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Taylor Series

A Taylor series is aseries expansion of afunction about a point. A one-dimensional Taylor series is an expansion of areal function f(x) about a point x=a 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 a=0 , the expansion is known as aMaclaurin 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 usingSeries[f,x,a,n]. Theth term of a Taylor series of a function can be computed in theWolfram Language usingSeriesCoefficient[f,x,a,n] and is given by the inverseZ-transform

(2)

Taylor series of some common functions include

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

To derive the Taylor series of a function f(x) , note that the integral of the (n+1) stderivative f^((n+1)) of f(x) from the point x_0 to an arbitrary point is given by

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

(9)

where f^((n))(x_0) is theth derivative of f(x) evaluated at x_0 , 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 f^((k))(x_0) 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 n+1 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, R_n is a remainder term known as theLagrange remainder, which is given by

R_n=int...int_(x_0)^x_()_(n+1)f^((n+1))(x)(dx)^(n+1).

(15)

Rewriting therepeated integral then gives

R_n=int_(x_0)^xf^((n+1))(t)((x-t)^n)/(n!)dt.

(16)

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

(17)

for some x^* in [x_0,x] . Therefore, integrating n+1 times gives the result

R_n=((x-x_0)^(n+1))/((n+1)!)f^((n+1))(x^*)

(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 x^* in [x_0,x] . Note that the Lagrange remainder R_n is also sometimes taken to refer to the remainder when terms up to the (n-1) st power are taken in the Taylor series (Whittaker and Watson 1990, pp. 95-96).

Taylor series can also be defined for functions of acomplexvariable. By theCauchy integral formula,

			(19)
			(20)
			(21)

In the interior of,

(22)

so, using

(23)

it follows that

			(24)
			(25)

Using theCauchy integral formula for derivatives,

f(z)=sum_(n=0)^infty(z-z_0)^n(f^((n))(z_0))/(n!).

(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

f(x_0+Deltax)=f(x_0)+Deltaxf^'(x_0)+1/(2!)(Deltax)^2f^('')(x_0)+....

(29)

A Taylor series of areal function in two variables f(x,y) 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 areal 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 n=2 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 n=3 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, invector form

f(r+a)=sum_(j=0)^infty[1/(j!)(a·del _(r^'))^jf(r^')]_(r^'=r).

(36)

The zeroth- and first-order terms are f(r) and (a·del _(r^'))f(r^')|_(r^'=r) , 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 inMathematical 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. Monthly103, 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 inMethods 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 inA 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." FromMathWorld--A Wolfram Web Resource.https://mathworld.wolfram.com/TaylorSeries.html

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