Taylor Series

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 about a point is given by

f(x)=f(a)+f^'(a)(x-a)+(f^('')(a))/(2!)(x-a)^2+(f^((3))(a))/(3!)(x-a)^3+...+(f^((n))(a))/(n!)(x-a)^n+....

(1)

If , 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 about a point up to order may be found using Series[f, x, a, n]. The th term of a Taylor series of a function can be computed in the Wolfram Language using SeriesCoefficient[f, x, a, n] and is given by the inverse Z-transform

(2)

Taylor series of some common functions include

			(3)
			(4)
			(5)
			(6)
			(7)
			(8)

To derive the Taylor series of a function , note that the integral of the st derivative of from the point to an arbitrary point is given by

(9)

where is the th derivative of evaluated at , and is therefore simply a constant. Now integrate a second time to obtain

int_(x_0)^x[int_(x_0)^xf^((n+1))(x)dx]dx
=int_(x_0)^x[f^((n))(x)-f^((n))(x_0)]dx
=[f^((n-1))(x)]_(x_0)^x-(x-x_0)f^((n))(x_0)
=f^((n-1))(x)-f^((n-1))(x_0)-(x-x_0)f^((n))(x_0),

(10)

where is again a constant. Integrating a third time,

int_(x_0)^xint_(x_0)^xint_(x_0)^xf^((n+1))(x)(dx)^3=f^((n-2))(x)-f^((n-2))(x_0)
-(x-x_0)f^((n-1))(x_0)-((x-x_0)^2)/(2!)f^((n))(x_0),

(11)

and continuing up to integrations then gives

int...int_(x_0)^x_()_(n+1)f^((n+1))(x)(dx)^(n+1)=f(x)-f(x_0)-(x-x_0)f^'(x_0)
-((x-x_0)^2)/(2!)f^('')(x_0)-...-((x-x_0)^n)/(n!)f^((n))(x_0).

(12)

Rearranging then gives the one-dimensional Taylor series

			(13)
			(14)

Here, 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 , it must be true that

(17)

for some . Therefore, integrating times gives the result

(18)

(Abramowitz and Stegun 1972, p. 880), so the maximum error after terms of the Taylor series is the maximum value of (18) running through all . Note that the Lagrange remainder is also sometimes taken to refer to the remainder when terms up to the st power are taken in the Taylor series (Whittaker and Watson 1990, pp. 95-96).

Taylor series can also be defined for functions of a complex variable. By the Cauchy integral formula,

			(19)
			(20)
			(21)

In the interior of ,

(22)

so, using

(23)

it follows that

			(24)
			(25)

Using the Cauchy integral formula for derivatives,

(26)

An alternative form of the one-dimensional Taylor series may be obtained by letting

(27)

so that

(28)

Substitute this result into (◇) to give

(29)

A Taylor series of a real function in two variables is given by

f(x+Deltax,y+Deltay)=f(x,y)+[f_x(x,y)Deltax+f_y(x,y)Deltay]+1/(2!)[(Deltax)^2f_(xx)(x,y)+2DeltaxDeltayf_(xy)(x,y)+(Deltay)^2f_(yy)(x,y)]+1/(3!)[(Deltax)^3f_(xxx)(x,y)+3(Deltax)^2Deltayf_(xxy)(x,y)+3Deltax(Deltay)^2f_(xyy)(x,y)+(Deltay)^3f_(yyy)(x,y)]+....

(30)

This can be further generalized for a real function in variables,

$f(x_1,...,x_n)=sum_(j=0)^infty{1/(j!)[sum_(k=1)^n(x_k-a_k)partial/(partialx_k^')]^jf(x_1^',...,x_n^')}_(x_1^'=a_1,...,x_n^'=a_n).$

(31)

Rewriting,

$f(x_1+a_1,...,x_n+a_n)=sum_(j=0)^infty{1/(j!)(sum_(k=1)^na_kpartial/(partialx_k^'))^jf(x_1^',...,x_n^')}_(x_1^'=x_1,...,x_n^'=x_n).$

(32)

For example, taking in (31) gives

$f(x_1,x_2)=sum_(j=0)^(infty){1/(j!)[(x_1-a_1)partial/(partialx_1^')+(x_2-a_2)partial/(partialx_2^')]^jf(x_1^',x_2^')}_(x_1^'=a_1,x_2^'=a_2)$	(33)
	(34)

Taking in (32) gives

$f(x_1+a_1,x_2+a_2,x_3+a_3) =sum_(j=0)^infty{1/(j!)(a_1partial/(partialx_1^')+a_2partial/(partialx_2^')+a_3partial/(partialx_3^'))^jf(x_1^',x_2^',x_3^')}_(x_1^'=x_1,x_2^'=x_2,x_3^'=x_3),$

(35)

or, in vector form

(36)

The zeroth- and first-order terms are and , respectively. The second-order term is

			(37)
			(38)

so the first few terms of the expansion are

f(r+a)=f(r)+(a·del _(r^'))f(r^')|_(r^'=r)+1/2a·[a·del _(r^')(del _(r^')f(r^'))]|_(r^'=r).

(39)

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 880, 1972.Arfken, G. "Taylor's Expansion." §5.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 303-313, 1985.Askey, R. and Haimo, D. T. "Similarities between Fourier and Power Series." Amer. Math. Monthly 103, 297-304, 1996.Comtet, L. "Calcul pratique des coefficients de Taylor d'une fonction algébrique." Enseign. Math. 10, 267-270, 1964.Morse, P. M. and Feshbach, H. "Derivatives of Analytic Functions, Taylor and Laurent Series." §4.3 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 374-398, 1953.Whittaker, E. T. and Watson, G. N. "Forms of the Remainder in Taylor's Series." §5.41 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 95-96, 1990.

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Taylor Series

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Weisstein, Eric W. "Taylor Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TaylorSeries.html

Taylor Series

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