Exponential function and
natural logarithm:
Geometric series:
Binomial theorem:
Trigonometric functions:
where the
Bs are Bernoulli numbers.
Hyperbolic functions:
Lambert's W function:
The numbers Bk appearing in the summation expansions of tan and tanh are the Bernoulli numbers. The binomial expansion uses binomial coefficients. The Ek in the expansion of sec are Euler numbers.
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
-
for which we want a Taylor series about 0.
We have:
We can simply substitute the second series into the first. Doing so,
Expanding by using multinomial coefficients gives the requisite Taylor series.
Or, for example, consider
-
We have
-
-
Then,
-
Assume the power series is
-
Then
-
-
Collecting the terms up to fourth order yields
-
Comparing coefficients finally yields the Taylor series for the function
-
Taylor series as definitions
Classically the above functions are defined by some property that hold 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.
See also
References
External links
