Online "book" about solution of y=x^x
L
I have some items on my web site that might be of interest to those
researching equations in the form of y=x^x and related functions. This is
a topic that appears from time to time in SCI.MATH. Myself and a friend
began researching the solution of y=x^x back in 1975 while we were still
in high school. The ASCII-based "book" on my web site is a compilation
of our results from 1975 to 1995. It is written in a very informal style
that we hope is informative to users of this news group. We have included
two FORTRAN-77 files, SKRFIT.F and WEXZAL.F which are included with the
book as they implement many of the topics in the book. The web site is:
http://www.networkone.net/~believe
Questions and comments can be sent to bel...@networkone.net and we will
try to answer questions as time permits.
The outline of the book is:
Chapter 01 - Basic discussion of y=x^x.
Chapter 02 - Inverse of x^x. This is called the Coupled Root [crt(x)].
Chapter 03 - Coupled Root of large numbers. The "Wexzal" = crt(10^x).
Chapter 04 - Closed form solution of equations via Wexzals.
Chapter 05 - Integrals involving Wexzals.
Chapter 06 - Asymptotics and limits with Wexzals.
Chapter 07 - Numerical computation of wzl(x). Calculator fans... take note!
Chapter 08 - Misc items involving Wexzals. Incomplete.
Chapter 09 - Curve-fitting with Wexzals. Incomplete.
Chapter 10 - New type of graphs involving 1/wzl(1/x).
Chapter 11 - Math model of projectile deacceleration via Wexzals.
Chapter 12 - Barrel length vs. velocity question answered with Wexzals.
Chapter 13 - Car acceleration model via Wexzals.
Chapter 14 - Equalities and inequalities of Coupled Roots & other functions.
Chapter 15 - Closed form solution of equations via Coupled Roots.
Chapter 16 - Table of integrals involving Wexzals & Coupled Roots.
Chapter 17 - Table of Asymptotic expansions and limits.
Chapter 18 - Special values and table of functions from Chapters 11-13.
The file WEXZAL.F is a collection of "home-brewed" functions and routines
that are discussed in the book. Included is the computation of the
solution of y=x^x and y=x^(x^x) along with the first 4 derivatives of
y=wzl(x).
The program SKRFIT is a non-linear 2 parameter curve fitting program that
was written to solve the program of barrel lengths vs. muzzle velocity of
firearms (See chapter 12). It uses routines from WEXZAL.F which contains
the computation of WZL(x) and other items as discussed in the book.
To use the program SKRFIT after it has been compiled, one creates a
datafile that contains free-formatted (x,y) data. This data is to be fitted
in the form of:
A
y = ------
B
WZL(-)
x
where A and B are to be determined. An initial value of B is entered. This
is called B0 by the program. The value of B0 should be higher than the
expected value of B otherwise the program might not converge. An example
data set and a run of the program follows.
[~/wotan]$ cat d881.dat
2.0 380.0
4.0 520.0
6.0 595.0
8.0 650.0
10.0 685.0
[~/wotan]$ ./skrfit
WHAT IS THE NAME OF THE INPUT FILE?
d881.dat
[OK]
[THERE WERE 5 ITEMS READ]
B0?
2.2
.501210771833945
.508314611965324
.508452940471791
.508452991457064
.508452991457076
.508452991457080
.508452991457066
.508452991457072
.508452991457077
.508452991457079
.508452991457077
.508452991457070
.508452991457068
.508452991457075
.508452991457074
.508452991457074
Y= 953.0774331 /WZL( 1.966750155 /X)
RMS= 2.616469773
[~/wotan]$ exit
Anyway, we hope you find the "book" informative and entertaining.
-Jay
Parker W.
I am hoping to write a paper about it soon, but i need a little more
material. A lot of my work could simplify yours and fit it all
together, moreover, expanding the idea a lot more than you have shown
in your book.
Parker W.
PS I hope i can write the paper within 6 months...