In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Taylor series or Taylor expansion of a
function is an
infinite sum
In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...
of terms that are expressed in terms of the function's
derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
s 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 terms of a Taylor series is a
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
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
In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a polynomial of degree k, called the k-th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation a ...
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
In mathematics, a real interval is the set (mathematics), set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without ...
(or
open disk in the
complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
) containing . This implies that the function is analytic at every point of the interval (or disk).
Definition
The Taylor series of a
real or
complex-valued function , that is
infinitely differentiable at a
real or
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
, is the
power series
Here, denotes the
factorial
In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times ...
of . The function denotes the th
derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
of evaluated at the point . The derivative of order zero of is defined to be itself and and
are both defined to be 1. This series can be written by using
sigma notation, as in the right side formula. With , the Maclaurin series takes the form:
Examples
The Taylor series of any
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
is the polynomial itself.
The Maclaurin series of is the
geometric series
In mathematics, a geometric series is a series (mathematics), series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, 1/2 + 1/4 + 1/8 + 1/16 + ⋯, the series \tfrac12 + \tfrac1 ...
So, by substituting for , the Taylor series of at is
By integrating the above Maclaurin series, we find the Maclaurin series of , where denotes the
natural logarithm
The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
:
The corresponding Taylor series of at is
and more generally, the corresponding Taylor series of at an arbitrary nonzero point is:
The Maclaurin series of the
exponential function is
The above expansion holds because the derivative of with respect to is also , and equals 1. This leaves the terms in the numerator and in the denominator of each term in the infinite sum.
History
The
ancient Greek philosopher
Ancient Greek philosophy arose in the 6th century BC. Philosophy was used to make sense of the world using reason. It dealt with a wide variety of subjects, including astronomy, epistemology, mathematics, political philosophy, ethics, metaphysics ...
Zeno of Elea
Zeno of Elea (; ; ) was a pre-Socratic Greek philosopher from Elea, in Southern Italy (Magna Graecia). He was a student of Parmenides and one of the Eleatics. Zeno defended his instructor's belief in monism, the idea that only one single en ...
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
Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...
proposed a philosophical resolution of the paradox, but the mathematical content was apparently unresolved until taken up by
Archimedes
Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
, as it had been prior to Aristotle by the Presocratic Atomist
Democritus
Democritus (, ; , ''Dēmókritos'', meaning "chosen of the people"; – ) was an Ancient Greece, Ancient Greek Pre-Socratic philosophy, pre-Socratic philosopher from Abdera, Thrace, Abdera, primarily remembered today for his formulation of an ...
. 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
Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu ( The Nine Chapters on the Mathematical Art).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state ...
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 Indian mathematician
Madhava of Sangamagrama. Though no record of his work survives, writings of his followers in the
Kerala school of astronomy and mathematics
The Kerala school of astronomy and mathematics or the Kerala school was a school of Indian mathematics, mathematics and Indian astronomy, astronomy founded by Madhava of Sangamagrama in Kingdom of Tanur, Tirur, Malappuram district, Malappuram, K ...
suggest that he found the Taylor series for the
trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
s of
sine,
cosine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that ...
, 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
Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
, 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 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, 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 functions
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 function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
s sine and cosine, are examples of entire functions. Examples of functions that are not entire include the
square root
In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
, the
logarithm
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
, the
trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
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, 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
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
is uniquely extended to a
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
on an open disk in the
complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
. This makes the machinery of
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
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 convergence
Pictured is an accurate approximation of around the point . The pink curve is a polynomial of degree seven:
The error in this approximation is no more than . For a full cycle centered at the origin () the error is less than 0.08215. In particular, for , the error is less than 0.000003.
In contrast, also shown is a picture of the natural logarithm function and some of its Taylor polynomials around . These approximations converge to the function only in the region ; outside of this region the higher-degree Taylor polynomials are ''worse'' approximations for the function.
The ''error'' incurred in approximating a function by its th-degree Taylor polynomial is called the ''remainder'' or ''
residual'' 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 does not equal its Taylor series around the origin. Thus, is an example of a
non-analytic smooth function.
In
real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include co ...
, this example shows that there are
infinitely differentiable functions whose Taylor series are ''not'' equal to even if they converge. By contrast, the
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
s studied in
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
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 , however, does not approach 0 when 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
coefficient
In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
s 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.
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 ; see
Laurent series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansio ...
. For example, can be written as a Laurent series.
Generalization
The generalization of the Taylor series does converge to the value of the function itself for any
bounded continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
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
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of , where is a
Poisson-distributed random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
that takes the value with probability . Hence,
The
law of large numbers implies that the identity holds.
List of Maclaurin series of some common functions
Several important Maclaurin series expansions follow. All these expansions are valid for complex arguments .
Exponential function

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:
Natural logarithm
The
natural logarithm
The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
(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 .)
Geometric series
The
geometric series
In mathematics, a geometric series is a series (mathematics), series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, 1/2 + 1/4 + 1/8 + 1/16 + ⋯, the series \tfrac12 + \tfrac1 ...
and its derivatives have Maclaurin series
All are convergent for
. These are special cases of the
binomial series given in the next section.
Binomial series
The
binomial series is the power series
whose coefficients are the generalized
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s
(If , this product is an
empty product
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplication, multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operat ...
and has value 1.) 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
In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
function and its
inverse:
When only the
linear term is retained, this simplifies to the
binomial approximation.
Trigonometric functions
The usual
trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
s and their inverses have the following Maclaurin series:
All angles are expressed in
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...
s. The numbers appearing in the expansions of are the
Bernoulli numbers. The in the expansion of are
Euler numbers.
Hyperbolic functions
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.
Polylogarithmic functions
The
polylogarithm
In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natur ...
s 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.
Elliptic functions
The complete
elliptic integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising i ...
s 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 P(n) has this generating function:
The strict partition number sequence Q(n) has that generating function:
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 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
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivati ...
. Particularly convenient is the use of
computer algebra system
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The de ...
s to calculate Taylor series.
First example
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
Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a memb ...
)
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 7th-degree polynomial:
Since the cosine is an
even function, the coefficients for all the odd powers are zero.
Second example
Suppose we want the Taylor series at 0 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
Third example
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,
Taylor series as definitions
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
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
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
In mathematics, the matrix exponential is a matrix function on square matrix, square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exp ...
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
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 derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...
s.
Second-order Taylor series in several variables
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
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
of evaluated at and is the
Hessian matrix. Applying the
multi-index notation the Taylor series for several variables becomes
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.
Example
In order to compute a second-order Taylor series expansion around 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
Comparison with Fourier series
The trigonometric
Fourier series
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
enables one to express a
periodic function
A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the t ...
(or a function defined on a closed interval ) as an infinite sum of
trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
s (
sines and
cosine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that ...
s). 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 about the point are all exactly equal to at . 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
A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neighbourh ...
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, could be a
Weierstrass function
In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function (mathematics), function that is continuous function, continuous everywhere but Differentiable function, differentiab ...
.)
* 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 In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some Function (mathematics), function f. An important class of pointwise concepts are the ''pointwise operations'', that ...
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 C
1 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 also
*
Asymptotic expansion
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation ...
*
Newton polynomial
*
Padé approximant – best approximation by a rational function
*
*
Approximation theory
In mathematics, approximation theory is concerned with how function (mathematics), functions can best be approximation, approximated with simpler functions, and with quantitative property, quantitatively characterization (mathematics), characteri ...
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Function approximation
In general, a function approximation problem asks us to select a function (mathematics), function among a that closely matches ("approximates") a in a task-specific way. The need for function approximations arises in many branches of applied ...
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External links
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{{Authority control
Real analysis
Complex analysis
Series expansions