## 26. C. Explicit Algebraical Functions.

The next important class of functions is that of *explicit algebraical functions*. These are functions which can be generated from \(x\) by a finite number of operations such as those used in generating rational functions, together with a finite number of operations of root extraction. Thus \[\frac{\sqrt{1 + x} – \sqrt[3]{1 – x}} {\sqrt{1 + x} + \sqrt[3]{1 – x}},\quad \sqrt{x} + \sqrt{x +\sqrt{x}},\quad \left(\frac{x^{2} + x + \sqrt{3}}{x\sqrt[3]{2} – \pi}\right)^{\frac{2}{3}}\] are explicit algebraical functions, and so is \(x^{m/n}\) ( \(\sqrt[n]{x^{m}}\)), where \(m\) and \(n\) are any integers.

It should be noticed that there is an ambiguity of notation involved in such an equation as \(y = \sqrt{x}\). We have, up to the present, regarded () \(\sqrt{2}\) as denoting the *positive* square root of \(2\), and it would be natural to denote by \(\sqrt{x}\), where \(x\) is any positive number, the positive square root of \(x\), in which case \(y = \sqrt{x}\) would be a one-valued function of \(x\). It is however often more convenient to regard \(\sqrt{x}\) as standing for the two-valued function whose two values are the positive and negative square roots of \(x\).

The reader will observe that, when this course is adopted, the function \(\sqrt{x}\) differs fundamentally from rational functions in two respects. In the first place a rational function is always defined for all values of \(x\) with a certain number of isolated exceptions. But \(\sqrt{x}\) is undefined for a *whole range* of values of \(x\) ( all negative values). Secondly the function, when \(x\) has a value for which it is defined, has generally two values of opposite signs.

The function \(\sqrt[3]{x}\), on the other hand, is one-valued and defined for all values of \(x\).

## 27. D. Implicit Algebraical Functions.

It is easy to verify that if \[y = \frac{\sqrt{1 + x} – \sqrt[3]{1 – x}} {\sqrt{1 + x} + \sqrt[3]{1 – x}},\] then \[\left(\frac{1 + y}{1 – y}\right)^{6} = \frac{(1 + x)^{3}}{(1 – x)^{2}};\] or if \[y = \sqrt{x} + \sqrt{x + \sqrt{x}},\] then \[y^{4} – (4y^{2} + 4y + 1)x = 0.\] Each of these equations may be expressed in the form \[\begin{equation*} y^{m} + R_{1}y^{m-1} + \dots + R_{m} = 0, \tag {1}\end{equation*}\] where \(R_{1}\), \(R_{2}\), …, \(R_{m}\) are rational functions of \(x\): and the reader will easily verify that, if \(y\) is any one of the functions considered in the last set of examples, \(y\) satisfies an equation of this form. It is naturally suggested that the same is true of any explicit algebraic function. And this is in fact true, and indeed not difficult to prove, though we shall not delay to write out a formal proof here. An example should make clear to the reader the lines on which such a proof would proceed. Let \[y = \frac{x + \sqrt{x} + \sqrt{x + \sqrt{x}} + \sqrt[3]{1 + x}} {x – \sqrt{x} + \sqrt{x + \sqrt{x}} – \sqrt[3]{1 + x}}.\] Then we have the equations \[\begin{gathered} y = \frac{x + u + v + w} {x – u + v – w}, \\ u^{2} = x,\quad v^{2} = x + u,\quad w^{3} = 1 + x,\end{gathered}\] and we have only to eliminate \(u\), \(v\), \(w\) between these equations in order to obtain an equation of the form desired.

We are therefore led to give the following definition: *a function \(y = f(x)\) will be said to be an algebraical function of \(x\) if it is the root of an equation such as , the root of an equation of the \(m\) ^{th} degree in \(y\), whose coefficients are rational functions of \(x\)*. There is plainly no loss of generality in supposing the first coefficient to be unity.

This class of functions includes all the explicit algebraical functions considered in § 26. But it also includes other functions which cannot be expressed as explicit algebraical functions. For it is known that in general such an equation as cannot be solved explicitly for \(y\) in terms of \(x\), when \(m\) is greater than \(4\), though such a solution is always possible if \(m = 1\), \(2\), \(3\), or \(4\) and in special cases for higher values of \(m\).

The definition of an algebraical function should be compared with that of an algebraical number given in the last chapter (Misc. Exs. 32).

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