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Its late at night, and its 5 years since I was doing complex calc. and 7 since
I was doing real calc., but at a quick look and a hunch I would say it is
because 1/(a+bx^2) has a discontinuity at x^2 = -a/b and that the
two fomulae are the results for the restrictions of the function to one side
or the other of this point.
/bevin
PS: Yes, another mathematician making a living playing with computers. My
favorite areas were group theory and algebraic topology - the results in the
later are some of the most obscure and beautiful around...
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| A little deeper analysis, and an explanation...
1
--------- has a discontinuity at x^2 = -a/b, x = +/-(a^0.5*b^-0.5)
a + b*x^2
^
call this side K |
Sure enough, the two formulae given have the following behaviour, ignoring
irrelevant constants...
-----
a+x\/ -ab -ve +ve +ve +ve
---------- --------(-K)----------0----------(+K)---------
-----
a-x\/ -ab +ve +ve +ve -ve
division result: -ve +ve +ve -ve
section where LOG defined: ************************
-----
bx-\/ -ab -ve -ve -ve +ve
---------- --------(-K)----------0----------(+K)---------
-----
bx+\/ -ab -ve +ve +ve +ve
division result: +ve -ve -ve +ve
section where LOG defined: ******** **********
Okay, so that explains why the two formulae are 'different', now why does
this happen since there is only one discontinuity???? Well it turns out that
there is a natural 'center' on the complex plane where the integration 'begins'
and a circle around this center through the discontinuity. Different formula
are required inside and outside this circle. For example if we viewed this
problem from above the complex plane, what would be seen is the following, with
the working inside the circle, and working outside!
a+... bx-...
log ------ log ------
a-... bx+...
iY circle around center through K
| /
| /
-------------- discontinuity at K
/ | \ /
/ | \ /
/ | \ /
| | |/
----------------------------+-----------*---------------------X
| | |
\ | /
\ | /
\ | /
---------------
|
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So, while my first reply that it was caused by actually crossing the
discontinuity was 'wrong', it hinted at the truth, it is caused by crossing the
discontinuities 'interference'. This stuff often explains the breakdown of
Taylor and MacLaurin series, since even a discontinuity that is not on the real
number line causes this effect!!! If memory serves me correctly, the series
for LOG itself shows this.
/Bevin
PS: In case you hadn't even got the following...
-------
\/ -ab -a
_________________ = ----------------
b --------
\/ -ab
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| Bevin probably remembers a lot more complex analysis than I do,
but I will give my analysis anyhow and let readers comment some more.
It is my feeling that both answers are correct and that they both
apply to the same region (ab<0).
The reason that they can both be correct even though they are not
the same is because THEY DIFFER BY A CONSTANT. (Remember the +C
that everyone omits when evaluating indefinite integrals...)
To verify that they differ by a constant, note that log P - log Q
is the same as log (P/Q), so it suffices to show that P/Q is a constant.
In our case, P=(bx-sqrt(-ab))/(bx+sqrt(-ab)) and
Q=(a+x sqrt(-ab))/(a-x sqrt(-ab)). Dividing P by Q, clearing
fractions and simplifying results in P/Q=-1, a constant.
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Stan, I think you've missed a critical issue. The DOMAINS of the two solutions
are different. Consider the case where a=1, b=-1, x=0. The two formulae are
then
1 + 0*...
log ---------
1 + 0*... ie. log(1) which is defined,
0 - k
log ---------
0 + k ie. log(-1) which is NOT defined
/Bevin
PS: This is midnight math again, so...
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