Joe,

Thank you for the correction.

I sense that Federico has a point but I think that I am unable to appreciate it.  Can you help me?  I don't seem to understand what he means when he uses the word "indetermined" and I think that if his English were a little better he might have said undetermined.  In either case, I don't think that I'm getting the idea that he is trying to communicate.  Can anybody interpret the following for me?

"… it is very plausible how problematic it will be to determine the nature of (x irr) ^ ^{(x irr)}, which means that for the Irrationals the curve is mainly undetermined."

By the way, Federico's English is far better than my Italian and I very much appreciate his effort.

 

Here is my 2nd attempt at the paragraph that I messed up in my last posting:

 

The problem seems to be that the function x^x is closed for the natural number system but not for other number systems.

Federico,

Thank you for your comment.

There are seven or more number systems.

http://en.wikipedia.org/wiki/Template:Numbers

I'm not sure what you mean by "undetermined".  Maple will plot x^x without complaints. 

View 4937_Page 344 Federico C. Zabetta.mw on MapleNet or Download 4937_Page 344 Federico C. Zabetta.mw
View file details

(1/2)^(1/2) is approximately 0.7071067811865475244008443621048490392848

x is rational but x^x is irrational and so I can't know all its digits.

The problem seems to be that the function x^x is not closed in any number system other than the natural number system.  Is that it?

I wish I could think of a question to ask you that would help me better understand what you are telling me.

Thank you again for trying to help me understand.

 

Will,

Thank you.  That's good enough for me.  Thanks again.

What is there about this equation:

that would cause anyone (probably not Euler) to think that it supports the hypothesis that "God exists"?

Is there anything particularly interesting about this equation?  Are there any implied assumptions about the variables?  Is n assumed to be an integer?

Will,

Jacques Carette thinks that you have the power to move this thread to my blog.  If this is so, please make it happen.

Please move this thread to my blog.

Thank you.

Sir, I disagree.  You are the one that is extremely kind.

From where I'm at, many years of experience in math, CS and Maple is indistinguishable from astonishing genius.

A bit of magic that blew me away was given to me by Robert Israel.  It plots the phase of a complex functions, F(x), that I was interested in.  Here is Robert's Matlab:

de := diff(a(x), x) = 180*evalc(Im((diff(F(x), x))/F(x)))/Pi:

DEtools[DEplot](de, a(x), x = -.1 .. 2, [a(1) = 180*argument(F(1))/Pi], linecolour = black, arrows = none, axes = boxed)

I didn't know that what I wanted was a function, a(x), that is the imaginary part of the natural log of F(x).  I also had no idea how to get a(x).  That is way way cool!

Arthur C. Clark's third law: Any sufficiently advanced technology is indistinguishable from magic.

I will reply to Will to see if he can work some magic.

Because I'm inexperienced at this.  I hardly know what a blog is and I don’t know when to use a forum rather than a blog.

The question came to me because I’ve been trying to read Dr. Euler’s Fabulous Formula by Nahin and that equation is on page 344 of that book.

I'm sorry if I have made a mistake.  I'll try to do better next time.  It would help if I could get some guidance as to where to put what.

I and other students have found that Jacques Carette and others like you are often kind enough to consider our questions when posted here. It’s almost like being able to asking Euler himself.

I love it when there is someone around to clean up my act.  Thank you.

How did I make that mess?  I tried.

View 4937_Page 344.mw on MapleNet or Download 4937_Page 344.mw
View file details

Thank you.

Maple likes this:

int(-f(t)*(t-x), t = 0 .. x)

I like this:

int(f(t)*(x-t), t = 0 .. x)

Can you tell me how to get Maple to display it the way I like it?

View 4937_Page 99 integration Alex Vogt.mw on MapleNet or Download 4937_Page 99 integration Alex Vogt.mw
View file details

Thank you.

Maple likes this:

int(-f(t)*(t-x), t = 0 .. x)

I like this:

int(f(t)*(x-t), t = 0 .. x)

Can you tell me how to get Maple to display it the way I like it?

View 4937_Page 99 integration Alex Vogt.mw on MapleNet or Download 4937_Page 99 integration Alex Vogt.mw
View file details

Thank you, but Maple still doesn't seem to agree that the last of your equations is always true.

Here is my attempt to repeat what you posted:

View 4937_Page 99 integration Alex Vogt.mw on MapleNet or Download 4937_Page 99 integration Alex Vogt.mw
View file details

I'm sure that I'm doing something that you did not intend.  Can you straighten me out?

Thank you, but Maple still doesn't seem to agree that the last of your equations is always true.

Here is my attempt to repeat what you posted:

View 4937_Page 99 integration Alex Vogt.mw on MapleNet or Download 4937_Page 99 integration Alex Vogt.mw
View file details

I'm sure that I'm doing something that you did not intend.  Can you straighten me out?

Thank you. That is a lot simpler. Here is a file that contains an improved version of my original solution, a small improvement, plus other machinations and your solution which is the best. View 4937_IntroductionToMaplePage63Exercise4b.mw on MapleNet or Download 4937_IntroductionToMaplePage63Exercise4b.mw
View file details I was aware of nextprime() but I was unaware of prevprime(). Looking at ?nextprime this is hard even for me to understand at this point. Still, there seems to be no need for numtheory. I wonder what Andre Heck was looking for. Perhaps numtheory was needed to solve this problem is the context of Maple 8 but is not needed in the context of Maple 11. That if statement is cool too. I am very entertained whenever I read what you type. Thanks again.

Thank you. That is a lot simpler. Here is a file that contains an improved version of my original solution, a small improvement, plus other machinations and your solution which is the best. View 4937_IntroductionToMaplePage63Exercise4b.mw on MapleNet or Download 4937_IntroductionToMaplePage63Exercise4b.mw
View file details I was aware of nextprime() but I was unaware of prevprime(). Looking at ?nextprime this is hard even for me to understand at this point. Still, there seems to be no need for numtheory. I wonder what Andre Heck was looking for. Perhaps numtheory was needed to solve this problem is the context of Maple 8 but is not needed in the context of Maple 11. That if statement is cool too. I am very entertained whenever I read what you type. Thanks again.