# Determining the relationship to elementary functions?

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How does one determine with 100% certainty that a certain function can't ever possibly be put into terms of elementary functions, or really any arbitrary list of functions?

Take the gamma function for instance, $\Gamma(x)$. I am fairly confident the gamma function cannot be put into terms of elementary functions, but on the other hand, I've never ever seen anything remotely resembling a proof of that statement or any statement of that nature about any special function. How would I determine it can't be put into terms of other non-special functions or a mix of special functions and elementary functions?

Edited by SFNQuestions

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How does one determine with 100% certainty that a certain function can't ever possibly be put into terms of elementary functions, or really any arbitrary list of functions?

Take the gamma function for instance, $\Gamma(x)$. I am fairly confident the gamma function cannot be put into terms of elementary functions, but on the other hand, I've never ever seen anything remotely resembling a proof of that statement or any statement of that nature about any special function. How would I determine it can't be put into terms of other non-special functions or a mix of special functions and elementary functions?

I take it this question is related to your other one.

The sort of proof you are looking for is known as an existence and uniqueness theorem, of which there are several each pertaining to a particular area of mathematics.

It looks as though you are studying what is known as the functions of a real variable.

Even in this limited area the E&U theorems involve some highly abstract maths which fills large textbooks.

A few 'standards' are

Hurewicz Dimension Theory

Hobson The Theory of Functions of a Real Variable (2 vols)

Titchmarsh The Theory of Functions

Burkhill A second Course in Mathematical Analysis

Graves Theory of Functions of Real Variables

Verblunsky An introduction to the Theory of Functions of a Real Variable

Littlewood The Elements of the Theory of Real Functions

Some of these are rather old.

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I take it this question is related to your other one.

The sort of proof you are looking for is known as an existence and uniqueness theorem, of which there are several each pertaining to a particular area of mathematics.

It looks as though you are studying what is known as the functions of a real variable.

Even in this limited area the E&U theorems involve some highly abstract maths which fills large textbooks.

A few 'standards' are

Hurewicz Dimension Theory

Hobson The Theory of Functions of a Real Variable (2 vols)

Titchmarsh The Theory of Functions

Burkhill A second Course in Mathematical Analysis

Graves Theory of Functions of Real Variables

Verblunsky An introduction to the Theory of Functions of a Real Variable

Littlewood The Elements of the Theory of Real Functions

Some of these are rather old.

I mean I appreciate you trying to help, but I don't see any indication my question is answered. I'm not referring to a uniqueness theorem as a solution to a functional equation, this is different. And it doesn't need to be limited to functions of a real variable. What I mean is: how can you test different functions against another function to determine if one function can somehow be put into terms of another function at its simplest level? For instance, take the cosine function. For the longest time, no one had any idea about complex exponents, people pre-hand made the assumption that you simply couldn't represent the cosine function in a fundamentally different way as a function of a variable, only to later discover it could be represented as functions of complex variables. That's the mistake I want to avoid. I may have a function or Taylor series that may not look like anything I recognize, but is it its own fundamental combination of operators? Or is it a complicated culmination of pre-existing functions? That's what I want to determine.

Edited by SFNQuestions

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I know there's an answer to this, because if there wasn't, every single person who ever said "cannot be put into terms of elementary functions" is automatically discretited, so I know they definitely must have done something to verify that.

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Yes, there is an answer to this and studiot directed you to texts that will lead you to the answer. The problem is that you expect simple answer! The answer is very deep involving deep mathematics. We have no idea what math background you have. Have you taken an upper level class in "abstract algebra"? Have you taken graduate courses in algebra?