Taylor series

# Taylor series

Overview

In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Taylor series is a representation of a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

as an infinite sum
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

of terms that are calculated from the values of the function's derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

s at a single point.

The concept of a Taylor series was formally introduced by the English mathematician Brook Taylor
Brook Taylor
Brook Taylor FRS was an English mathematician who is best known for Taylor's theorem and the Taylor series.- Life and work :...

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
Colin Maclaurin
Colin Maclaurin was a Scottish mathematician who made important contributions to geometry and algebra. The Maclaurin series, a special case of the Taylor series, are named after him....

, who made extensive use of this special case of Taylor series in the 18th century.

It is common practice to approximate a function by using a finite number of terms of its Taylor series.
Discussion

Encyclopedia

In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Taylor series is a representation of a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

as an infinite sum
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

of terms that are calculated from the values of the function's derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

s at a single point.

The concept of a Taylor series was formally introduced by the English mathematician Brook Taylor
Brook Taylor
Brook Taylor FRS was an English mathematician who is best known for Taylor's theorem and the Taylor series.- Life and work :...

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
Colin Maclaurin
Colin Maclaurin was a Scottish mathematician who made important contributions to geometry and algebra. The Maclaurin series, a special case of the Taylor series, are named after him....

, who made extensive use of this special case of Taylor series in the 18th century.

It is common practice to approximate a function by using a finite number of terms of its Taylor series. Taylor's theorem
Taylor's theorem
In calculus, Taylor's theorem gives an approximation of a k times differentiable function around a given point by a k-th order Taylor-polynomial. For analytic functions the Taylor polynomials at a given point are finite order truncations of its Taylor's series, which completely determines the...

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
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

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
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

) is known as an analytic function
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, categories that are similar in some ways, but different in others...

.

## Definition

The Taylor series of a real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

or complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

function ƒ(x) that is infinitely differentiable in a neighborhood of a real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

or complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

a is the power series

which can be written in the more compact sigma notation as

where n! denotes the factorial
Factorial
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

of n and ƒ (n)(a) denotes the nth derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

of ƒ evaluated at the point a. The zeroth derivative of ƒ is defined to be ƒ itself and and 0! are both defined to be 1. In the case that , the series is also called a Maclaurin series.

## Examples

The Maclaurin series for any polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

is the polynomial itself.

The Maclaurin series for for |x| < 1 is the geometric series

so the Taylor series for x−1 at is

By integrating the above Maclaurin series we find the Maclaurin series for , where log denotes the natural logarithm
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

:

and the corresponding Taylor series for log(x) at is

The Taylor series for the exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

ex at a = 0 is

The above expansion holds because the derivative of ex with respect to x is also ex and e0 equals 1. This leaves the terms in the numerator and n! in the denominator for each term in the infinite sum.

## History

The Greek philosopher Zeno
Zeno of Elea
Zeno of Elea was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known for his paradoxes, which Bertrand Russell has described as "immeasurably subtle and profound".- Life...

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
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...

proposed a philosophical resolution of the paradox, but the mathematical content was apparently unresolved until taken up by Democritus
Democritus
Democritus was an Ancient Greek philosopher born in Abdera, Thrace, Greece. He was an influential pre-Socratic philosopher and pupil of Leucippus, who formulated an atomic theory for the cosmos....

and then Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

. It was through Archimedes's method of exhaustion
Method of exhaustion
The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the n-th polygon and the containing shape will...

that an infinite number of progressive subdivisions could be performed to achieve a finite result. Liu Hui
Liu Hui
Liu Hui was a mathematician of the state of Cao Wei during the Three Kingdoms period of Chinese history. In 263, he edited and published a book with solutions to mathematical problems presented in the famous Chinese book of mathematic known as The Nine Chapters on the Mathematical Art .He was a...

independently employed a similar method a few centuries later.

In the 14th century, the earliest examples of the use of Taylor series and closely related methods were given by Madhava of Sangamagrama
Mādhava of Sañgamāgrama was a prominent Kerala mathematician-astronomer from the town of Irińńālakkuţa near Cochin, Kerala, India. He is considered the founder of the Kerala School of Astronomy and Mathematics...

. Though no record of his work survives, writings of later Indian mathematicians
Indian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...

suggest that he found a number of special cases of the Taylor series, including those for the trigonometric function
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

s of sine
Sine
In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....

, cosine, 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.

In the 17th century, James Gregory
James Gregory (astronomer and mathematician)
James Gregory FRS was a Scottish mathematician and astronomer. He described an early practical design for the reflecting telescope – the Gregorian telescope – and made advances in trigonometry, discovering infinite series representations for several trigonometric functions.- Biography :The...

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
Brook Taylor
Brook Taylor FRS was an English mathematician who is best known for Taylor's theorem and the Taylor series.- Life and work :...

, after whom the series are now named.

The Maclaurin series was named after Colin Maclaurin
Colin Maclaurin
Colin Maclaurin was a Scottish mathematician who made important contributions to geometry and algebra. The Maclaurin series, a special case of the Taylor series, are named after him....

, a professor in Edinburgh, who published the special case of the Taylor result in the 18th century.

## Analytic functions

If f(x) is given by a convergent power series in an open disc (or interval in the real line) centered at b, it is said to be analytic
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, categories that are similar in some ways, but different in others...

in this disc. Thus for x in this disc, f is given by a convergent power series
Differentiating by x the above formula n times, then setting x=b gives:
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.

If f(x) is equal to its Taylor series everywhere it is called entire
Entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...

. The polynomials and the exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

ex and the trigonometric function
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

s sine and cosine are examples of entire functions. Examples of functions that are not entire include the logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

, the trigonometric function
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

tangent, and its inverse arctan. For these functions the Taylor series do not converge if x is far from a. Taylor series can be used to calculate the value of an entire function in 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:
1. The partial sums (the Taylor polynomials) of the series can be used as approximations of the entire function. These approximations are good if sufficiently many terms are included.
2. Differentiation and integration of power series can be performed term by term and is hence particularly easy.
3. An analytic function
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, categories that are similar in some ways, but different in others...

is uniquely extended to a holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

on an open disk in the complex plane
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

. This makes the machinery of complex analysis
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 useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

available.
4. 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
Clenshaw algorithm
In numerical analysis, the Clenshaw algorithm is a recursive method to evaluate a linear combination of Chebyshev polynomials. In general it applies to any class of functions that can be defined by a three-term recurrence relation.-Clenshaw algorithm:...

).
5. Algebraic operations can be done readily on the power series representation; for instance the Euler's formula
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...

follows from Taylor series expansions for trigonometric and exponential functions. This result is of fundamental importance in such fields as harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...

.

## Approximation and convergence

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:

The error in this approximation is no more than |x|9/9!. 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 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
Runge's phenomenon
In the mathematical field of numerical analysis, Runge's phenomenon is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree...

.

The error incurred in approximating a function by its nth-degree Taylor polynomial is called the remainder or residual and is denoted by the function Rn(x).
Taylor's theorem
Taylor's theorem
In calculus, Taylor's theorem gives an approximation of a k times differentiable function around a given point by a k-th order Taylor-polynomial. For analytic functions the Taylor polynomials at a given point are finite order truncations of its Taylor's series, which completely determines the...

can be used to obtain a bound on the size of the remainder.

In general, Taylor series need not be convergent at all. And in fact the set of functions with a convergent Taylor series is a meager set in the Fréchet space
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces...

of smooth functions. 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
is infinitely differentiable at , and has all derivatives zero there. Consequently, the Taylor series of f(x) about 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.

In real analysis
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...

, this example shows that there are infinitely differentiable functions f(x) whose Taylor series are not equal to f(x) even if they converge. By contrast in complex analysis
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 useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

there are no holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

s f(z) whose Taylor series converges to a value different from f(z). The complex function e−z−2 does not approach 0 as z approaches 0 along the imaginary axis, and its Taylor series is thus not defined there.

More generally, every sequence of real or complex numbers can appear as coefficients in the Taylor series of an infinitely differentiable function defined on the real line, a consequence of Borel's lemma
Borel's lemma
In mathematics, Borel's lemma is an important result about partial differential equations named after Émile Borel.Suppose U is an open set in the Euclidean space Rn, and suppose that f_0, f_1, ... is a sequence of smooth, complex-valued functions on U...

In mathematics, the radius of convergence of a power series is a quantity, either a non-negative real number or ∞, that represents a domain in which the series will converge. Within the radius of convergence, a power series converges absolutely and uniformly on compacta as well...

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.

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
Laurent series
In mathematics, the Laurent series of a complex function f 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...

. For example, f(x) = e−x−2 can be written as a Laurent series.

### Generalization

There is, however, a generalization of the Taylor series that does converge to the value of the function itself for any bounded
Bounded function
In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M...

continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

on (0,∞), using the calculus of finite differences. Specifically, one has the following theorem, due to Einar Hille, that for any t > 0,
Here Δ 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.

In general, for any infinite sequence ai, the following power series identity holds:
So in particular,
The series on the right is the expectation value of f(a + X), where X is a Poisson distributed
Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...

random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

that takes the value jh with probability e−t/h(t/h)j/j!. Hence,
The law of large numbers
Law of large numbers
In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times...

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 x.

Exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

:

Natural logarithm
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

:

Finite geometric series:

Infinite geometric series:

Variants of the infinite geometric series:

Square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

:

Binomial series
Binomial series
In mathematics, the binomial series is the Taylor series at x = 0 of the function f given by f =  α, where is an arbitrary complex number...

(includes the square root for α = 1/2 and the infinite geometric series for α = −1):

with generalized binomial coefficient
Binomial coefficient
In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...

s

Trigonometric function
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

s:

where the Bs are Bernoulli numbers.

Hyperbolic function
Hyperbolic function
In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh" , and the hyperbolic cosine "cosh" , from which are derived the hyperbolic tangent "tanh" and so on.Just as the points form a...

s:

Lambert's W function:

The numbers Bk appearing in the summation expansions of tan(x) and tanh(x) are the Bernoulli numbers. The Ek in the expansion of sec(x) 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
Integration by parts
In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...

. Particularly convenient is the use of computer algebra system
Computer algebra system
A computer algebra system is a software program that facilitates symbolic mathematics. The core functionality of a CAS is manipulation of mathematical expressions in symbolic form.-Symbolic manipulations:...

s to calculate Taylor series.

### First example

Compute the 7th degree Maclaurin polynomial for the function.

First, rewrite the function as.
We have for the natural logarithm (by using the big O notation
Big O notation
In mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann-Landau notation, or...

)
and for the cosine function
The latter series expansion has a zero constant term
Constant term
In mathematics, a constant term is a term in an algebraic expression has a value that is constant or cannot change, because it does not contain any modifiable variables. For example, in the quadratic polynomialx^2 + 2x + 3,\ the 3 is a 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:

Since the cosine is an even function, the coefficients for all the odd powers x, x3, x5, x7, .. have to be zero.

### Second example

Suppose we want the Taylor series at 0 of the function
.

We have for the exponential function

and, as in the first example,

Assume the power series is

Then multiplication with the denominator and substitution of the series of the cosine yields

Collecting the terms up to fourth order yields

Comparing coefficients with the above series of the exponential function yields the desired Taylor series

### 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
.

We know the Taylor series of function is:

Thus,

## Taylor series as definitions

Classically, algebraic function
Algebraic function
In mathematics, an algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves polynomials with rational coefficients. For example, an algebraic function in one variable x is a solution y for an equationwhere the coefficients ai are polynomial...

s are defined by an algebraic equation, and transcendental function
Transcendental function
A transcendental function is a function that does not satisfy a polynomial equation whose coefficients are themselves polynomials, in contrast to an algebraic function, which does satisfy such an equation...

s (including those discussed above) are defined by some property that holds for them, such as a differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

. For example, the exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

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
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, categories that are similar in some ways, but different in others...

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 define analytical functions of matrices and operators, such as the matrix exponential
Matrix exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group....

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, x and y, the Taylor series to second order about the point (a, b) is:

where the subscripts denote the respective partial derivative
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...

s.

A second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

of evaluated at and is the Hessian matrix
Hessian matrix
In mathematics, the Hessian matrix is the square matrix of second-order partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named...

. Applying the multi-index notation
Multi-index notation
The mathematical notation of multi-indices simplifies formulae used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices....

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, again in full analogy to the single variable case.

### Example

Compute a second-order Taylor series expansion around point of a function
Firstly, we compute all partial derivatives we need

The Taylor series is

which in this case becomes

Since is analytic in |y| < 1, we have
for |y| < 1.

## Fractional Taylor series

With the emergence of fractional calculus
Fractional calculus
Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation operator.and the integration operator J...

, a natural question arises about what the Taylor Series expansion would be. Odibat and Shawagfeh answered this in 2007. By using the Caputo fractional derivative, , and indicating the limit as we approach from the right, the fractional Taylor series can be written as

• Laurent series
Laurent series
In mathematics, the Laurent series of a complex function f 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...

• Newton's divided difference interpolation
Newton polynomial
In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is the interpolation polynomial for a given set of data points in the Newton form...