Can e^{e^{e^{x}}} Be Simplified or Related to Hypergeometric Functions?

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In summary, there is no simple or clear way to simplify or clarify e^e^e^x. Some have suggested an equivalence to a hypergeometric, but this may not be helpful. It is important to clarify the order of operations, as the expression without parentheses is ambiguous. One possible simplification is ((e^e)^e)^x = (e^e)^{ex} = e^{e^2x}.
  • #1
PlasticOh-No
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Is there any way to simplify or clarify e^e^e^x ?
Some sort of equivalence to a hypergeometric?
How about e^e^x?
Thanks
 
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  • #2


Assuming that these associate the usual way (right-to-left), they don't really simplify.
 
  • #3


What you have written, without parentheses,is ambiguous. That is why CRGreathouse said "Assuming that these associate the usual way (right-to-left)".

[tex]e^{e^{e^x}}[/tex]
does not simplify.

[tex]((e^e)^e)^x= (e^e)^{ex}= e^{e^2x}[/tex]
 
  • #4


PlasticOh-No said:
Is there any way to simplify or clarify e^e^e^x ?
Some sort of equivalence to a hypergeometric?
How about e^e^x?
Thanks

Assume it's the way I think you want it to be:

[tex]e^{e^{e^{x}}}=\sum_{n=0}^{\infty} a_n x^n[/tex]

and you know how to figure out what each a_n is.
 
  • #5


Thanks everyone for your help. Yes, I meant

[tex]e^{e^{e^{x}}}[/tex]
 
Last edited:

FAQ: Can e^{e^{e^{x}}} Be Simplified or Related to Hypergeometric Functions?

1. What is the meaning of "e" in the expression e^e^e^x?

The symbol "e" in this context represents the mathematical constant, approximately equal to 2.71828. It is also known as Euler's number.

2. How do you simplify e^e^e^x?

To simplify this expression, we can use the rule that e^a^b is equal to e^(a*b). Therefore, e^e^e^x can be rewritten as e^(e*e^x). From here, we can use the rule that e^a*e^b is equal to e^(a+b). So, e^(e*e^x) simplifies to e^(e+e^x).

3. Can e^e^e^x be rewritten in a different form?

Yes, e^e^e^x can also be written as (e^x)^e*e. This is known as the power tower form of the expression.

4. What is the purpose of simplifying e^e^e^x?

Simplifying this expression can make it easier to work with and understand in mathematical equations and calculations. It can also help in finding the derivative or antiderivative of the expression.

5. Is there a limit to how many times e can be raised to a power in this expression?

No, there is no limit to how many times e can be raised to a power in e^e^e^x. The expression can continue with as many layers as needed.

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