I think all these inter-relationships would be much clearer if you
talked about polar notation, orthogonal vectors, complex numbers and
what trigonometric functions actually are.
Donna
[EMAIL PROTECTED]
On 30-Jun-06, at 5:43 PM, John Randall wrote:
Dan Bron wrote:
Can you now explain why I was taught, in elementary math, to
express %:4
as (+,-)2 instead of just 2 ? If I have to write (%:4) = (+,-)2
why do I not have to write (^._1) = 0 j. 1p1 * 1 + 2 * i: _ ?
Is it
because the former has only 2 elements, but the latter infinitely
many?
By the way, I'm pretty sure 0 j. 1p1 * 1 + 2 * i: N (scalar
positive
integer N ) is the right expression for generating for the logs
of _1
but J doesn't agree:
Henry's non-mathematician's response is right on the money.
Mathematicians used to have a more inclusive idea of function,
including
"multivalued functions", which are now called "relations" (the
terminology
being most obvious in relational databases). Square root is the first
obvious example of a multivalued function, and it may be treated
this way
in elementary curricula. Multivalued functions are not functions in
modern terminology, and they are systematically eliminated from
calculus
curricula, except in implicit differentiation.
You are absolutely correct that the complex numbers z such that ^z
is _1
are given by odd multiples of j.1p1 . This can be seen easily from
Euler's
formula. J is using a principal domain that is something like the
complex
numbers r*^j.x where r>0 and x is a real number in the interval
(_1p1,1p1)
(with possibly one end closed). The complex log is then (^.r)
+j.x . As
Henry points out, if x is very close to 1p1 (where the cut in the
complex
plane is made), it may go either way.
Best wishes,
John
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