Re: What is the integral of x^^x?

On Aug 6, 11:35 pm, Ray Johnstone <r...@xxxxxxxxxxxx> wrote:
On Mon, 06 Aug 2007 17:23:20 -0700, mike3 <mike4...@xxxxxxxxx> wrote:
On Aug 4, 10:30 am, Ray Johnstone <r...@xxxxxxxxxxxx> wrote:
What is the integral of x^^x?
r...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx/~ray

If you mean the power function x raised to the xth power, then
there is no way to express it with any "familiar" operations,
even familiar to a good mathematician (!) You sure can't do it
with just +, -, *, /, e^x and ln(x). Even the Mathematica
computer algebra system chokes on the integral -- it's available
operations just can't cut it.

You could of course invent one that would do it but that seems
like a "cheat", now don't it?

Even writing it in an "open" form (ie. infinite summations,
products, or other infinite formulas) instead of "closed" form
requires the (possibly unfamiliar, depending on what you've
encountered) Gamma function and is a real eye-glazer to look
at:

int x^x dx = -Gamma(1, -ln(x))x + Gamma(2, -ln(x))x + 3/2 Gamma(3, -
ln(x))x + sum_{n=3...inf} (-1)^n a_(2,n) Gamma(n+1, -ln(x)).

and

a_(m,n) = 1 iff n = 0
a_(m,n) = 1/(n!) iff m = 1
a_(m,n) = 1/n sum_{j=1...n} j a_{m,n-j) a_(m-1,j-1) otherwise

(whew!)

(Memorize that)

(source:http://mathworld.wolfram.com/PowerTower.html)

(It's possible to decompose the upper incomplete Gamma functions
there into simple sums, but the result is awful
looking when you plug everything together. If you can do it I'll
suggest you next try to start memorizing it so well you can
write it down on everything. :) )

However, the notation x^^x might mean a tetration instead of
an exponentiation. But how would you define, say, 4 tetrated
to five sixteenths or nine twelfths, or even pi?! There isn't
yet a widely-accepted definition of y^^x for real hyper
exponents x. If you are thinking of some definition in
particular, you'll need to state it. Without that definition
there is nothing that can be said about the integral!

Thanks. I've liked the gamma function ever since I learned it was
related to factorials. That may be the reason I was able to scrape a
bare pass in maths. But I think it much more likely it was due to a
miracle.


Obviously you didn't have a good time with math, then.

r...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx/~ray

.

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