y^x = x^y

[Dave]

It basically shows that the graph is discontinuous; i.e. there isn't a value of y for every value of x.

[bloodhound]

I tried to graph the function implicitly in Maple, which is a very good maths software and weird things begun to happen.

 

heres the commands.

 

>with(plots);

>implicitplot(x^y=y^x,x=0..10,y=0..10);

 

 

This is the result. Note the graph looks similar to the one above but its all wavy.

 

Now the strange things happens

 

i try

 

>implicitplot(x^y=y^x,x=0..100,y=0..100);

 

and this is what i get

 

 

its all gone wrong.

 

IF anyone has maple , can you repeat what i have done and see if u get the same result. cheers

[ydoaPs]

ok, i'm dumb. i went through 3 minutes of manipulation and got back to y^x=x^y

[DQW]

Someone has plotted the solution out at this site.

[cronxeh]

Well back in Physicsforums.com we've done a little discussion on this exact subject, and the graph I've got is this:

[ydoaPs]

isn't that what post #80 said?

[DQW]

It is. I did, however, fail to acknowledge cronxeh by "name". I apologize.

 

That is a spectacularly interesting looking plot, cronx. Thanks !

[cronxeh]

Oh geez. I only skimmed through the replies looking at a plot but only seen Q1 plot in post ~ #76 ? I see DQW posted it, its cool. I'm particularly interested in similarities between this particular function and the gamma function - since it does have some similar regions as well as curves which could be a little 'massaged' to make alike. What would be the purpose of that? Well maybe no really good purpose but just for a visual satisfaction, but then again so are most interesting functions in math

 

 

Sees an angry mob emerging on the horizon.. Booyakasha!

[ydoaPs]

it isn't a function

[cronxeh]

z=f(x,y)=x^y-y^x=0

[ydoaPs]

line test...there is more than one y value for every x value, therefore, not a function.

[cronxeh]

welcome to.. the real world! Here I've enclosed the plot

[ydoaPs]

explain?

[Dave]

Defining f as a function of two variables is fine, as you can see from the 3D plot. It's just when you attempt to solve for either of the variables that you have problems. It should be fairly obvious by looking at the plot that you can't re-arrange for x or y, though.

 

But yes, there looks to be a similarity (of sorts) between the gamma function over negative reals and the singularities you can see in the lower right and upper left quadrants. However, the gamma function has singularities at negative integers, and this doesn't seem to follow the same trend.

[bmaxwell]

Defining f as a function of two variables is fine' date=' as you can see from the 3D plot. It's just when you attempt to solve for either of the variables that you have problems. It should be fairly obvious by looking at the plot that you can't re-arrange for x or y, though.

 

But yes, there looks to be a similarity (of sorts) between the gamma function over negative reals and the singularities you can see in the lower right and upper left quadrants. However, the gamma function has singularities at negative integers, and this doesn't seem to follow the same trend.[/quote']

 

 

My brain Hurts NOW

[ydoaPs]

i'v just blind and didn't see that he added a variable. i was thinking in two dimensions.