Proving the non-existence of a function.

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In summary, the conversation discusses the non-existence of the elementary function N(x) and how it is defined. One way to show its impossibility is through the existence of another function I(x) and the integral of x^x being irreducible to elementary terms. The conversation also mentions the possibility of N(x) existing under different specifications.
  • #1
m84uily
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Hello! I want to know more ways to show the non-existence of the elementary function N(x).
Here's how N(x) is defined:
g(x) : Any elementary function.
[tex] N'(g(x)) = \frac{g(x)}{e^{N(g(x))}} [/tex]

I've only thought of a single way to show this impossibility and it doesn't really develop my understanding of the "why" behind it very well. Here's how I went about it:

If such a function N(x) were to exist then there could also be a function I(x) which could be defined as:
[tex]I(g(x)) = e^{N(g(x))} [/tex]
Which would mean:
[tex]I'(g(x)) = g(x) [/tex]
[tex]\int x^x dx = I(x^x) + C[/tex]
The integral of x^x has been proven irreducible to elementary terms by Marchisotto and Zakeri and therefore N(x) cannot exist for it allows the existence of I(x).

Would anyone else mind posting other lines of reasoning please?
 
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  • #2
m84uily said:
The integral of x^x has been proven irreducible to elementary terms by Marchisotto and Zakeri and therefore N(x) cannot exist for it allows the existence of I(x).

Do not confuse " irreducible to elementary terms" with "cannot exist".
Obviuosly x^x exists, even if it is irreducible to elementary terms.
Similary, N(x) possibly exists even if it is irreducible to elementary terms.
 
  • #3
Similary, N(x) possibly exists even if it is irreducible to elementary terms.

Well I had said
I want to know more ways to show the non-existence of the elementary function N(x).
Under the specification that N(x) is an elementary function, it cannot exist as I have shown.
 

1. Can you prove that a certain function does not exist?

Yes, it is possible to prove the non-existence of a function through various methods such as proof by contradiction or counterexample.

2. How do you prove that a function is non-existent?

The most common method is to show that the function leads to a contradiction or does not satisfy certain mathematical properties, thus rendering it non-existent.

3. Is it difficult to prove the non-existence of a function?

It depends on the complexity of the function and the available mathematical tools. Some functions can be easily proven to be non-existent, while others may require advanced techniques and extensive mathematical knowledge.

4. Can a function be proven to be non-existent in all cases?

No, there may be some specific cases where a function may exist, even though it is non-existent in most cases. Therefore, it is important to carefully analyze the function in question and consider all possible scenarios before making a conclusion.

5. What is the significance of proving the non-existence of a function?

Proving the non-existence of a function can help in understanding the limitations of mathematics and identifying any inconsistencies or flaws in a particular theory. It can also lead to the development of new mathematical concepts and theories.

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