Thought of 2 Difficult Math Problems

  • Context: Graduate 
  • Thread starter Thread starter Izzhov
  • Start date Start date
Click For Summary

Discussion Overview

The discussion revolves around two challenging mathematical problems: the first involves solving the equation y^x = x^y + 1 for y as a function of x, and the second concerns the exact value of the infinite series sum \sum_{x=1}^\infty \frac{1}{x^3}, given that \sum_{x=1}^\infty \frac{1}{x^2} = \frac{\pi^2}{6} is known.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose defining a function FOO(x) such that FOO(x)^x = x^FOO(x) + 1, but question whether this function exists.
  • Others argue that simply changing the representation of the variable does not simplify the problem of solving for y.
  • A participant suggests that proving the existence of FOO(x) would resolve the issue, while another emphasizes the need to express y explicitly as a function of x.
  • Some participants discuss the implications of domain restrictions on the ability to express y in the desired form.
  • There is mention of the derivative of FOO(x) and its implications for approximating the function near a specific point.
  • Participants note that certain functions, like the error function or Bessel functions, cannot be expressed in a simple closed form.
  • Discussion includes the known value of Apéry's constant related to the second problem, with no closed form for the sum \sum_{x=1}^\infty \frac{1}{x^3} being established.

Areas of Agreement / Disagreement

Participants express differing views on the feasibility of solving the first problem for y as a function of x, with some believing it can be done under certain conditions, while others maintain that it may not be possible. The second problem remains unresolved regarding its exact value, with participants acknowledging the known approximation but no consensus on a closed form.

Contextual Notes

Limitations include the unresolved nature of the existence of FOO(x) and the lack of a closed form for the sum \sum_{x=1}^\infty \frac{1}{x^3}. The discussion also highlights the complexity of expressing certain functions in a simple form.

  • #31
Dragonfall said:
Did I wander into bizzaro world, or are you guys particularly harsh on the OP?

I agree. As JonF posted, it seems pretty clear he was asking for an elementary solution of y in terms of x...

Does the question want y as a "relation of" x... or does it have to be a function of x? As arildno posted, x = 0 satisfies the equation with any y... So y isn't a function of x and can't be written as such, since a function would need one y value per x value (but you have (x = 0, y = 0) which is a solution or (x=0, y=1) as a solution etc)
 
Mathematics news on Phys.org
  • #32
Ah, it's the 'you are mean, you're supposed to be psychic and guess what the OP meant' moment.
 
  • #33
Izzhov said:
Thank you very much! :biggrin:

Your skill at communication is truly unmatched.
Your eloquent "y=blah" says it all..
 
  • #34
learningphysics said:
I agree. As JonF posted, it seems pretty clear he was asking for an elementary solution of y in terms of x...

Does the question want y as a "relation of" x... or does it have to be a function of x? As arildno posted, x = 0 satisfies the equation with any y... So y isn't a function of x and can't be written as such, since a function would need one y value per x value (but you have (x = 0, y = 0) which is a solution or (x=0, y=1) as a solution etc)
Hmm, you can find local regions where y can be regarded as a function foo(x), as I have shown.

However, not even an explicit linear approximation to foo(x) in the vicinity of (1,2) is enough for the blah-blah guys of the world..
 
  • #35
arildno said:
Your skill at communication is truly unmatched.
Your eloquent "y=blah" says it all..

I actually thought it was fairly self-explanatory. I guess I didn't realize that you thought through every problem at the calculus level. :-p I was looking for a more algebraic approach to it, but I'm fine, since I now have the solution. :smile:
 
  • #36
Is there some heirarchy that says 'the calculus level' is lower than the 'algebraic level'? You had the solution before, but you just weren't happy with it. The solution was perfectly correct.
 
Last edited:
  • #37
matt grime said:
Is there some heirarchy that says 'the calculus level' is lower than the 'algebraic level'? You had the solution before, but you just weren't happy with it. The solution was perfectly correct.

Actually, the whole problem in our communication is that "calculus level" is higher than algebraic level; it's a classic case of the answerer thinking more deeply about the question than the asker was expecting, and answering it at a higher level (i.e. using calculus) than the asker wanted (i.e. using algebra).
 
  • #38
Izzhov said:
Actually, the whole problem in our communication is that "calculus level" is higher than algebraic level; it's a classic case of the answerer thinking more deeply about the question than the asker was expecting, and answering it at a higher level (i.e. using calculus) than the asker wanted (i.e. using algebra).

Whichever way you want it, dear, you won't find any other best linear approximation to foo(x) about (1,2) than the one I posted..
 
  • #39
arildno said:
Whichever way you want it, dear, you won't find any other best linear approximation to foo(x) about (1,2) than the one I posted..

'K, thanks. :smile:
 

Similar threads

Graduate A global version of the implicit function theorem
  • · Replies 0 ·
Replies
0
Views
644
Undergrad Alternating Harmonic Numbers are cool, spread the word!
  • · Replies 4 ·
Replies
4
Views
3K
Graduate Are these two optimization problems equivalent?
  • · Replies 1 ·
Replies
1
Views
2K
High School A symmetric problem has an asymmetric solution - why?
  • · Replies 21 ·
Replies
21
Views
2K
Help understanding modal projection in PDE with assumed solution form
  • · Replies 4 ·
Replies
4
Views
2K
Please tell me where I am going wrong in this integral
  • · Replies 2 ·
Replies
2
Views
859
MHB Why Do Normalized B-Splines Always Sum to One?
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad Transforming coordinates between Cartesian and spherical
  • · Replies 1 ·
Replies
1
Views
2K
Uniform Continuity of functions
  • · Replies 40 ·
2
Replies
40
Views
5K
High School Can We Simplify the Taylor Series Expansion for e^(f(x,y))?
  • · Replies 3 ·
Replies
3
Views
2K