[OT??] Does raising an arbitrary number to a complex power "mean" anything?

This is a question that comes up while daydreaming on hot summer afternoon.

I know that an phasor can be described by e**(j*theta)
I know the rules well enough to use it by rote. Don't have the math 
insight to have ever discovered it.

The initial question that crossed my mind was
     "What does it mean to take the j'th root of a real number?"

That could be generalized to
     "What does it mean to raise an arbitrary number to a complex power?"

On Jul 27, 5:45 pm, Richard Owlett <rowl...@atlascomm.net> wrote:
> This is a question that comes up while daydreaming on hot summer afternoon.
>
> I know that an phasor can be described by e**(j*theta)
> I know the rules well enough to use it by rote. Don't have the math
> insight to have ever discovered it.
>
> The initial question that crossed my mind was
>      "What does it mean to take the j'th root of a real number?"
>
> That could be generalized to
>      "What does it mean to raise an arbitrary number to a complex power?"

A complex number z to a complex power c can be written as z^c =
exp(c*log(z)) where log(z) = ln(|z|) + i*(angle(z) + 2npi). The result
is multiple-valued. The "principal value" is obtained by setting n=0.

John

On Jul 27, 4:45&#4294967295;pm, Richard Owlett <rowl...@atlascomm.net> wrote:
> This is a question that comes up while daydreaming on hot summer afternoon.

Ah...daydreaming..my favorite pasttime. :)

> I know that an phasor can be described by e**(j*theta)
> I know the rules well enough to use it by rote. Don't have the math
> insight to have ever discovered it.

Mathematicians, doing what they always do, fiddling around with
various things need to be fiddled with, asked what  e**x would look
like when x=jw. Just so happens that it looks just like Taylor
expansion for cos(w) and jsin(w) added together:

http://en.wikipedia.org/wiki/Euler%27s_formula

> The initial question that crossed my mind was
> &#4294967295; &#4294967295; &#4294967295;"What does it mean to take the j'th root of a real number?"

What does it mean to take square-root of 2? :) We can only know by
performing the operation.

> That could be generalized to
> &#4294967295; &#4294967295; &#4294967295;"What does it mean to raise an arbitrary number to a complex power?"

You're undoubtedly asking for an intuitive explanation.  I
sympathize.  I used to go mad when professors would gloss over the
details of Fourier Transforms and go straight to the cookbook
(symmetry, shift-in-time, etc), until a teaching assistant finally
told me, and rest of class, that deep insight will only come after
study of Theory of Distributions.

There is a simpler way to approach analysis and gain the intuition
that you seem to seek, and it has more to do with physics than math:

The continuum is a farce. All we have is the quantum.  So in some
sense, analysis is a farce. If you throw out all your math, and start
back at the very beginning, with an imaginary empty dark space with
nothing in it, then add quantized particles (marbles work for me),
then you have discrete math.  And then you realize that quantities
must always be discrete (more or less). We humans imagined the
continuum, going from whole parts, to rationals of whole parts, to
irrationals, to convergent and divergent series, using limits as
necessary. It helps the intuition, IMHO, to never surrender to the
fictious notion of the continuum...but, at the same time, appreciate
and respect its limits (no pun intended).

You then realize that you have never held a negative number of cookies
in your hand. You realize that if there is such thing as negative
numbers, there might be other kinds of numbers, neither negative nor
positive but something else truly weird, like complex numbers, and if
you were allowed to create a fictious field of negative number without
complaint, why not create a fictitous field of other numbers that
happen to behave like complex numbers? After all, the only criteria
for all of math is that you find a nice set of rules (axioms) that
make operations in this new field of number consistent and regular.

And then you can stop, because at that point, you essentially know
what you need to know, which is that you are working with things that
are not real any way (that's why imaginary part is called imaginary),
and the only thing that matters is how well the rules work for your
values and operations.  The beauty of complex numbers in analysis is
that they take things that we have, real continuous functions, allow
us to enter a world that does not exist, do some operations in that
world, then come back to reality, knowing that, as long as we have
been consistent, we are allowed to do this.

Then you can raise complex numbers to complex numbers as much as you
want...or take Fourier Transform of a Fourier Transform, etc.

Thinking about the quantum also has benefits beyond math, IMHO. For
example, once it is accepted that particles propagate according to
quantum mechanics, Zeno's Paradox need no longer be considered,
because it, and many paradoxes like it, presume the existence of a
continuum. http://en.wikipedia.org/wiki/Zeno's_paradoxes.

Only the marbles are real.

-Le Chaud Lapin-

John wrote:
> On Jul 27, 5:45 pm, Richard Owlett <rowl...@atlascomm.net> wrote:
> 
>>This is a question that comes up while daydreaming on hot summer afternoon.
>>
>>I know that an phasor can be described by e**(j*theta)
>>I know the rules well enough to use it by rote. Don't have the math
>>insight to have ever discovered it.
>>
>>The initial question that crossed my mind was
>>     "What does it mean to take the j'th root of a real number?"
>>
>>That could be generalized to
>>     "What does it mean to raise an arbitrary number to a complex power?"
> 
> 
> A complex number z to a complex power c can be written as z^c =
> exp(c*log(z)) where log(z) = ln(|z|) + i*(angle(z) + 2npi). The result
> is multiple-valued. The "principal value" is obtained by setting n=0.
> 
> John

Thanks.
Almost got there but blocked myself with unwarranted presumptions.
Will have think about imaginary component of log(z).

Le Chaud Lapin wrote:

> On Jul 27, 4:45 pm, Richard Owlett <rowl...@atlascomm.net> wrote:
> 
>>This is a question that comes up while daydreaming on hot summer afternoon.
> 
> 
> Ah...daydreaming..my favorite pasttime. :)
> 
> 
>>I know that an phasor can be described by e**(j*theta)
>>I know the rules well enough to use it by rote. Don't have the math
>>insight to have ever discovered it.
> 
> 
> Mathematicians, doing what they always do, fiddling around with
> various things need to be fiddled with, asked what  e**x would look
> like when x=jw. Just so happens that it looks just like Taylor
> expansion for cos(w) and jsin(w) added together:
> 
> http://en.wikipedia.org/wiki/Euler%27s_formula
> 
> 
>>The initial question that crossed my mind was
>>     "What does it mean to take the j'th root of a real number?"
> 
> 
> What does it mean to take square-root of 2? :) We can only know by
> performing the operation.

Yeah, BUT ;)
What if I had phrased it as "What does it mean to raise a real number to 
the j'th power?"

As far as I know the "definition" of raising x to the power n requires 
requires multiplying x by x n times. How to do it j times ;/
Don't think that can be done except by assuming z^c = exp(c*log(z)) as 
John did in his post.

> 
> 
>>That could be generalized to
>>     "What does it mean to raise an arbitrary number to a complex power?"
> 
> 
> You're undoubtedly asking for an intuitive explanation.

That would be ideal. But we know how often ideal happens.

 > I sympathize.  I used to go mad when professors would gloss over the
> details of Fourier Transforms and go straight to the cookbook
> (symmetry, shift-in-time, etc), until a teaching assistant finally
> told me, and rest of class, that deep insight will only come after
> study of Theory of Distributions.

"Theory of Distributions" ??? Took a look at 
http://en.wikipedia.org/wiki/Distribution_(mathematics) and went away 
babbling. You must have had a lot more pure math than I did. We got it 
as the limit of Fourier Series IIRC. That made "logical" sense. I may 
have helped that one job involved maintaining an analog computer for 
experimentally determining the coefficients of a Fourier Series.


> 
> There is a simpler way to approach analysis and gain the intuition
> that you seem to seek, and it has more to do with physics than math:
> 
> The continuum is a farce. All we have is the quantum.  So in some
> sense, analysis is a farce. If you throw out all your math, and start
> back at the very beginning, with an imaginary empty dark space with
> nothing in it, then add quantized particles (marbles work for me),
> then you have discrete math.  And then you realize that quantities
> must always be discrete (more or less). We humans imagined the
> continuum, going from whole parts, to rationals of whole parts, to
> irrationals, to convergent and divergent series, using limits as
> necessary. It helps the intuition, IMHO, to never surrender to the
> fictious notion of the continuum...but, at the same time, appreciate
> and respect its limits (no pun intended).
> 
> You then realize that you have never held a negative number of cookies
> in your hand.

Careful, I could recognize a positive (ie a real) number of cookies in 
my brother's hand.

 > You realize that if there is such thing as negative
> numbers, there might be other kinds of numbers, neither negative nor
> positive but something else truly weird, like complex numbers, and if
> you were allowed to create a fictious field of negative number without
> complaint, why not create a fictitous field of other numbers that
> happen to behave like complex numbers? After all, the only criteria
> for all of math is that you find a nice set of rules (axioms) that
> make operations in this new field of number consistent and regular.
> 
> And then you can stop, because at that point, you essentially know
> what you need to know, which is that you are working with things that
> are not real any way (that's why imaginary part is called imaginary),
> and the only thing that matters is how well the rules work for your
> values and operations.  The beauty of complex numbers in analysis is
> that they take things that we have, real continuous functions, allow
> us to enter a world that does not exist, do some operations in that
> world, then come back to reality, knowing that, as long as we have
> been consistent, we are allowed to do this.
> 
> Then you can raise complex numbers to complex numbers as much as you
> want...or take Fourier Transform of a Fourier Transform, etc.
> 
> Thinking about the quantum also has benefits beyond math, IMHO. For
> example, once it is accepted that particles propagate according to
> quantum mechanics, Zeno's Paradox need no longer be considered,
> because it, and many paradoxes like it, presume the existence of a
> continuum. http://en.wikipedia.org/wiki/Zeno's_paradoxes.
> 
> Only the marbles are real.
> 
> -Le Chaud Lapin-

INTERESTING. Thanks. We do look at the universe(s) around us from 
different perspectives.

"Richard Owlett" <rowlett@atlascomm.net> wrote in message 
news:j9-dnSAQrZ0XWBDVnZ2dnUVZ_g-dnZ2d@supernews.com...
> John wrote:
>> On Jul 27, 5:45 pm, Richard Owlett <rowl...@atlascomm.net> wrote:
>>
>>>This is a question that comes up while daydreaming on hot summer 
>>>afternoon.
>>>
>>>I know that an phasor can be described by e**(j*theta)
>>>I know the rules well enough to use it by rote. Don't have the math
>>>insight to have ever discovered it.
>>>
>>>The initial question that crossed my mind was
>>>     "What does it mean to take the j'th root of a real number?"
>>>
>>>That could be generalized to
>>>     "What does it mean to raise an arbitrary number to a complex power?"
>>
>>
>> A complex number z to a complex power c can be written as z^c =
>> exp(c*log(z)) where log(z) = ln(|z|) + i*(angle(z) + 2npi). The result
>> is multiple-valued. The "principal value" is obtained by setting n=0.
>>
>> John
>
> Thanks.
> Almost got there but blocked myself with unwarranted presumptions.
> Will have think about imaginary component of log(z).
>
Try i^i just for chuckles...
   Best wishes,
   --Phil 

Le Chaud Lapin wrote:
(snip)

> Mathematicians, doing what they always do, fiddling around with
> various things need to be fiddled with, asked what  e**x would look
> like when x=jw. Just so happens that it looks just like Taylor
> expansion for cos(w) and jsin(w) added together:

That works, too, but you should read Feynman's explanation.

Much more fun to read than my explanation of it....

First he does non-integer powers, I believe in a way
that has some history behind it.   Historically, one
knew that 10**0.5 was sqrt(10), that 10**0.25 was
sqrt(sqrt(10)), down to 10**0.0009765625 was
sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(10))))))))))

Given the calculation of 10 square roots, you can compute
fractional powers of 10, with the power within 1/1024 of
the desired value.  That is, make a three digit table of
powers of 10.  Invert the table for a three digit log table.
(Probably with some interpolation.)

Then he goes onto 10**(i/1024) using the small x
approximation for 10**x.  Using that, he goes on to
show that 10**ix is periodic in x, and when graphed looks
a lot like sin(x) but the scale is wrong.

After reading that, read the last chapter of Volume 1,
on the non-conservation of parity.

-- glen

Philip Martel wrote:
> "Richard Owlett" <rowlett@atlascomm.net> wrote in message 
> news:j9-dnSAQrZ0XWBDVnZ2dnUVZ_g-dnZ2d@supernews.com...
> 
>>John wrote:
>>
>>>On Jul 27, 5:45 pm, Richard Owlett <rowl...@atlascomm.net> wrote:
>>>
>>>
>>>>This is a question that comes up while daydreaming on hot summer 
>>>>afternoon.
>>>>
>>>>I know that an phasor can be described by e**(j*theta)
>>>>I know the rules well enough to use it by rote. Don't have the math
>>>>insight to have ever discovered it.
>>>>
>>>>The initial question that crossed my mind was
>>>>    "What does it mean to take the j'th root of a real number?"
>>>>
>>>>That could be generalized to
>>>>    "What does it mean to raise an arbitrary number to a complex power?"
>>>
>>>
>>>A complex number z to a complex power c can be written as z^c =
>>>exp(c*log(z)) where log(z) = ln(|z|) + i*(angle(z) + 2npi). The result
>>>is multiple-valued. The "principal value" is obtained by setting n=0.
>>>
>>>John
>>
>>Thanks.
>>Almost got there but blocked myself with unwarranted presumptions.
>>Will have think about imaginary component of log(z).
>>
> 
> Try i^i just for chuckles...
>    Best wishes,
>    --Phil 
> 
> 

OK group.
Will I be masochist, he a sadist, or a a informative exercise? ????
Just DO NOT remember "A chartreuse Naugahyde noose" nor associated "pink 
elephants" ;/

Whacily urs

glen herrmannsfeldt wrote:

> Le Chaud Lapin wrote:
> (snip)
> 
>> Mathematicians, doing what they always do, fiddling around with
>> various things need to be fiddled with, asked what  e**x would look
>> like when x=jw. Just so happens that it looks just like Taylor
>> expansion for cos(w) and jsin(w) added together:
> 
> 
> That works, too, but you should read Feynman's explanation.

OK already ;)
A URL please.
Please don't just tweak my curiosity bump.

Richard Owlett wrote:

> Mathematicians, doing what they always do, fiddling around with
> various things need to be fiddled with, asked what  e**x would look
> like when x=jw. Just so happens that it looks just like Taylor
> expansion for cos(w) and jsin(w) added together:

There are a number of ways to see it.  Note that sin(x) and
cos(x) are two functions equal to the negative of their second
derivative.  Note that the second derivative of exp(ax) is
a**2*exp(ax), so if a is i, it is -exp(ix).  cos(x) and sin(x)
have to be (proportional to) the even and odd components of
exp(ix).  (A second order differential equation has only
two independent solutions.)

Then again, the properties of the derivative are directly
related to the taylor series expansion, so it isn't really
another way.

Maybe less obvious is the connection between the sin and
cos of algebra (and calculus) and the sin and cos of geometry.

-- glen