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Taylor Series

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

a_n=Z^(-1)[f(1/(z-a))](n).
(2)

Taylor series of some common functions include

1/(1-x)=1/(1-a)+(x-a)/((1-a)^2)+((x-a)^2)/((1-a)^3)+...
(3)
cosx=cosa-sina(x-a)-1/2cosa(x-a)^2+1/6sina(x-a)^3+...
(4)
e^x=e^a[1+(x-a)+1/2(x-a)^2+1/6(x-a)^3+...]
(5)
lnx=lna+(x-a)/a-((x-a)^2)/(2a^2)+((x-a)^3)/(3a^3)-...
(6)
sinx=sina+cosa(x-a)-1/2sina(x-a)^2-1/6cosa(x-a)^3+...
(7)
tanx=tana+sec^2a(x-a)+sec^2atana(x-a)^2+sec^2a(sec^2a-2/3)(x-a)^3+....
(8)

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

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 the nth 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

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)+R_n
(13)
=sum_(k=0)^(n)((x-x_0)^kf^((k))(x_0))/(k!)+R_n.
(14)

Here, R_n is a remainder term known as the Lagrange remainder, which is given by

R_n=int...int_(x_0)^x_()_(n+1)f^((n+1))(x)(dx)^(n+1).
(15)

Rewriting the repeated integral then gives

R_n=int_(x_0)^xf^((n+1))(t)((x-t)^n)/(n!)dt.
(16)

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

int_(x_0)^xg(x)dx=(x-x_0)g(x^*)
(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 n 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 a complex variable. By the Cauchy integral formula,

f(z)=1/(2pii)int_C(f(z^')dz^')/(z^'-z)
(19)
=1/(2pii)int_C(f(z^')dz^')/((z^'-z_0)-(z-z_0))
(20)
=1/(2pii)int_C(f(z^')dz^')/((z^'-z_0)(1-(z-z_0)/(z^'-z_0))).
(21)

In the interior of C,

(|z-z_0|)/(|z^'-z_0|)<1
(22)

so, using

1/(1-t)=sum_(n=0)^inftyt^n,
(23)

it follows that

f(z)=1/(2pii)int_Csum_(n=0)^(infty)((z-z_0)^nf(z^')dz^')/((z^'-z_0)^(n+1))
(24)
=1/(2pii)sum_(n=0)^(infty)(z-z_0)^nint_C(f(z^')dz^')/((z^'-z_0)^(n+1)).
(25)

Using the Cauchy 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

x-x_0=Deltax
(27)

so that

x=x_0+Deltax.
(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 a real 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 a real function in n 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)
=f(a_1,a_2)+[(x_1-a_1)(partialf)/(partialx_1)+(x_2-a_2)(partialf)/(partialx_2)]+1/(2!)[(x_1-a_1)^2(partial^2f)/(partialx_1^2)+2(x_1-a_1)(x_2-a_2)(partial^2f)/(partialx_1partialx_2)+(x_2-a_2)^2(partial^2f)/(partialx_2^2)]+....
(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, in vector 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

1/2(a·del _(r^'))(a·del _(r^'))f(r^')|_(r^'=r)=1/2a·del _(r^')[a·(del f(r^'))]_(r^'=r)
(37)
=1/2a·[a·del _(r^')(del _(r^')f(r^'))]|_(r^'=r),
(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)

See also

Cauchy Remainder, Fourier Series, Generalized Fourier Series, Lagrange Inversion Theorem, Lagrange Remainder, Laurent Series, Maclaurin Series, Newton's Forward Difference Formula, Taylor's Inequality, Taylor's Theorem Explore this topic in the MathWorld classroom

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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

Cite this as:

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

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