mm-629
===
Subject: Re: numeric series question
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9EMtTJ20288;
Start with a geometric sequence with the 1st term 1 and
quotient -x^k:
1/(1 + x^k) = 1 - x^k + x^2k - x^3k +- ...
For k = 3:
1/(1 + x^3) = 1 - x^3 + x^6 - x^9 +- ...
The radius of convergence is 1. Inside of the radius, we can
integrate
term by term. The radius of convergence does not change in the
process. At the radius of convergence, the series diverges of
converges relatively. For a series Sum (-1)^n*|a_n| with
alternating
signs, |a_n| -> 0 for n -> inf is sufficient for convergence.
S dx/(1 + x^3) = x - x^4/4 + x^7/7 - x^10/10 +- ...
S dx/(1 + x^3) = 1/3*S dx/(x + 1) - 1/3*S (x - 2)/(x^2 - x +
1)dx =
= 1/3*ln(x + 1) - 1/6*S (2x - 1)/(x^2 - x + 1) +
+ 1/2 S dx/(x^2 - x + 1) =
= 1/3*ln(x + 1) - 1/6*ln(x^2 - x + 1)
+ 1/2*S dx/[(x-1/2)^2 + 3/4] =
= 1/3*ln(x + 1) - 1/6*ln(x^2 - x + 1) +
+ 1/2*sqrt(3)/2 * S dt/(t^2 + 1) (where t = 2/sqrt(3)(x -
1/2)) =
= 1/3*ln(x + 1) - 1/6*ln(x^2 - x + 1) +
+ sqrt(3)/3 * atan(2/sqrt(3)*(x - 1/2) + C
F(x) = 1/3*ln(x + 1) - 1/6*ln(x^2 - x + 1) +
+ sqrt(3)/3 * atan[(2x - 1)*sqrt(3)/3] + C
Since F(0) = sqrt(3)/3*atan(-sqrt(3)/3) + C = 0,
C = sqrt(3)/3*atan(sqrt(3)/3) = sqrt(3)/3 * pi/6
F(1) = 1/3*ln(2) - 1/6*ln(1) + 2*sqrt(3)/3 * atan(sqrt(3)/3) =
= 1/3*ln(2) - 0 + 2*sqrt(3)/3 * pi/6 = (approximately)
= 0.69314718/3 + 3.14159265 * 1.73205081/9 =
= 0.23104906 + 0.60459979 = 0.83564885
Generally, x^k + 1 has one real root -1 for odd k. The
remaining
quadratic factors are:
[x - e^(i*2pi/k)]*[x - e^(-i*2pi/k)] = x^2 - 2*cos(2pi/k)*x +
1
and this allows to expand x^k + 1 into fractions, the
integrals of
which are easy to calculate.
===
Subject: Re: numeric series question
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9F28pi03332;
>...
>Generally, x^k + 1 has one real root -1 for odd k. The
remaining
>quadratic factors are:
>[x - e^(i*2pi/k)]*[x - e^(-i*2pi/k)] = x^2 - 2*cos(2pi/k)*x
+ 1
>and this allows to expand x^k + 1 into fractions, the
integrals of
>which are easy to calculate.
Sorry for the error.
x^k + 1 has one real root -1 for k odd (leading to the ln(2)
term in
the series sum). The remaining quadratic factors are:
[x + e^(i*2pi*m/k)]*[x + e^(-i*2pi*m/k)] = x^2 +
2*cos(2pi*m/k)*x + 1
m = 1, 2, 3, ..., (k - 1)/2
x^k does not have a real root for k even. All the quadratic
factors
are:
[x + e^(i*2pi*(m - 1/2)/k]*[x + e^(-i*2pi*(m - 1/2)/k] =
= x^2 + 2*cos(2pi*(m - 1/2)/k)*x + 1
m = 1, 2, 3, ..., k/2
and this allows to expand 1/(x^k + 1) into fractions, the
integrals of
which are easy to calculate.
===
Subject: Re: Descrete Math help with question
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9EMtTB20297;
>I'v been trying to figure out this question and 
i don't even
know
>where to begin.
>Define two Sets A and B as follows: A = {(2n+1)^3|n E Z} and
>B = {(2n+1)|n E Z}
>a) prove that A is a proper subset of B.
>b) suppose we redifine A and B, replacing Z by R . What is 
the
>relation between these two sets? State and prove your answer.
You should probably talk with your instructor or your TA if
you
have no idea where to begin. But:
(a) To say A is a subset of B means, by definition, that
any element of A is an element of B. Now look at B: it's 
just
the set of odd integers. A is the set of cubes of odd
integers.
So the statement is that the cube of any odd integer is an odd
integer, which of course makes sense.
To *prove* it, let (2n + 1)^3 be any element of A, and write
it
out so that it has the form 2m + 1 of an element of B (please
note that the m need not be the same as n):
(2n + 1)^3 = 8n^3 + 12n^2 + 6n + 1 = 2m + 1
where m = ? (you do this part).
Now: to say A is a *proper* subset means that also not every
element of B belongs to A. Here it means that not every odd
integer is the cube of an odd integer. To prove this, all you
have to do is give an example of an odd integer which isn't 
a
cube or third power of an integer. I'll let you do this.
(b) OK, now replace Z by R (the set of real numbers) and try
to understand what B is. Using different letters may help:
it's the set of all real numbers which are of the form 2x + 
1,
where x is real. But any real number is of that form: if y
is any real, there's an x such that y = 2x + 1 -- namely,
x = (1/2)(y - 1). So B is just the set of all real numbers.
By the same token, A is the set of all numbers of the form
x^3 where x is real. But any real number is of that form:
if y is any real, there's an x such that y = x^3 -- namely,
what?
So A is the set of all reals and B is the set of all reals.
Now you should be able to answer questions like: is A a subset
of B? Is A a proper subset of B?
I don't know if you are required to give proofs in your
course,
but if you are, you should get in the habit of working
straight
from definitions. Good luck with your course.
Todd Trimble
===
Subject: Re: Truth Tables Help
> I have a four part question in a tutorial, parts 1 & 2 I
don't have any
> problem with. Parts 3 & 4 I have no idea what they mean.
Can someone
explain
> to me what parts 3 & 4 mean and how can I answer the
questions.
> Alternatively if you can point me to a website that might
be of some
help.
> The question:
> Draw the truth tables for each of the following expressions:
> 1. AB'CD' + 
ABC'D' + A'BC'D + 
A'B'CD
> 2. (A + B' + C' + D)(A' + B + 
C + D')
> 3. Sm(2.5.6.9)
> 4. ?M(0,7,10,11)
> Appreciate any help.
> TIA
Whenever I note that the use of language skill in engineering
is a valuable
asset, I get put down. Here is an example of a problem that
is poorly posed
and communicated. I have no idea what anything after the
colon signifies.
I can only guess that earlier in the tutorial there was a
discussion of
symbology to be used in the remainder of the tutorial.
Without knowing what
that is, I have no idea on how to help.
Bill
===
Subject: Re: Truth Tables Help
Problem with is parts 3 & 3.
Parts 1 & 2 read
1. (A and not B and C and not D) or (A and B and not C and
not D) or ( not
A
and B and not C and D) or (not A and not B and C and D)
2. (A or not B or not C or D)(not A or B or C or not D)
> I have a four part question in a tutorial, parts 1 & 2 I
don't have any
> problem with. Parts 3 & 4 I have no idea what they mean.
Can someone
explain
> to me what parts 3 & 4 mean and how can I answer the
questions.
> Alternatively if you can point me to a website that might
be of some
help.
> The question:
> Draw the truth tables for each of the following
expressions:
> 1. AB'CD' + 
ABC'D' + A'BC'D + 
A'B'CD
> 2. (A + B' + C' + D)(A' + B 
+ C + D')
> 3. Sm(2.5.6.9)
> 4. ?M(0,7,10,11)
> Appreciate any help.
> TIA
> Whenever I note that the use of language skill in
engineering is a
valuable
> asset, I get put down. Here is an example of a problem that
is poorly
posed
> and communicated. I have no idea what anything after the
colon signifies.
> I can only guess that earlier in the tutorial there was a
discussion of
> symbology to be used in the remainder of the tutorial.
Without knowing
what
> that is, I have no idea on how to help.
> Bill
===
Subject: Re: Truth Tables Help
> Problem with is parts 3 & 4.
Exactly, so what, according to your text or notes, do
Sigma(2.5.6.9) and
Pi(0,7,10,11) mean?
===
Subject: Re: Truth Tables Help
First line should have read 3& 4
> Problem with is parts 3 & 3.
> Parts 1 & 2 read
> 1. (A and not B and C and not D) or (A and B and not C and
not D) or (
not
A
> and B and not C and D) or (not A and not B and C and D)
> 2. (A or not B or not C or D)(not A or B or C or not D)
at
> I have a four part question in a tutorial, parts 1 & 2
I don't have
any
> problem with. Parts 3 & 4 I have no idea what they
mean. Can someone
> explain
> to me what parts 3 & 4 mean and how can I answer the
questions.
> Alternatively if you can point me to a website that
might be of some
> help.
> The question:
> Draw the truth tables for each of the following
expressions:
> 1. AB'CD' + 
ABC'D' + A'BC'D + 
A'B'CD
> 2. (A + B' + C' + D)(A' + 
B + C + D')
> 3. Sm(2.5.6.9)
> 4. ?M(0,7,10,11)
> Appreciate any help.
> TIA
>
>
> Whenever I note that the use of language skill in
engineering is a
> valuable
> asset, I get put down. Here is an example of a problem
that is poorly
> posed
> and communicated. I have no idea what anything after the
colon
signifies.
> I can only guess that earlier in the tutorial there was a
discussion of
> symbology to be used in the remainder of the tutorial.
Without knowing
> what
> that is, I have no idea on how to help.
> Bill
===
Subject: Re: Truth Tables Help
X-RFC2646:  Original
If I recall correctly Sigma is usually successive addition
and the big PI is
usually a successive multiplication.
> My earlier post should read:
> I have a four part question in a tutorial, parts 1 & 2 I
don't have any
> problem with. Parts 3 & 4 I have no idea what they mean.
Can someone
> explain
> to me what parts 3 & 4 mean and how can I answer the
questions.
> Alternatively if you can point me to a website that might
be of some
> help.
> The question:
> Draw the truth tables for each of the following
expressions:
> 1. AB'CD' + 
ABC'D' + A'BC'D + 
A'B'CD
> 2. (A + B' + C' + D)(A' + B 
+ C + D')
> 3. Sigma m(2.5.6.9)
> 4. Pi M(0,7,10,11)
> 3 & 4 should read:
> _
>
> /_ m (2.5.6.9)
> __
> I I M (0,7,10,11)
> What do your texts or lecture notes say these notations
mean? Guess:
> Sigma indicates + over some set, and Pi indicates . over
some set--but
> what sets?
===
Subject: Re: Truth Tables Help
> If I recall correctly Sigma is usually successive addition
and the big PI
is
> usually a successive multiplication.
Yes, but addition of what in this case, and multiplication of
what in
this case?
>
> My earlier post should read:
> I have a four part question in a tutorial, parts 1 & 2 I
don't have
any
> problem with. Parts 3 & 4 I have no idea what they mean.
Can someone
> explain
> to me what parts 3 & 4 mean and how can I answer the
questions.
> Alternatively if you can point me to a website that
might be of some
> help.
> The question:
> Draw the truth tables for each of the following
expressions:
> 1. AB'CD' + 
ABC'D' + A'BC'D + 
A'B'CD
> 2. (A + B' + C' + D)(A' + 
B + C + D')
> 3. Sigma m(2.5.6.9)
> 4. Pi M(0,7,10,11)
>
> 3 & 4 should read:
> _
>
> /_ m (2.5.6.9)
>
> __
> I I M (0,7,10,11)
> What do your texts or lecture notes say these notations
mean? Guess:
> Sigma indicates + over some set, and Pi indicates . over
some set--but
> what sets?
===
Subject: Probability Theory -- Serious help needed!!
Can someone please help me with this problem:
The average density of a forest is 16 trees on every 100
square yards.
The tree trunks can be considered as cylinders of a diameter
of 0.2
yards. We are standing inside the forest, 120 yards from its
edge.
If we shoot a gun bullet out of the forest without aiming,
what is the
probability that it will hit a tree trunk? (Ignore the
marginal fact
that centers of the tree trunks can not be closer than 0.2
yards to
each other.)
===
Subject: Re: Probability Theory -- Serious help needed!!
X-RFC2646:  Original
First , even not aiming, assume that ur shooting in a
horizontal direction
and at a height less than the height of all cylinders ( trees
), and fix
that height.
Second, assume that the bullet is a circle of diameter c.
Third, assume that the forest is a cylinder of radius 120
yards and ur
shooting from the middle of that circle.
Now you can imagine that there is a horizontal circle and the
bullet will
pass from the center of that circle to the edge of it, so
whats the
probability that the bullet does not hit the edge (hit a tree
trunk). it is
easier for you to find the probability that the bullet does
hit the edge,
say it is p, then (1 - p) is the probability ur looking for
(hitting a tree
trunk).
now lets solve for the probability that the bullet does hit
the edge. ?
(a) the bullet does hit the edge, that means that there is a
line of width
c which has nothing in it, ( no trees intersect that line ).
(b) Ignoring the marginal fact that centers of the tree
trunks can not be
closer than 0.2 yards to each other,
***Consider the following examples:
1) Suppose there are two slots, say A and B, and there are
three different
balls, say a , b, and c; and we want to drop the balls in
those two slots
with no aiming, Then one possible outcome is that we have the
three balls in
slot A, and nothing in slot B, or another outcome is that we
have ball a in
A, and balls b and c are in B. There will be 8 different
outcomes, this is
done by taking (2^3) where 2 is the number of slots and 3 is
the number of
different balls. All these 8 outcomes are equally probable
and thats a major
idea you have to understand, so probabilty of having a in
slot A is 1/8, and
probabilty of having a and b and c in A is also 1/8;
Then the number of different outcomes is (V^N), u can think
of this as V is
the number of slots and N is the number of balls, here we are
assuming that
Remember that number of outcomes does not depend on the shape
of V.
a volume V, and we want to compute the probability that there
are no
different number of outcomes in ((V-v)^N),
v in V is ((V-v)^N)/(V^N), thats a major idea you have to
understand
too.***
average 16 trees in 100 square yards, then in the forest u
have an average
of Pi * (120 ^ 2) * 0.16, set it equal to N ( number of trees
in the
forest).
Total accessible Area for the trees is : Pi (120 - 0.1)^2,
set it equal to
S.
Surface of a line of width c and length 120 is 120c.
ok, now you all information to solve the problem, the
probability that we
have N trees in (S - 120c) is ((S - 120c)^N)/(S^N),
> Can someone please help me with this problem:
> The average density of a forest is 16 trees on every 100
square yards.
> The tree trunks can be considered as cylinders of a
diameter of 0.2
> yards. We are standing inside the forest, 120 yards from
its edge.
> If we shoot a gun bullet out of the forest without aiming,
what is the
> probability that it will hit a tree trunk? (Ignore the
marginal fact
> that centers of the tree trunks can not be closer than 0.2
yards to
> each other.)
===
Subject: Re: Probability Theory -- Serious help needed!!
> Can someone please help me with this problem:
> The average density of a forest is 16 trees on every 100
square yards.
> The tree trunks can be considered as cylinders of a
diameter of 0.2
> yards. We are standing inside the forest, 120 yards from
its edge.
> If we shoot a gun bullet out of the forest without aiming,
what is the
> probability that it will hit a tree trunk? (Ignore the
marginal fact
> that centers of the tree trunks can not be closer than 0.2
yards to
> each other.)
Probability and statistics aren't my forte, but since no 
one's
said anything, I'll take a stab at getting you started. If 
I'm
not mistaken, this is a problem in using the Poisson
distribution.
The bullet will hit a tree if the centre of the tree is within
0.2 yards of the bullet's line of ßight. Thus, the bullet 
will
make it to the edge of the forest if no tree centre is within
0.2
yards of that line, i.e., if a strip 120 yards long and 0.4
yards
wide contains no tree centre. That strip has an area of 48 sq.
yds. The mean density of tree centres is 0.16 per sq. yd., or
7.68 centres per 48 sq. yds., and from here it should be
pretty
straightforward.
Brian
===
Subject: Re: Uniform Convergence
> Is there any easy way to show that the function:
> f(c,h) = (c^h - 1)/h
> is uniformly convergent as h tends to 0?
> I am trying to develop logarithms in a different way from
the
> classical way of considering the integral of 1/x. So I
differentiate
> c^x with respect to x, and get the limit as h tends to 0 of
f(c,h)
> multiplied by c^x. Then I just need to define ln(c) as that
limit, and
> then I need to show it is continuous. The reason why I need
to do
> that, is that I can then explicitly calculate ln(c)-1 at 2
and 3, show
> that one of them is negative and one positive, so there
must be a
> number e between 2 and 3 for which ln(e)-1=0 (since ln is
continuous),
> and then go from there.
> However to show it is continuous I need to show two things:
a) that
> the individual functions are cts (this is easy because they
are
> clearly differentiable), and b) that the sequence of
functions
> converges uniformly, and this is what I am struggling with.
> Any help would be appreciated.
> Adam.
OK, let's assume that for c > 0, c^x has been 
defined for all
real x, that
the usual power properties hold, and that for fixed p in R the
map x -> x^p
is differentiable on (0,oo), with derivative = p*x^(p-1). Now
you want to
consider f(c,h) = (c^h - 1)/h to get at the derivative of x
-> c^x without
use of ln(x). Unfortunately the simplest and most natural
route to this
involves the integral of 1/x. However, you don't need to
define ln(x)!
Here's what I mean: We simply note that
(*) (c^h - 1)/h = int[1,c] x^(h-1) dx.
And as h -> 0, x^(h-1) -> 1/x uniformly on [1,c] (here taking
c > 1).
Therefore (*) -> int[1,c] 1/x dx as h -> 0. And we haven't
mentioned a log
yet. As the last integral is a continuous function of c,
we're done.
Does this do you any good?
===
Subject: Re: Uniform Convergence
> Define c^x in the standard way as follows:
> Suppose x is a natural number, then c^x is defined
inductively as c^1
> = c and c^(x+1)=c*c^x (the proof that this is well defined
follows
> from the principle of induction trivially).
> Suppose x is an integer. If x is 0, c^x is 1. If x is
negative, c^x is
> 1/c^(-x).
> Suppose x is rational. Then x = p/q, p and q integers and q
non-zero.
> Then c^x is the solution y of the equation y^q = c^p.
> Suppose x is real. Then there is a sequence of rationals
x_n tending
> to x. So c^x is the limit as n tends to infinite of c^(x_n).
> Suppose x is complex. Then c^x is exp(x*ln(c)).
> The function c^x is clearly differentiable WITH RESPECT TO
c, which is
> what we want in the continuity proof, because we simply
form the
> newton quotient and use the binomial theorem.
I'm not sure how you're getting the 
differentiablity. You'll
have things
like
[(Pi+h)^sqrt(2) - Pi^sqrt(2)]/h.
So which binomial theorem did you have in mind? (I know how
to show, pretty
simply, that if r is rational and x -> x^r is differentiable
on (0,oo) with
the usual formula holding, then so is x^r for r irrational,
but it's a
different idea.)
===
Subject: Re: Uniform Convergence
> Suppose x is real. Then there is a sequence of rationals
> x_n tending to x. So c^x is the limit as n tends to
> infinite of c^(x_n).
>To make this rigorous you have to prove that the limit
>exists and is independent of the particular sequence <x_n>
>chosen. You also have to show that this is consistent with
>the earlier parts of the definition.
> [...]
> In terms of your earlier points - I had thought of most
of these, but
> what do you mean for real numbers where you say you need
to prove that
> the limit exists?
> You have to prove that if <x_n : n in N> converges to x,
> then <c^{x_n} : n in N> actually converges. This is not
> trivial.
> [...]
It's pretty elementary isn't it? Assuming c^r 
has been defined
for rational
r, and that the usual power properties have been proven
(c^(r+s) = c^r*c^s,
...), I don't see any problem. One thing you need is that
c^(1/n) -> 1 as n
-> oo, but this is easy.
===
Subject: Re: Uniform Convergence
in alt.math.undergrad:
> Suppose x is real. Then there is a sequence of rationals
> x_n tending to x. So c^x is the limit as n tends to
> infinite of c^(x_n).
>To make this rigorous you have to prove that the limit
>exists and is independent of the particular sequence <x_n>
>chosen. You also have to show that this is consistent with
>the earlier parts of the definition.
> [...]
> In terms of your earlier points - I had thought of most
of these, but
> what do you mean for real numbers where you say you need
to prove that
> the limit exists?
> You have to prove that if <x_n : n in N> converges to x,
> then <c^{x_n} : n in N> actually converges. This is not
> trivial.
> [...]
> It's pretty elementary isn't it? Assuming 
c^r has been
defined for
rational
> r, and that the usual power properties have been proven
(c^(r+s) =
c^r*c^s,
> ...), I don't see any problem. One thing you need is that
c^(1/n) -> 1 as
n
> -> oo, but this is easy.
Depends on how much you're assuming about R. If you get to
assume completeness (in the form of convergence of Cauchy
sequences), I suppose that it's not bad, but I 
don't think
that it qualifies as trivial: there's a fair bit 
of grunt
work.
Brian
===
Subject: Re: Shawn's Limit
>Really? What if f(x) = e^x + 1?
> If O(x) is defined as my mystery function, then
> It evaluates to -1/(e^x+1) which by the way ramps straight
down to 0.
> O(9)=.0001
> O(20)= -.00000000206
> so probably a better way to say it is lim(x-->infinity)O(x)
= 0
> good for just about everything other than tan(x) or
derivations
> thereof.
> A bunch of things are identically, and sometimes trivially
= 0, others
> get there really fast.
> so..O(x)=0+
What if f(x) = e^(-x) + 1. Then O(x) -> 1 as x -> oo.
===
Subject: Re: Phrase Structure
<b20iwgkh2d55.1afghvdfua0nx$.dlg@40tude.net>
> Construct phrase-structure grammars to generate the
following set
> {0^n1^2n | n >= 0}
> (I assume that this should be {0^n 1^(2n) : n >= 0}.) This
> is really extremely straightforward; you should be able to
> do it yourself. Here are a few hints, though. You can do
> it with a single non-terminal symbol; the main production
> should generate one new 0 and two new 1's. Have you seen a
> grammar that generates {0^n 1^n : n >= 0}? If so, just
> modify it slightly.
Oh, a phrase-structure grammar is a regular grammar?
===
Subject: Re: Phrase Structure
[...]
> Oh, a phrase-structure grammar is a regular grammar?
Every regular grammar is a phrase-structure grammar, but not
conversely. The language in question isn't regular, so a
regular
grammar won't be possible; it is context-free, though.
Brian
===
Subject: Re: factoring without a calculator
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9FFbwL05552;
>I forget how to facor w/o a calculator, and my math 113 book
does not
>explain how to factor. If somebody could show me how to
factor
>x^2 + x + 1 > 0 without using the quadratic formula i would
be very
>appreciative.
You can say:
x^2 + x + 1 = 0
(x + 1/2)^2 - 1/4 + 1 = 0
(x + 1/2)^2 = -3/4
x + 1/2 = +/- sqrt(-3/4)
x = [sqrt(-3/4) - 1/2] or [-sqrt(-3/4) - 1/2]
and so these two values, when put into the equation, give x^2
+ x + 1
= 0
which means when these values are taken from x, the result is
zero:
so the factorisation is:
(x + sqrt(-3/4)+1/2)(x - sqrt(-3/4)+1/2)
I think.
===
Subject: Better late than never
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9FKGNs00622;
>I forget how to facor w/o a calculator, and my math 113
book does
not
>explain how to factor. If somebody could show me how to
factor
>x^2 + x + 1 > 0 without using the quadratic formula i would
be very
>appreciative.
>You can say:
>x^2 + x + 1 = 0
>(x + 1/2)^2 - 1/4 + 1 = 0
>(x + 1/2)^2 = -3/4
>x + 1/2 = +/- sqrt(-3/4)
>x = [sqrt(-3/4) - 1/2] or [-sqrt(-3/4) - 1/2]
>and so these two values, when put into the equation, give
x^2 + x + 1
>= 0
>which means when these values are taken from x, the result
is zero:
>so the factorisation is:
>(x + sqrt(-3/4)+1/2)(x - sqrt(-3/4)+1/2)
>I think.
michael,
Although your post came almost two years after the OP, I was
heartened
by it. Hopefully dangerman and G.E. Ivey are long gone from
this
forum. Unfortunately, the likes of them are far too common in
this
forum. I am glad to see someone who is willing to offer
constructive
help rather than denigrating comments.
Keep up the good works.
- MO