Power expansions

The powers of a variable x appeared originally purely in algebraic problems. With the development of calculus the great importance of power expansions became evident. The expansion discovered by Taylor (1715) and by Maclaurin (1742) enables us to predict the course of a function if we know the value of the function and all its derivatives in one particular point. The "Taylor series" thus became one of the cornerstones of analytical research and was particularly useful in establishing the existence of solutions of different equations.

If the series has no other purpose than numerical evaluation of the function, the degree of convergence has to be investigated. The Taylor expansion may converge in the entire complex plane or within a given circle only, and it may diverge even at every point. It was recognized, however, that a more liberal formulation of the question of convergence greatly increases the usefulness of an expansion.

One can make good use, for example, of "semiconvergent expansions" which actually diverge if we increase the number of terms to infinity, but converge in the beginning, thus allowing evaluation of the function with a certain limited accuracy which cannot be surpassed, since the error of the truncated series decreases to a certain minimum and then increases again. Much attention was paid also to the problem of inventing methods of summing a series in such a way that it shall become convergent, although the original series, if added term by term, increased to infinity.

With the development of the theory of orthogonal expansions the realization came that occassionally power expansions, whose coefficients are not determined according to the scheme of Taylor, can operate much more effectively than the Taylor series itself. Such expansions are not based on the process of successive differentiation but on integration. A large class of functions which are not sufficiently analytic to allow a Taylor expansion can be represented by such orthogonal expansions.

The realm of power expansions is thus extended far beyond the family of analytic functions. But even for analytical functions we may gain in convergence if we do not employ the powers directly but in the form of polynomials which are members of an orthogonal set of functions. These expansions belong to a given definite real realm of the variable x, and our aim is to approximate a function in such a way that the error shall not become too small or too large at any particular point of the range, but rather of the same order of magnitude all over the range. The gain in comparison with the Taylor series arises from the fact that we sacrifice in accuracy at the point where the Taylor series gave very accurate results but reduce the error in the peripheral regions where the error of the Taylor expansion became intolerably large.