# Series expansion for a function.

[section]1. INTRODUTION

It is known that the Taylor's series expansion as given in  for a function y(x) about the point x = a , where y(x) is continuous and possesses continuous derivatives of order (n+1) in an interval that contains the point x = a, is

y(x) = y(a) + (x - a)y'(a) + [(x - a).sup.2]/2! y''(a) + ... + [(x - a).sup.n]/n! [y.sup.(n)] (a) + [R.sub.n+1](x) ... (1.1)

where [R.sub.n+1](x) is the remainder term which will be of the form

[R.sub.n](x) = [(x - a).sup.(n+1)]/(n + 1)! [y.sup.(n+1)([xi]), a < [xi] < x

From (1.1), one can easily deduce the Maclaurin's expansion as given in  of the form

y(x) = y(0) + xy'(0) + [x.sup.2]/2! y''(0) + ... + [x.sup.n]/n! [y.sup.(n)](0) + ... (1.2)

The series expansion for many transcendental functions can be obtained from the Maclaurin's series. As a matter of fact, for example;

[e.sup.x] = 1 + x + [x.sup.2]/2! + ... + [x.sup.n]/n! + ... (13)

In this paper, an attempt is made to derive a series expansion for [PHI](x) where [PHI] is any function of x, when the values of the unknown functions are known corresponding to the values of the independent variable in a certain interval.

[section]2. DERIVATION

Let [y.sub.0], [y.sub.1], [y.sub.2], ..., [y.sub.n] and [[PHI].sub.0], [[PHI].sub.1], [[PHI].sub.2], ..., [[PHI].sub.n] be 2(n+1) values of the functions y(x) and [PHI](x) respectively, corresponding to the values of [x.sub.0], [x.sub.1], [x.sub.2], ..., [x.sub.n] where [x.sub.i] - [x.sub.i-1] = h (constant) for i = 1, 2, 3, ..., n.

We shall now try to fit a polynomial of the form

y(x) = [a.sub.0] [PHI](x) + [a.sub.1] (x - [x.sub.0]) + [a.sub.2] (x - [x.sub.0]) (x - [x.sub.1]) + ... + [a.sub.n] (x - [x.sub.0]) (x - [x.sub.1]) ... (x - [x.sub.n-1]) + ... (2.1)

satisfying the given data.

By putting x = [x.sub.0], [x.sub.1], [x.sub.2], ..., [x.sub.n] in (2.1) and equating those values with that of the functional of y, we obtain the values of the coefficients [a.sub.0], [a.sub.1], [a.sub.2], [a.sub.3], ..., [a.sub.n] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

Where [DELTA][y.sub.0] = [y.sub.1] - [y.sub.0], [[DELTA].sup.2] [y.sub.0] = [y.sub.2] - 2[y.sub.1] + [y.sub.0],

[[DELTA].sup.3] [y.sub.0] = [y.sub.3] - 3[y.sub.2] + 3[y.sub.1] - [y.sub.0], ..., and

[[DELTA].sup.n] [y.sub.0] = [y.sub.n] - n [y.sub.n-1] + ... + [(- 1).sup.n] [y.sub.0]

are first, second, third and so on up to nth order forward differences respectively.

Substituting the values of the coefficients as given in (2.2) in (2.1) , one can get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

where u = {x - [x.sub.0]/h}.

Rewriting the above equation (2.3) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

It can be easily seen that the polynomial enclosed in the first square brackets fits the curve y(x) satisfying the given data, which is the Newton's forward difference interpolation formula. And hence, the other part of (2.4) can be directly equated to zero.

Thus, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

By multiplying both sides of (2.5) with [[PHI].sub.0] and expressing the differences of [PHI]'s in forward difference operator notation, one can have Newton's forward formula for [PHI](x).

Now, the series expansions for various functions can be had in different form from (2.5). For instance,

taking [PHI](x) = [e.sup.x] in (2.5), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

(or)

Choosing proper h for any x>0, we can have

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.8)

CONCLUSIONS

The main aim of this paper is to have a series expansion for a given transcendental function other than the conventional one usually obtained from the Maclaurin's series. That can be seen as a matter of fact, from the expansions of [e.sup.x] in (1.3) and (2.8).

REFERENCE

[1.] Sastry, S.S. 2005, 'Introductory Methods of Numerical Analysis', Prentice--Hall of India, New Delhi, India.

V. B. Kumar Vatti., P.V. Reddy and M.S. Kumar Mylapalli

Dept. of Engg. Mathematics, Andhra University, Visakhapatnam--530 003 Andhra Pradesh, India.

Email : drvattivbk@yahoo.co.in, vasucrypto@yahoo.com, maniraj9@yahoo.co.in
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