# Is there a proof that there is no general method to solve transcendental equations?

Being motivated by this post, I was wondering if there is a proof (analogous to the case of Diophantine equations) that there is no general method for solving transcendental equations? It seems pretty clear, intuitively, that there can be no general method; but the only reason I feel strongly about that is because I can't conceive that transcendental equations have a general method, but Diophantine equations don't. I was never able to understand the proof for the case of Diophantine equations, so I am not in a position to even know where to begin thinking about this. Has any work been done on this problem?

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Try encoding the halting of a general turing machine into a transcendental equation. –  anon Aug 30 '10 at 16:15
Uh, this is an old question, I agree, and there is no sense in posting something after all the answers are well-accepted and voted, but I see that none of the answers below mention that this question essentially asks for the Constant problem. Just sayin'. –  Balarka Sen Jan 21 '14 at 13:54

Yes. The problem is that the term "transcendental function" is absurdly general. For example, pick a computable bijection between the integers and the set of Turing machines and consider the function $f : \mathbb{R} \to \mathbb{R}$ which is $0$ on the integers corresponding to halting Turing machines, $1$ on the integers corresponding to non-halting Turing machines, and smoothly interpolated in between (for example via bump functions) such that if $x$ is not an integer then $0 < f(x) < 1$. Then the problem of determining the solutions to $f(x) = 1$ is equivalent to the halting problem. (And $f$ is even smooth.)
As you point out, the problem is that -- wikipedia article notwithstanding -- there is not an agreed upon definition of "transcendental function". I was thinking about this question a bit myself and had decided that a transcendental function should be a complex analytic function on $\mathbb{C}^n$ which is not algebraic. Suppose we ignore the non-algebraic part and just ask whether there is an algorithm to tell whether a system of (recursive?) complex-analytic functions has a common solution. What do we think about that? –  Pete L. Clark Aug 31 '10 at 21:40