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In a numerical methods class I'm taking, it was claimed that the equation $A = \frac{R^2}{2} \left(\theta - \sin\theta \right)$ cannot be analytically solved for $\theta$. I don't doubt that this is true, but I'm curious how it could be proved that this is true.

In general, how would one go about proving that a solution to an equation in terms of elementary functions does not exist?

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  • $\begingroup$ Really what you mean to say is that there is no expression in terms of elementary functions. The word "analytic" gets tossed around too much and leads to confusion. Surely there is a solution to that equation however it's not expressible in terms of elementary functions like trigonometric functions, exponentials, powers and roots. $\endgroup$
    – Cameron Williams
    Jul 8 '13 at 23:00
  • $\begingroup$ Interesting, thanks for pointing that out. Am I correct in understanding that if an expression does not exist in terms of elementary functions then it is not possible to obtain an exact value either by hand or with a computer? $\endgroup$
    – Gordon Bailey
    Jul 8 '13 at 23:10
  • $\begingroup$ @GordonBailey: What do you mean by exact value? If you mean by "arbitrarily (but finitely) accurate", then no, you're not correct. There are many computable functions which are not elementary. On the other hand, you can't really expect functions which are not integer (or at least rational-) valued to be computed exactly, as real numbers can't be exactly represented. The set of elementary functions is in many ways, quite arbitrary, and their (relative) significance is historical more than anything, as far as I know. $\endgroup$
    – tomasz
    Jul 8 '13 at 23:16
  • $\begingroup$ @GordonBailey No. The Gamma Function provides a simple counterexample: $\Gamma(\frac{1}{2}) = \sqrt{\pi}$. Also, trivial counterexamples exist such as the Lambert W function: $W(0) = 0$. $\endgroup$
    – Gamma Function
    Jul 8 '13 at 23:16
  • $\begingroup$ Thank you both for your answers. I'm curious what the right wording for my original question would be. Basically I want to know how you can determine if an expression for (in this case) $\theta$ can be found in terms of $A$ and $R$. Is this actually a mathematically meaningful question? $\endgroup$
    – Gordon Bailey
    Jul 8 '13 at 23:37
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About a century and a half ago, Liouville developed a criterion to determine whether the solution to certain types of differential equation could be expressed in elementary terms. This criterion was powerful enough to show for example that there is no elementary function whose derivative is $e^{-x^2}$.

The ideas of Liouville led to the field of differential algebra.

Much later, Risch produced an algorithm that will determine, for a fairly wide class of elementary functions, whether a function has an elementary antiderivative. There have been improvements since, and improved algorithms have been at least partly implemented.

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Mainly, Liouville and Risch treated integration in finite terms and answered, with their method, the problem of integration of elementary functions by elementary functions.

For the problem of solving equations in terms of elementary functions, Liouville's representation of elementary functions as composition of explicit algebraic functions and elementary standard functions, or, as composition of explicit algebraic functions, $\exp$ and $\ln$, is necessary.

Solving an equation in terms of elementary functions means that the function which is set constant by this equation has an inverse function and that this inverse function is expressable in terms of elementary functions.

Assume that your equation exists: A and R, not dependent on $\theta$, are choosen so that the equation depends on its solving variable ($\theta$).

Write your equation as zeroing equation. Then look at the structure of this equation. The solving variable ($\theta$) is argument of algebraic component functions (The identity function ($\theta\mapsto \theta$) is an algebraic function.) and of transcendental component functions ($\sin$). The component functions are the arguments of an outer function which is an algebraic function. This algebraic function can be represented as a unary algebraic function only for the values of $\theta$ where the values of the component functions are algebraically dependent. An algebraic function $A$ can have an inverse function which is expressable in terms of elementary functions only if $A$ is unary. Therefore your equation can have a solution in terms of elementary functions only if a value of $\theta$ where the values of the component functions are algebraically dependent is a solution of the equation. The equation from your example can have a solution which is expressable in terms of elementary functions only when $0$ is a solution of this equation.

Look for transcendental number theory, e.g. for Schanuel's conjecture and its precursors, and for the expression of elementary functions as composition of explicit algebraic functions, $\exp$ and $\ln$.

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