Taylor's and Laurent Series | Basic Physics for IIT JAM PDF Download

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 x₀ to an arbitrary point x is given by 

(9)

where f⁽ⁿ⁾ (x₀) is the n^th derivative of f(x) evaluated at x₀, and is therefore simply a constant. Now integrate a second time to obtain 

  (10)

where f^(k) (x₀) 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 K₁ and k₂ centered at z=a and of radii r₁ and r₂<r₁ 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 c₂ and c₁, with the radius of c₁ larger than that of c₂. Let z₀ be at the center of c₁ and c₂, and  be between c₁ and c₂. Now create a cut line c_c between c₁ and c₂, 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 c₁ and c₂, the integrals are independent of the path of integration in that region. If we replace paths of integration c₁ and c₂ by a circle c of radius r with  , then 

 (33)

 (34)

 (35)

Generally, the path of integration can be any path ϒ that lies in the annular region and encircles z₀ 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 r₁ and decreasing r₂ until singularities of f(z) that lie just outside c₁ or just inside c₂ are reached. If f(x) has no singularities inside c₂, then all the b_k terms in (◇) equal zero and the Laurent series of (◇) reduces to a Taylor series with coefficients a_k.