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In general, it's not true that a given transcendental function $f(z)$ will give you a transcendental output for countably infinite algebraic inputs, but is there a condition that can prove that property is or isn't true for a specific function? The proofs that $e^{a}$ is transcendental are complicated and unique, but what about other transcendental functions? Isn't there something that can generalize the results by this point?

TeXnichal
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  • Among others, Diego Marques has written a lot of papers on this topic, for example. By the way, I don't know what you mean by "countably infinite algebraic inputs". The inputs to these functions are complex numbers, and the adjective phrase "countably infinite" applies to SETS, not to individual numbers, but you are applying the phrase to numbers, and thus what you are saying is similar to saying "the population of Albert Einstein". – Dave L. Renfro Sep 14 '19 at 18:27
  • Thank you, I will see if I can make use of the papers. However, the premise of the question is consistent because you can of course perform operations on sets, that is the basis for functions to begin with. The exponential function of a set of numbers that are algebraic is a set of numbers which are transcendental, similar to how if you take the entire real line, then exponentiate it, you get another set of numbers that are all greater than 0. – TeXnichal Sep 14 '19 at 20:21
  • Unfortunately Marques' work does not appear useful because they rely on the Schanuel conjecture being true, and I do not want to rely on unproven theorems for specific results since that is redundant. – TeXnichal Sep 14 '19 at 20:35
  • because they rely on the Schanuel conjecture being true --- I suspect you did not look at many of his papers, because many of his papers do not involve Schanuel's conjecture (and, yes, many do). Look at his papers (and other people's papers) that deal with aspects related to Hilbert's, Strauss's, and Stäckel's late 1800s results involving possible distributions of arithmetical values (rational, algebraic, transcendental) of transcendental functions at rational (or algebraic) values, such as is surveyed in this paper. – Dave L. Renfro Sep 14 '19 at 21:46
  • I am not aware of a general rule, but other examples are $\ln(z)$ , $\sin(z)$ and $\cos(z)$ – Peter Sep 16 '19 at 21:17
  • see the answers in https://math.stackexchange.com/questions/2000638/on-the-behavior-of-transcendental-functions/3889240? – IV_ Jan 13 '21 at 21:28

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