51. Interpolation

The problem of determining a function of \(x\) which shall assume, for all positive integral values of \(x\), values agreeing with those of a given function of \(n\), is of extreme importance in higher mathematics. It is called the problem of functional interpolation.

Were the problem however merely that of finding some function of \(x\) to fulfil the condition stated, it would of course present no difficulty whatever. We could, as explained above, simply fill in the missing values as we pleased: we might indeed simply regard the given values of the function of \(n\) as all the values of the function of \(x\) and say that the definition of the latter function failed for all other values of \(x\). But such purely theoretical solutions are obviously not what is usually wanted. What is usually wanted is some formula involving \(x\) (of as simple a kind as possible) which assumes the given values for \(x = 1\), \(2\), ….

In some cases, especially when the function of \(n\) is itself defined by a formula, there is an obvious solution. If for example \(y = \phi(n)\), where \(\phi(n)\) is a function of \(n\), such as \(n^{2}\) or \(\cos n\pi\), which would have a meaning even were \(n\) not a positive integer, we naturally take our function of \(x\) to be \(y = \phi(x)\). But even in this very simple case it is easy to write down other almost equally obvious solutions of the problem. For example \[y = \phi(x) + \sin x\pi\] assumes the value \(\phi(n)\) for \(x = n\), since \(\sin n\pi = 0\).

In other cases \(\phi(n)\) may be defined by a formula, such as \((-1)^{n}\), which ceases to define for some values of \(x\) (as here in the case of fractional values of \(x\) with even denominators, or irrational values). But it may be possible to transform the formula in such a way that it does define for all values of \(x\). In this case, for example, \[(-1)^{n} = \cos n\pi,\] if \(n\) is an integer, and the problem of interpolation is solved by the function \(\cos x\pi\).

In other cases \(\phi(x)\) may be defined for some values of \(x\) other than positive integers, but not for all. Thus from \(y = n^{n}\) we are led to \(y = x^{x}\). This expression has a meaning for some only of the remaining values of \(x\). If for simplicity we confine ourselves to positive values of \(x\), then \(x^{x}\) has a meaning for all rational values of \(x\), in virtue of the definitions of fractional powers adopted in elementary algebra. But when \(x\) is irrational \(x^{x}\) has (so far as we are in a position to say at the present moment) no meaning at all. Thus in this case the problem of interpolation at once leads us to consider the question of extending our definitions in such a way that \(x^{x}\) shall have a meaning even when \(x\) is irrational. We shall see later on how the desired extension may be effected.

Again, consider the case in which \[y = 1 \cdot 2 \dots n = n!.\] In this case there is no obvious formula in \(x\) which reduces to \(n!\) for \(x = n\), as \(x!\) means nothing for values of \(x\) other than the positive integers. This is a case in which attempts to solve the problem of interpolation have led to important advances in mathematics. For mathematicians have succeeded in discovering a function (the Gamma-function) which possesses the desired property and many other interesting and important properties besides.