Unsolvable Exponential: Solving x = y^y

[soandos]

and for negative irrational?

 

[JANm]

Yes the negative irrationals, so the algebraic numbers, which are solutions of polynomal equations and the trancendentals like -e and -pi will be the numbers for which the discontinuity of x^x arises. Whether one can say of irrationals that they are even or not I don't know.

 

[soandos]

I think (based on a a mathematica Table[If[Re[x^x]==x^x,x,0],{x,-200,-.01,.01}]
but am not sure that the only time that it will be continuous is if the number is an integer.
Can someone find a counter-example or prove this?

 

[Ed Aboud]

Where would an equation like this arise?

 

[CRGreathouse]

I think (based on a a mathematica Table[If[Re[x^x]==x^x,x,0],{x,-200,-.01,.01}]
but am not sure that the only time that it will be continuous is if the number is an integer.
Can someone find a counter-example or prove this?

I really don't know what you mean. *What* will be continuous when *what* is an integer?

1. \Re(x^x)-x^x is continuous exactly at the integers
2. [\Re(x^x)=x^x] is continuous exactly at the integers, where [] is the Iverson Bracket
3. \Re(x^x)=x^x iff x\in\mathbb{Z}
4. \Re(x^x)=x^x implies x\in\mathbb{Z}
5. Something else...?

 

[JANm]

The real part of x^x= Re(x^x) = x^x - Im(x^x).
Function: If Re(x^x)=x^x then function = x else 0?

 

[soandos]

sorry i was really unclear.
what i meant was that i think that Im[x^x]=0 iff x is an integer, and if not, then it is some other value (assuming that domain of x in this case is x<0)
i thought that this is pretty much intuitive as you are taking a non-odd root of a negative number.
the table was just to check the assumptions.
with respect to

1. LaTeX Code: \\Re(x^x)-x^x is continuous exactly at the integers
2. LaTeX Code: [\\Re(x^x)=x^x] is continuous exactly at the integers, where [] is the Iverson Bracket
3. LaTeX Code: \\Re(x^x)=x^x iff LaTeX Code: x\\in\\mathbb{Z}
4. LaTeX Code: \\Re(x^x)=x^x implies LaTeX Code: x\\in\\mathbb{Z}

i think that all of them are true, and about the iverson bracket, which i think can be strengthened by saying that it is equal to 1 exactly at the integers.

 

[CRGreathouse]

I really don't know what you mean. *What* will be continuous when *what* is an integer?

1. \Re(x^x)-x^x is continuous exactly at the integers
2. [\Re(x^x)=x^x] is continuous exactly at the integers, where [] is the Iverson Bracket
3. \Re(x^x)=x^x iff x\in\mathbb{Z}
4. \Re(x^x)=x^x implies x\in\mathbb{Z}

i think that all of them are true, and about the iverson bracket, which i think can be strengthened by saying that it is equal to 1 exactly at the integers.

I think that all four are false.

 

[soandos]

how could that be? is there any other way that one can take a non-odd root and end up with a number of the same sign if the domain is restricted to negative numbers?

 

[CRGreathouse]

how could that be? is there any other way that one can take a non-odd root and end up with a number of the same sign if the domain is restricted to negative numbers?

You didn't say that the domain was restricted to negative numbers.

On the negative numbers, I'd think that the function in #2 would be discontinuous precisely at the integers.

For #3 and #4, what about noninteger rational numbers with odd denominators and even numerators?

I'm not entirely sure about #1 on the negative reals, but it might be true. The others look false even on the negative reals.

 

[soandos]

To be honest, i am not sure, and that is why i ran the mathimatica table. the only numbers that showed up were the integers. even with odd denominators and even numerators, there are still irrational.can't prove it, just going off mathematica here, in fact is there a possibility that it is wrong? as a test case it says that (-1)^(1/3) has an imaginary part, though to the best of my knowledge, it is entirely real. could it be that because there is more than one solution to a given expression that it just picks the imaginary ones?

 

[CRGreathouse]

can't prove it, just going off mathematica here, in fact is there a possibility that it is wrong?

It's hard to imagine a numerical calculation of an irrational number being right.

For me, (-1)^(1/3) = -1 since (-1)^3 = -1 and I take the principal root.

 

[JANm]

' Made a graph of y=abs(x)^x.
I thought that it would be much more jumpy. It looks like some third degree with a relative maximum at (-,3; 1,4) and a relative minimum at (0,2;0,8). The difference with the third degree is the horizontal assymptote y=0 for small x.
So for positive y there can be three, two or one solution.
If there is a solution for negative x, x has to be a even rational.
For uneven rationals there are solutions with y>-rel max_y=-1,4 approximately.

 

[soandos]

I plotted the same thing that you did, and it only intersected the x-axis in one place, but
ln(abs(x)^x) gives the intersections with the x-axis at -1,1 and with point symmetry at the origin. i am not sure, but i think that the original (with out the log) did not have this symmetry (just by looking at the graphs). how is that possible?

 

[JANm]

I plotted the same thing that you did, and it only intersected the x-axis in one place, but
ln(abs(x)^x) gives the intersections with the x-axis at -1,1 and with point symmetry at the origin. i am not sure, but i think that the original (with out the log) did not have this symmetry (just by looking at the graphs). how is that possible?

How can this intersection take place? I have an horizontal assymptote at the line y=0.

From the derivative =0 I calculated the x-values of the local/relative extrema.
x_relmax=-e^-1 and x_relmin=e^-1. Should I call it local or relative I am not sure of...

I also found a formula closer to the end result:

y=sgn(x)*abs(x)^x.

 

[soandos]

that gets an entirely differnt plot, where the negative is very messed up

 

[JANm]

Dear Soandos
Don't know why you say that. This function is well defined. We still need swap (x,y) after it is well understood, and that would give two valuedness for x as a function of y. My mathlab gave a good drawing exept from 0 to 1/e it didnot understand the heaviside jump it had to make. We have an exponential function of a positive value and multiply it with the sign function to make it negative for x<0. What the value is at x=0 we still have to decide, but more important this solution has values which work and other values that don't work for the asked function in the start of this thread: x =y^y .
Assumed was that x negative had no solutions at all, but I claim that for y=odd rational this function is well defined and that the sign thing is not important there...
A uneven root of a negative number gives a negative number so to say...
So after the former function is properly inverted (by selecting an area of one valuedness)
we have to submit all the values where y is not a uneven rational.

Measure theory says that rational numbers are countable (so certainly the odd ones) and therefore a null set from all the possible values arosen. Around each odd rational a interval can be taken in such a manner that the sum of all the intervals converges and moreover can be made as small a one wishes.

A null set seems small but another fact is that rational lie close to any number on the value-line. It needs to be proven that that also counts for odd rationals only, but I see no problem in that proof. So let's call that the thesis of JANm. This means that for any given negative number x and a presision factor delta there can be found an y (odd rational) with abs(x-y^y) < delta.

That is as far as we got thusfar...
greetings

 

[remix220]

Is it possible to solve x = y^y for y?

y must represent a number therefor it cannot be multiplied or divided, no?
I'm bad in math!
I could see where x = y, but I don't see where you get a answer from exponent, if you don't know what y represents. Hmmmm...
Where did you come a crossed this problem?

http://www.simonsingh.net/images/465.jpg

 

[soandos]

I do not understand your first two lines.
logs can be used to reduce exponential eqations, like 4^x=250 (x=Log[4]/Log[250]), and the
x th roots of both sides are not taken.
the first time that I saw something like it was solving for the period in a compound interest forumla.

 

[rscosa]

Hi!
it is impossible to find a explicit expression for the solutions of you equation because that means that you can solve the integral x^x which is impossible to express in tems of elementary functions (read wikipedia about this topic) so the answer is: You only can get numerical approximation of you solution if it exists.

 

[soandos]

this has been realized, and that is why the thread talks about Newtons method, and other methods of approximation. there are also problems with these methods (in general) when the function is not continuous. (i.e. for negative x)

also, somebody correct me if i am wrong but the expression (-1)^(2/7) for example can have more than one value, right?