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A Taylor series is a series expansion of a function about a point. A one-dimensional Taylor series is an expansion of a real function f(x) about a point x=a is given by

f(x) = (1)

If a=0, the expansion is known as a Maclaurin series.

Taylor's theorem (actually discovered first by Gregory) states that any function satisfying certain conditions can be expressed as a Taylor series.

The Taylor (or more general) series of a function f(x) about a point up to order may be found using Series[f, {x, a, n}]. The n^{th} term of a Taylor series of a function f can be computed in the Wolfram Language using Series Coefficient[f, {x, a, n}] and is given by the inverse Z-transform

(2)

Taylor series of some common functions include

To derive the Taylor series of a function f(x), note that the integral of the (n+1)st derivative f^{(n+1)} of f(x) from the point x_{0} to an arbitrary point x is given by

(9)

where f^{(n)} (x_{0}) is the n^{th} derivative of f(x) evaluated at x_{0}, and is therefore simply a constant. Now integrate a second time to obtain

(10)

where f^{(k)} (x_{0}) is again a constant. Integrating a third time,

(11)

and continuing up to n+1 integrations then gives

(12)

Rearranging then gives the one-dimensional Taylor series

Here, R_{n} is a remainder term known as the Lagrange remainder, which is given by

(15)

Rewriting the repeated integral then gives

(16)

Now, from the mean-value theorem for a function g(x), it must be true that

(17)

for some Therefore, integrating n+1 times gives the result

(18)

If f(z) is analytic throughout the annular region between and on the concentric circles K_{1}_{ }and k_{2} centered at z=a and of radii r_{1} and r_{2}<r_{1} respectively, then there exists a unique series expansion in terms of positive and negative powers of (z-a),

(19)

where

(20)

(21)

(Korn and Korn 1968, pp. 197-198).

Let there be two circular contours c_{2} and c_{1}, with the radius of c_{1} larger than that of c_{2}. Let z_{0 }be at the center of c_{1} and c_{2}, and be between c_{1} and c_{2}. Now create a cut line c_{c} between c_{1} and c_{2}, and integrate around the path , so that the plus and minus contributions of c_{c} cancel one another, as illustrated above. From the Cauchy integral formula,

(22)

(23)

(24)

Now, since contributions from the cut line in opposite directions cancel out,

(25)

(26)

(27)

For the first integral, . For the second, . Now use the Taylor series (valid for)

(28)

to obtain

(29)

(30)

(31)

where the second term has been re-indexed. Re-indexing again,

(32)

Since the integrands, including the function f(z), are analytic in the annular region defined by c_{1} and c_{2}, the integrals are independent of the path of integration in that region. If we replace paths of integration c_{1} and c_{2} by a circle c of radius r with , then

(33)

(34)

(35)

Generally, the path of integration can be any path Ï’ that lies in the annular region and encircles z_{0} once in the positive (counterclockwise) direction.

The complex residues a_{n} are therefore defined by (36)

Note that the annular region itself can be expanded by increasing r_{1} and decreasing r_{2 }until singularities of f(z) that lie just outside c_{1} or just inside c_{2} are reached. If f(x) has no singularities inside c_{2}, then all the b_{k} terms in (â—‡) equal zero and the Laurent series of (â—‡) reduces to a Taylor series with coefficients a_{k}.

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