Difficulty with IMPOWER() Worksheet Function

[monirg]

Hello;

1) The cubic root of complex numbers "x+yi" are calculated using IMPOWER().
The w/s function appears to return incorrect values when the coefficients of the complex number are -ve.

2) For example:
A1:: -8+0i
B1::= IMPOWER(A1,1/3)
the function returns: 1. + 1.7320508i
instead of the correct value of -2.

3) The problem appears to be consistent with calculating theta [=atan(y/x)] as "pi" instead of "0" for the above example "-8+0i"

How to fix the problem ??

Thank you kindly.

 

[monirg]

Hi;

A more general example:
A1:: -3+4i
B1::= IMPOWER(A1,1/3)
the function returns...........:1.26495290635775+1.15061369838445i
instead of the correct value:1.62893714592218 -0.520174502304546i

Any suggestion ??

 

[c_m]

this is an awful solution but this is the fastest i came up with (cell A1 = "-8+0i")

=SQRT(SUMSQ(IMREAL(A1),IMAGINARY(A1)))^(1/3)*COS(1/3*ATAN(IMAGINARY(A1)/IMREAL(A1)))&"+i"&SQRT(SUMSQ(IMREAL(A1),IMAGINARY(A1)))^(1/3)*SIN(1/3*ATAN(IMAGINARY(A1)/IMREAL(A1)))

note that the above formula will produce the text in the cell instead of a number

there are other excel function which returns wrong value like binomdist...not sure why...so i usually prefer to write a UDF....you can do the same...reply to this if you need further help

 

[c_m]

Hi;

A more general example:
A1:: -3+4i
B1::= IMPOWER(A1,1/3)
the function returns...........:1.26495290635775+1.15061369838445i
instead of the correct value:1.62893714592218 -0.520174502304546i

Any suggestion ??

i think impower() works for -3+4i....you can kind of verify by taking the cube of impower("-3+4i",1/3)
so if A1 = "-3+4i"
B1 = impower(A1,1/3)
C1 = impower(B1,3)
you will see that C1 is same as A1

 

[c_m]

i think impower() works for -3+4i....you can kind of verify by taking the cube of impower("-3+4i",1/3)
so if A1 = "-3+4i"
B1 = impower(A1,1/3)
C1 = impower(B1,3)
you will see that C1 is same as A1

no you are right....the reverse is check for -8+0i but the answer is not correct

 

[tusharm]

One might prefer -2 as the answer but 1+Sqrt(3)I is a valid solution. Just test it. (1+sqrt(3)I)^3 = -8

Hello;

1) The cubic root of complex numbers "x+yi" are calculated using IMPOWER().
The w/s function appears to return incorrect values when the coefficients of the complex number are -ve.

2) For example:
A1:: -8+0i
B1::= IMPOWER(A1,1/3)
the function returns: 1. + 1.7320508i
instead of the correct value of -2.

3) The problem appears to be consistent with calculating theta [=atan(y/x)] as "pi" instead of "0" for the above example "-8+0i"

How to fix the problem ??

Thank you kindly.

 

[pgc01]

Hi monir

A more general example:
A1:: -3+4i
B1::= IMPOWER(A1,1/3)
the function returns...........:1.26495290635775+1.15061369838445i
instead of the correct value:1.62893714592218 -0.520174502304546i

I'm sure you know and it was just a lapse but you must remember that if you allow complex numbers, as is the case, a number has 2 square roots, 3 cubic roots, 4 fourth roots, etc.

It makes no sense to say THE cubic root of a number, you have to say ONE cubic root of a number, there are 2 more

In your example, the cubic roots of -3+4i are

1.26495290635775+1.15061369838445i
-1.62893714592217+0.520174502304545i
0.363984239564422-1.67078820068899i

and so ImPower() is not wrong, it gives you one of the possible answers.
Try taking the cube of these 3 numbers.

 

[tusharm]

Followup with "how to" information:

I don't know how much you know about roots and complex numbers -- my own knowledge is rather dated {grin} -- but the documentation that you quoted is correct only in an implicit manner. It states that theta is tan-inverse (y/x) where -pi < theta <= pi. The two together can be true only if the tan-inverse calculation takes into account the signs of the two arguments. The appropriate Excel function would be ATAN2, which would yield theta=pi!

For more on this (or most math/stats/engineering subjects) search google. In most cases you'll find your way to wikipedia. The general algorithm for finding all the roots for the N-th power of a complex number is documented at http://en.wikipedia.org/wiki/Nth_root

Follow that argument and you will find that the roots of -8 are -2 (corresponding to theta = pi), 1-SQRT(3)I -- corresponding to theta = 5Pi/3 -- and 1 + SQRT(3)I -- corresponding to theta = 8Pi/3.

Hello;

1) The cubic root of complex numbers "x+yi" are calculated using IMPOWER().
The w/s function appears to return incorrect values when the coefficients of the complex number are -ve.

2) For example:
A1:: -8+0i
B1::= IMPOWER(A1,1/3)
the function returns: 1. + 1.7320508i
instead of the correct value of -2.

3) The problem appears to be consistent with calculating theta [=atan(y/x)] as "pi" instead of "0" for the above example "-8+0i"

How to fix the problem ??

Thank you kindly.

 

[monirg]

Thank you all for your thoughtful replies. Here're some comments:

1) One would expect the w/s function IMPOWER("x+yi",1/3) to return the principal value, if any, similar to:
....=IMSQRT("-4+0i") returning +2i and not -2i
....=IMPOWER("8+0i",1/3) returning +2.0, and not -1+sqrt(3)i or -1-sqrt(3)i

2) Hence, IMPOWER("-8+0i",1/3) should return -2.0 and not one of the conjugate pair 1+sqrt(3)i or 1-sqrt(3)i.

3) My experience with complex numbers in XL environment is rather limited. However, one might reasonably argue that a complex number with zero imaginary coefficient is equivalent to a real number!
Now try =POWER(-8,1/3). You would correctly get -2.0 and not #NUM!

4) IMPOWER() actually relies on ATAN2(x,y) and not ATAN(y/x) to convert complex numbers to polar, contrary to the XL Help info on the function. This together with the always non-negative "r" maybe an internally-wired factor determining which value is returned by IMPOWER() and similar complex number functions.

Regards.

 

[pgc01]

Hi Monir

I'm sorry, but this time I don't agree with you on any of the points.

Points 1 and 2.

Root(a + bi, n) = Root(r × exp(φi), n) = (r ^ (1/n)) × exp((φ + 2 × π × k) i / n)
where k takes the values 0, 1, ... n-1
The principal value is defined as the one for k=0.

Let's calculate the principal value for your examples:

Example 1: -4+0i
r=4, φ=π, Root(-4 + 0i, 2) = (4^(1/2)) × exp((π + 2 × π × k) i / 2)
Principal value: 2 × exp(πi / 2) = 2i

Example 2: 8+0i
r=8, φ=0, Root(8 + 0i, 3) = (8^(1/3)) × exp((0 + 2 × π × k) i / 3)
Principal value: 2 × exp(0i / 3) = 2

Example 3: -8+0i
r=8, φ=π, Root(-8 + 0i, 3) = (8^(1/3)) × exp((π + 2 × π × k) i / 3)
Principal value: 2 × exp(πi / 3) = 2 × (1/2 + 3^(1/2)/2i) = 1 + 3^(1/2)i

Example 4: -3+4i
r=5, φ=ATAN2(-4/3), Root(-3 + 4i, 3) = (5^(1/3)) × exp((ATAN2(-4/3) + 2 × π × k) i / 3)
Principal value: (5^(1/3)) × exp(ATAN2(-4/3) i / 3) = 1.26495290635775 + 1.15061369838445i


Table with the results calculated here and the results calculated using ImPower():

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PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">Input</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">Principal</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">4</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">Input</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">Root order</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">Real</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">Imaginary</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">ImPower()</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">5</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">-4+0i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">2</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">0</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">2</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">1.22514845490862E-16+2i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">6</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">8+0i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">3</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">2</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">0</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">2</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">7</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">-8+0i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">3</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">1</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">1.73205080756888</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">1+1.73205080756888i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">8</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">-3+4i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">3</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">1.26495290635775</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">1.15061369838445</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: left; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888">1.26495290635775+1.15061369838445i</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #000000; BACKGROUND: #9cf; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #000000; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #000000; PADDING-TOP: 0.4em; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000">9</TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD><TD style="BORDER-TOP-WIDTH: 1px; PADDING-RIGHT: 0.5em; PADDING-LEFT: 0.5em; BORDER-LEFT-WIDTH: 1px; BORDER-LEFT-COLOR: #888888; BORDER-BOTTOM-WIDTH: 1px; BORDER-BOTTOM-COLOR: #888888; PADDING-BOTTOM: 0.25em; BORDER-TOP-COLOR: #888888; PADDING-TOP: 0.4em; TEXT-ALIGN: right; BORDER-RIGHT-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888"> </TD></TR><TR><TD style="PADDING-LEFT: 1em; BACKGROUND: #9cf" colSpan=10>[book2.xlsb]Sheet2</TD></TR></TBODY></TABLE>


You say that ImPower() doesn't always return the principal value of the root. I don't see where that's happening.


I will answer to the points 3 and 4 later. Let me tell you, however that I disagree completely with them.