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From: OsherD on 25 Feb 2010 22:47
From Osher Doctorow

It sometimes appears to us that the complex variable z argument of
complex functions f(z), where z = y + ix (or z = x + iy), is the most
difficult part of looking for a real "analog" of f(z) at least under +
and -. However, with Probable Causation/Influence P(A-->B) = 1 +
P(AB) - P(A) = 1 + y - x, y of z = y + ix goes over to y and ix goes
over to -x for addition and subtraction, so z is not the
complication.

Now let us examine the first "inverse" term of a Laurent series,
a_(-1)/(z - a), where constant a_(-1) is the symbol for the first
constant coefficient of the inverse term. We use the usual "change":

1) y/x --> 1 + y - x (--> denotes "conversion to" or "change to"
here)

from Conditional Probability y/x with x not 0 to Probable Causation/
Influence (PI) 1 + y - x, with a_(-1)/(z - a) being y/x. This
yields:

2) a_(-1)/(z - a) --> 1 + a_(-1) - (z - a) = 1 + (a_-1) + a - z = 1 +
[a_(-1) + a] - z

Now convert z to a real variable, which will be denoted x (not the x
of (1)), and for simplicity convert a_(-1) + a to a nonnegative real
constant. We get:

3) a_(-1)/(z - a) --> 1 + y - x, where z = x, y = a_(-1) + a

It turns out to be not much more difficult to convert remaining terms
of the "inverse" part of Laurent Series to PI, using such things as:

4) 1/(x - a)^2 = 1/(x^2 + a^2 - 2ax)

and then noticing that:

5) 1/(x - a)^2 --> 1 + 1 + 2ax - x^2 - a^2, where x^2 + a^2 > 2ax
from (x - a)^2 > 0.

From the last few posts, this ties in with the Memory scenario.

Osher Doctorow
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