agd@houem.UUCP (A.DEACON) (02/13/84)
In response to Seaman and Rentsch:
The value of i^i is very well defined. I gave
the values in a previous article and they are:
i -(4n+1)*pi/2
i = e for all integer n.
There is no problem in defining log(i) either:
for any complex number z
ln(z) = ln|z| + i(theta + 2n*pi) for all integer n
where -pi < theta <= pi and |z| is the modulus of z.
Theta is called the principle
argument of the ln. As you can see there are an infinite
number of values for ln(z). Of course the ln function
cannot be extended continuously to the entire complex
plane because of the pole. However, if you consider
the Riemann surface, it can be.
In the exp(x) sum, we need x=i*ln(i) to compute i^i,
so it does require something "funny". Being clever
has nothing to do with it. That's the way it is.
For Seaman:
i -(2n+1)*pi
(-1) = e for all integer n
and
i -(4n-1)*pi/2
(-i) = e for all integer n.
Art Deacon
AT&T Bell Labsags@pucc-i (Seaman) (02/14/84)
The problem with this discussion is that people don't know the difference
between singulars and plurals.
It makes no sense to say:
"The value of i**i is well defined. The values are..."
^^^^^ ^^^^^^
I never said that the VALUES of i**i are not well-defined.
I said the VALUE of i**i is not well-defined. That statement is correct.
Note that this differs from the situation with other functions that have
universally recognized inverses. The real square root function is defined
to take the positive branch -- hence sqrt(4) is +2 and not -2.
There is no universally recognized inverse to the complex exponential
function which is single-valued. No such inverse can be defined on
the complex plane (excluding the origin) unless you are willing to
put up with a jump discontinuity.
--
Dave Seaman
..!pur-ee!pucc-i:ags
"Against people who give vent to their loquacity
by extraneous bombastic circumlocution."andrew@inmet.UUCP (02/17/84)
#R:houem:-22900:inmet:5700001:000:601
inmet!andrew Feb 15 10:55:00 1984
As I mentioned last week (in net.kids), I was once kicked out of high school
for asking my math teacher if there could be such a thing as sqrt(i). Rather
than admit her ignorance, she accused me of "disrupting the class with smart-
aleck remarks" and attempted to have me suspended for a week. The chairman
of the Math. Dept. explained Euler's Theorem
exp(x*i) = cos(x) + i*sin(x)
to her (and me), using it to calculate the correct value
+/- (1+i)/sqrt(2)
and I was reinstated (much to her embarassment). This says a lot about the
teaching standards in American public high schools.holmes@dalcs.UUCP (Ray Holmes) (02/20/84)
[] >As I mentioned last week (in net.kids), I was once kicked out of high school >for asking my math teacher if there could be such a thing as sqrt(i). Rather >than admit her ignorance, she accused me of "disrupting the class with smart- >aleck remarks" and attempted to have me suspended for a week. Funny thing about that. I can remember being kicked out of a grade 6 (I think) class for telling another student that there was a "funny" number (called pi) that was realy the ratio of the circumference to the diameter and WAS NOT 22/7! Then again, that was the same teacher who kicked me out for objecting to her theory that an eclipse of the sun was when the sun was between the earth and the moon! Oh well.....
amigo2@ihuxq.UUCP (John Hobson) (02/21/84)
My father, an ex-(British) Royal Navy engineering officer, during his early days at the Royal Naval Engineering College, was told about pi by an elderly Chief Petty Officer. (Imagine strong east London accent) "Pi is a number promulgated by the Hadmirality to determine the di-a-meter of circles. Some uses 3.14159, but we uses 22 upon 7 as bein' quicker and more accurate." John Hobson AT&T Bell Labs Naperville, IL (312) 979-0193 ihnp4!ihuxq!amigo2