In mathematical analysis, 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 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 terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate as 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 if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing . This implies that the function is analytic at every point of the interval (or disk).
The Taylor series of a real or complex-valued function , that is infinitely differentiable at a real or complex number , is the power series
Here, denotes the factorial of . The function denotes the th derivative of evaluated at the point . The derivative of order zero of is defined to be itself and and are both defined to be . This series can be written by using sigma notation, as in the right side formula. With , the Maclaurin series takes the form:
Several important Maclaurin series expansions follow. All these expansions are valid for complex arguments .
The exponential function (with base ) has Maclaurin series
It converges for all .
The exponential generating function of the Bell numbers is the exponential function of the predecessor of the exponential function:
The natural logarithm (with base ) has Maclaurin series
The last series is known as Mercator series, named after Nicholas Mercator since it was published in his 1668 treatise Logarithmotechnia. Both of these series converge for . In addition, the series for converges for , and the series for converges for .
The geometric series and its derivatives have Maclaurin series
All are convergent for . These are special cases of the binomial series given in the next section.
The binomial series is the power series
whose coefficients are the generalized binomial coefficients
(If , this product is an empty product and has value .) It converges for for any real or complex number .
When , this is essentially the infinite geometric series mentioned in the previous section. The special cases and give the square root function and its inverse:
When only the linear term is retained, this simplifies to the binomial approximation.
The usual trigonometric functions and their inverses have the following Maclaurin series:
All angles are expressed in radians. The numbers appearing in the expansions of are the Bernoulli numbers. The in the expansion of are Euler numbers.
The hyperbolic functions have Maclaurin series closely related to the series for the corresponding trigonometric functions:
The numbers appearing in the series for are the Bernoulli numbers.
The polylogarithms have these defining identities:
The Legendre chi functions are defined as follows:
And the formulas presented below are called inverse tangent integrals:
In statistical thermodynamics these formulas are of great importance.
The complete elliptic integrals of first kind K and of second kind E can be defined as follows:
The Jacobi theta functions describe the world of the elliptic modular functions and they have these Taylor series:
The regular partition number sequence has this generating function:
The strict partition number sequence Q(n) has the generating function:
Several methods can be used to calculate Taylor series. One may apply the definition directly, although this often requires first identifying a general formula for the derivatives or coefficients. In many cases, Taylor series can also be obtained from known expansions by algebraic manipulations of power series, such as substitution, multiplication, division, addition, or subtraction. In some cases, they may also be derived by repeated integration by parts. In practice, Taylor series are often computed with the aid of computer algebra systems.
In order to compute the 7th-degree Maclaurin polynomial for the function
one may first rewrite the function as
the composition of two functions and . The Taylor series for the natural logarithm is (using big O notation)
and for the cosine function
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 polynomial of degree 7:
Since the cosine is an even function, the coefficients for all the odd powers are zero.
Given that the Taylor series at of the function . The Taylor series for the exponential function is
and the series for cosine is
Assume the series for their quotient is
Multiplying both sides by the denominator and then expanding it as a series yields
Comparing the coefficients of with the coefficients of ,
The coefficients of the series for can thus be computed one at a time, amounting to long division of the series for and :
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 as a Taylor series in , we use the known Taylor series of function :
Thus,
The error incurred in approximating a function by its degree Taylor polynomial is called the remainder and is denoted by the function . 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 does converge, its limit need not be equal to the value of the function . For example, the function
is infinitely differentiable at , and has all derivatives zero there. Consequently, the Taylor series of about is identically zero. However, is not the zero function, so it does not equal its Taylor series around the origin. Thus, is an example of a non-analytic smooth function. This example shows that there are infinitely differentiable functions in real analysis, whose Taylor series are not equal to 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 a meromorphic function, which might have singularities, never converges to a value different from the function itself. The complex function , however, does not approach when approaches along the imaginary axis, so it is not continuous in the complex plane and its Taylor series is undefined at .
Every sequence of real or complex numbers can appear more generally 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 everywhere.
A function cannot be written as a Taylor series centred at a singularity. In these cases, the function can still be expressed as a series expansion by allowing negative powers of the variable . Such a series is known as a Laurent series, which generalizes the Taylor series.
The generalization of the Taylor series does converge to the value of the function itself for any bounded continuous function on , and this can be done by using the calculus of finite differences. Specifically, the following theorem, due to Einar Hille, that for any ,
Here is the th finite difference operator with step size . 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 is analytic at , 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 , the following power series identity holds:
So in particular,
The series on the right is the expected value of , where is a Poisson-distributed random variable that takes the value with probability . Hence,
The law of large numbers implies that the identity holds.
If is given by a convergent power series in an open disk centred at in the complex plane (or an interval in the real line), it is said to be analytic in this region. Thus for in this region, is given by a convergent power series
Differentiating by the above formula times, then setting gives
and so the power series expansion agrees with the Taylor series. Thus, a function is analytic in an open disk centered at if and only if its Taylor series converges to the value of the function at each point of the disk.
If is equal to the sum of its Taylor series for all in the complex plane, it is called entire. The polynomials, exponential function , and the trigonometric functions of 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 is far from . That is, the Taylor series diverges at if the distance between and is larger than the radius of convergence. The Taylor series can be used to calculate the value of an entire function at every point, provided the value of the function and all its derivatives are known at a single point.
Uses of the Taylor series for analytic functions include:
The Taylor series may also be generalized to functions of more than one variable with
The last expression is the multivariate Taylor series in terms of multi-index notation with a full analogy to the single variable case.
For example, for a function that depends on two variables, and , the Taylor series to second order about the point is
where the subscripts denote the respective partial derivatives.
A second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as
where is the gradient of evaluated at and is the Hessian matrix.
In order to compute a second-order Taylor series expansion around the point of the function
one first computes all the necessary partial derivatives:
Evaluating these derivatives at the origin gives the Taylor coefficients
Substituting these values in to the general formula
produces
Since is analytic in , we have
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; 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. Liu Hui independently employed a similar method a few centuries later.
In the 14th century, the earliest examples of specific Taylor series (but not the general method) were given by the Indian mathematician Madhava of Sangamagrama. 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 (, , , and ) 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 , , , (the integral of ), (the integral of, the inverse Gudermannian function), , and (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.
In 1691âÂÂ1692, Newton wrote down an explicit statement of the Taylor and Maclaurin series in an unpublished version of his work De Quadratura Curvarum. It was the earliest explicit formulation of the general Taylor series. However, this work by Newton 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, 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.