Question about "e" exponent

Hi All,

As signal processing people, we have come across 'e' million times.
'e' is interesting when it has sqrt(-1) (j) to its power.

My question is, e^j can be expressed in cartesian coordinates as two
projections. But is it true that (any number)^j can also be expressed in
the cartesian coordinates. Intuitively it looks possible. Any thoughts?

Regards
Huzaifa

On Wed, 25 Jan 2012 04:35:56 -0600, "huzaifa"
<hskapasi@n_o_s_p_a_m.gmail.com> wrote:

>Hi All,
>
>As signal processing people, we have come across 'e' million times.
>'e' is interesting when it has sqrt(-1) (j) to its power.
>
>My question is, e^j can be expressed in cartesian coordinates as two
>projections. But is it true that (any number)^j can also be expressed in
>the cartesian coordinates. Intuitively it looks possible. Any thoughts?
>
>Regards
>Huzaifa

Uhmmmm maybe you should take some classes in basic mathematics again.
Start with x = exp( ln(x) ) and work it out from there.

Regards,

Mark DeArman

On Wed, 25 Jan 2012 03:12:45 -0800, Mac Decman
<dearman.mark@gmail.com> wrote:

>On Wed, 25 Jan 2012 04:35:56 -0600, "huzaifa"
><hskapasi@n_o_s_p_a_m.gmail.com> wrote:
>
>>Hi All,
>>
>>As signal processing people, we have come across 'e' million times.
>>'e' is interesting when it has sqrt(-1) (j) to its power.
>>
>>My question is, e^j can be expressed in cartesian coordinates as two
>>projections. But is it true that (any number)^j can also be expressed in
>>the cartesian coordinates. Intuitively it looks possible. Any thoughts?
>>
>>Regards
>>Huzaifa
>
>Uhmmmm maybe you should take some classes in basic mathematics again.
>Start with x = exp( ln(x) ) and work it out from there.
>
>Regards,
>
>Mark DeArman

Jeez, Sorry again, I always sound so condescending when I respond. I'm
not sure why my first pop off response sounds like such an ass always.
Take x = exp( ln(x) )
Then take an example of 2^i
2^I  = cos( ln(2) ) + i*sin(ln(2))
Fit this to your z = x+iy form.

Mark DeArman

huzaifa <hskapasi@n_o_s_p_a_m.gmail.com> wrote:

> As signal processing people, we have come across 'e' million times.
> 'e' is interesting when it has sqrt(-1) (j) to its power.

> My question is, e^j can be expressed in cartesian coordinates as two
> projections. But is it true that (any number)^j can also be expressed in
> the cartesian coordinates. Intuitively it looks possible. Any thoughts?

Yes.

(Well, cartesian coordinates is not the way I usually describe
it, but real and imaginary parts, yes.)

Many calculators that claim to do complex math won't do log10
or trig functions on complex values.

The TI-92 will do trig functions in both radians and degrees.

The TI-84 won't, but the TI-92 will, do arcsin(2), in both radians
and degrees, symbolic and numeric.

My first tries with complex exponentiation was a PL/I program
that I wrote in high school that would read in two complex numbers
and print out x**y. One of my first tries was i**i, which, surprising
to me at the time, and maybe still, is not complex.

-- glen

glen herrmannsfeldt wrote:
> huzaifa<hskapasi@n_o_s_p_a_m.gmail.com>  wrote:
>
>> As signal processing people, we have come across 'e' million times.
>> 'e' is interesting when it has sqrt(-1) (j) to its power.
>
>> My question is, e^j can be expressed in cartesian coordinates as two
>> projections. But is it true that (any number)^j can also be expressed in
>> the cartesian coordinates. Intuitively it looks possible. Any thoughts?
>
> Yes.
>
> (Well, cartesian coordinates is not the way I usually describe
> it, but real and imaginary parts, yes.)
>
> Many calculators that claim to do complex math won't do log10
> or trig functions on complex values.
>
> The TI-92 will do trig functions in both radians and degrees.
>
> The TI-84 won't, but the TI-92 will, do arcsin(2), in both radians
> and degrees, symbolic and numeric.
>
> My first tries with complex exponentiation was a PL/I program
> that I wrote in high school that would read in two complex numbers
> and print out x**y. One of my first tries was i**i, which, surprising
> to me at the time, and maybe still, is not complex.
>
> -- glen


It's still complex - the imaginary part is just zero.

(vector victor over ....)

--
Les Cargill

On Jan 25, 5:35&#4294967295;am, "huzaifa" <hskapasi@n_o_s_p_a_m.gmail.com> wrote:
> Hi All,
>
> As signal processing people, we have come across 'e' million times.
> 'e' is interesting when it has sqrt(-1) (j) to its power.
>
> My question is, e^j can be expressed in cartesian coordinates as two
> projections. But is it true that (any number)^j can also be expressed in
> the cartesian coordinates. Intuitively it looks possible. Any thoughts?
>
> Regards
> Huzaifa


I strongly recommend Paul Nahin's book, "Dr. Eulers Fabulous Formula"

I believe anyone working with DSP would enjoy and learn from this
book.

Clay

On Jan 25, 5:35&#4294967295;am, "huzaifa" <hskapasi@n_o_s_p_a_m.gmail.com> wrote:
> Hi All,
>
> As signal processing people, we have come across 'e' million times.
> 'e' is interesting when it has sqrt(-1) (j) to its power.
>
> My question is, e^j can be expressed in cartesian coordinates as two
> projections. But is it true that (any number)^j can also be expressed in
> the cartesian coordinates. Intuitively it looks possible. Any thoughts?
>
> Regards
> Huzaifa

e raised to the pi times i,
And plus 1 leaves you nought but a sigh.
This fact amazed Euler
That genius toiler,
And still gives us pause, bye the bye.

In article <jfp0lh$hv7$1@speranza.aioe.org>,
glen herrmannsfeldt  <gah@ugcs.caltech.edu> wrote:
>huzaifa <hskapasi@n_o_s_p_a_m.gmail.com> wrote:
>
>> As signal processing people, we have come across 'e' million times.
>> 'e' is interesting when it has sqrt(-1) (j) to its power.
>
>> My question is, e^j can be expressed in cartesian coordinates as two
>> projections. But is it true that (any number)^j can also be expressed in
>> the cartesian coordinates. Intuitively it looks possible. Any thoughts?
>
>Yes.
>
>(Well, cartesian coordinates is not the way I usually describe
>it, but real and imaginary parts, yes.)
>
>Many calculators that claim to do complex math won't do log10
>or trig functions on complex values.
>
>The TI-92 will do trig functions in both radians and degrees.
>
>The TI-84 won't, but the TI-92 will, do arcsin(2), in both radians
>and degrees, symbolic and numeric.
>

During my first few years at college, all my friends were getting
these new HP calculators that did all that complex math for you.  I stuck
with a cheap Radio Shack - it had some helper functions to convert between
polar and rectangular coordinates - but nothing else.  

I think this is why I learned Euler's rule so well.  Having to 
do it by hand, and all the benefits you get - was very valuable.  Heck, using
Euler's rule, many of the trigometric identities just fall out easily.  Would
have saved me from memorising them in high school trig...

My son's in Geometry now.  Schools let you use calculators soo early now.  I'm
not even sure I could find a calculator that didn't do all the complex math 
for you... I'd sure like it if his didn't.

--Mark

Mark Curry <gtwrek@sonic.net> wrote:
> In article <jfp0lh$hv7$1@speranza.aioe.org>,

(snip)
>>> As signal processing people, we have come across 'e' million times.
>>> 'e' is interesting when it has sqrt(-1) (j) to its power.

>>> My question is, e^j can be expressed in cartesian coordinates as two
>>> projections. But is it true that (any number)^j can also be expressed in
>>> the cartesian coordinates. Intuitively it looks possible. Any thoughts?

(snip, I wrote)
>>Many calculators that claim to do complex math won't do log10
>>or trig functions on complex values.

>>The TI-92 will do trig functions in both radians and degrees.

>>The TI-84 won't, but the TI-92 will, do arcsin(2), in both radians
>>and degrees, symbolic and numeric.

> During my first few years at college, all my friends were getting
> these new HP calculators that did all that complex math for you.  I stuck
> with a cheap Radio Shack - it had some helper functions to convert between
> polar and rectangular coordinates - but nothing else.  

When I was in college, calculators didn't yet do complex math.
We even had open calculator quizzes and exams, but without any 
numbers to calculate.

> I think this is why I learned Euler's rule so well.  Having to 
> do it by hand, and all the benefits you get - was very valuable.  Heck, using
> Euler's rule, many of the trigometric identities just fall out easily.  Would
> have saved me from memorising them in high school trig...

Well, they do like them, even require, graphing calculators, and maybe
seeing the graphs is useful. But yes, I do sometimes think they get
too used to them.

> My son's in Geometry now.  Schools let you use calculators soo early now.  I'm
> not even sure I could find a calculator that didn't do all the complex math 
> for you... I'd sure like it if his didn't.

The TI-92 will even do integrals. I have tried some, saying that no
calculator would do this one, and finding out that it did.

Though the TI-92 isn't allowed for the SAT or ACT, which is probably
good. The HP-28S, I believe, is allowed on the SAT. (Pretty good that
they get a 20 or so year old calculator on the list. They want
graphing calculators, the HP-28C being one of the first ones.)

I believe the TI-84 won't graph exp(ix)+exp(-ix), even if the real
part comes out zero, though possibly slightly off zero. 

But yes, I hardly ever do real problems with complex numbers, just
play around and see what the calculator does with them.

-- glen

On Jan 25, 9:36&#4294967295;am, Clay <c...@claysturner.com> wrote:
>
> e raised to the pi times i,
> And plus 1 leaves you nought but a sigh.
> This fact amazed Euler
> That genius toiler,
> And still gives us pause, bye the bye.

so Clay, whose poem is that?

r b-j