Hi,

I have begun a complex number tutorial which shows how e^jwt works. It
is located:

http://fourier-series.com/fourierseries2/complex_tutorial.html

I have more work to do on this, but come take a look.  I created these
programs for complex numbers in order to explain the complex
representaion of the fourier series.

Brent

On Jan 21, 7:48 am, buleg...@columbus.rr.com wrote:
> Hi,
>
> I have begun a complex number tutorial which shows how e^jwt works. It
> is located:
>
> http://fourier-series.com/fourierseries2/complex_tutorial.html
>
> I have more work to do on this, but come take a look.  I created these
> programs for complex numbers in order to explain the complex
> representaion of the fourier series.
>
> Brent

Hello Brent,

It appears you are having fun with this. You may wish to look at Paul
Nahin's book "Dr. Euler's Fabulous Formula Cures Many Mathematical
Ills." I think you will find a lot of neat thinks to graphically
demostrate.

And yes the world appreciates your tutorials.

Clay - a guy who had to learn things the old fashioned way before the
advent of the internet.

On Jan 21, 10:02 am, c...@claysturner.com wrote:
> On Jan 21, 7:48 am, buleg...@columbus.rr.com wrote:
>
> > Hi,
>
> > I have begun a complex number tutorial which shows how e^jwt works. It
> > is located:
>
> >http://fourier-series.com/fourierseries2/complex_tutorial.html
>
> > I have more work to do on this, but come take a look.  I created these
> > programs for complex numbers in order to explain the complex
> > representaion of the fourier series.
>
> > Brent
>
> Hello Brent,
>
> It appears you are having fun with this. You may wish to look at Paul
> Nahin's book "Dr. Euler's Fabulous Formula Cures Many Mathematical
> Ills." I think you will find a lot of neat thinks to graphically
> demostrate.
>
> And yes the world appreciates your tutorials.
>
> Clay - a guy who had to learn things the old fashioned way before the
> advent of the internet.

I think the last two of these programs are not loading correctly in
some computers.  My home computers load fine , but I am  seeing other
probs.  If the last two programs don't load, I am sorry.  I am trying
to figure out why.

On Jan 21, 10:26 am, buleg...@columbus.rr.com wrote:
> On Jan 21, 10:02 am, c...@claysturner.com wrote:
>
>
>
>
>
> > On Jan 21, 7:48 am, buleg...@columbus.rr.com wrote:
>
> > > Hi,
>
> > > I have begun a complex number tutorial which shows how e^jwt works. It
> > > is located:
>
> > >http://fourier-series.com/fourierseries2/complex_tutorial.html
>
> > > I have more work to do on this, but come take a look.  I created these
> > > programs for complex numbers in order to explain the complex
> > > representaion of the fourier series.
>
> > > Brent
>
> > Hello Brent,
>
> > It appears you are having fun with this. You may wish to look at Paul
> > Nahin's book "Dr. Euler's Fabulous Formula Cures Many Mathematical
> > Ills." I think you will find a lot of neat thinks to graphically
> > demostrate.
>
> > And yes the world appreciates your tutorials.
>
> > Clay - a guy who had to learn things the old fashioned way before the
> > advent of the internet.
>
> I think the last two of these programs are not loading correctly in
> some computers.  My home computers load fine , but I am  seeing other
> probs.  If the last two programs don't load, I am sorry.  I am trying
> to figure out why.- Hide quoted text -
>
> - Show quoted text -

On the last two programs, click to download and then close out the
program, then reclick , and it comes up correctly.  Don't know why ,
yet

On Mon, 21 Jan 2008 04:48:59 -0800, bulegoge wrote:

> Hi,
> 
> I have begun a complex number tutorial which shows how e^jwt works. It
> is located:
> 
> http://fourier-series.com/fourierseries2/complex_tutorial.html
> 
> I have more work to do on this, but come take a look.  I created these
> programs for complex numbers in order to explain the complex
> representaion of the fourier series.
> 
> Brent

Looks cool.

At this moment it's telling me that 2.5 * cos 2 * pi = 2.53.  Perhaps you 
need to make your rounding more consistent.

-- 
Tim Wescott
Control systems and communications consulting
http://www.wescottdesign.com

Need to learn how to apply control theory in your embedded system?
"Applied Control Theory for Embedded Systems" by Tim Wescott
Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html

On Jan 21, 6:48 am, buleg...@columbus.rr.com wrote:
> Hi,
>
> I have begun a complex number tutorial which shows how e^jwt works. It
> is located:
>
> http://fourier-series.com/fourierseries2/complex_tutorial.html
>
> I have more work to do on this, but come take a look.  I created these
> programs for complex numbers in order to explain the complex
> representaion of the fourier series.
>
> Brent

Cute, but you should explain why Euler needed "j" in his identity.
Explain why it's used as the vertical axis of the phasor plane. Many
engineers want a physical intuition that is defied by a reference to
the square root of -1. Most introductory texts just state Euler's
identity and leave it at that.

On Jan 21, 2:03 pm, Darol Klawetter <darol.klawet...@l-3com.com>
wrote:
> On Jan 21, 6:48 am, buleg...@columbus.rr.com wrote:
>
> > Hi,
>
> > I have begun a complex number tutorial which shows how e^jwt works. It
> > is located:
>
> >http://fourier-series.com/fourierseries2/complex_tutorial.html
>
> > I have more work to do on this, but come take a look.  I created these
> > programs for complex numbers in order to explain the complex
> > representaion of the fourier series.
>
> > Brent
>
> Cute, but you should explain why Euler needed "j" in his identity.
> Explain why it's used as the vertical axis of the phasor plane. Many
> engineers want a physical intuition that is defied by a reference to
> the square root of -1. Most introductory texts just state Euler's
> identity and leave it at that.

I intend to add some more explanation in the near future.  However,
this whole j thing is difficult because getting that physical
intuition has not really come to me yet.  I wonder if you must simply
accept, as an article of faith that Eulers identity works and that
(e^jwt+e^-jwt)/2 = cos(wt).  That is, the staring point is accepting
the identity.  The equivelent of an axiom in geometry, then build off
of the axiom.

On Mon, 21 Jan 2008 11:41:55 -0800 (PST), b...@columbus.rr.com
wrote:

>On Jan 21, 2:03 pm, Darol Klawetter <darol.klawet...@l-3com.com>
>wrote:
>> On Jan 21, 6:48 am, buleg...@columbus.rr.com wrote:
>>
>> > Hi,
>>
>> > I have begun a complex number tutorial which shows how e^jwt works. It
>> > is located:
>>
>> >http://fourier-series.com/fourierseries2/complex_tutorial.html
>>
>> > I have more work to do on this, but come take a look.  I created these
>> > programs for complex numbers in order to explain the complex
>> > representaion of the fourier series.
>>
>> > Brent
>>
>> Cute, but you should explain why Euler needed "j" in his identity.
>> Explain why it's used as the vertical axis of the phasor plane. Many
>> engineers want a physical intuition that is defied by a reference to
>> the square root of -1. Most introductory texts just state Euler's
>> identity and leave it at that.
>
>I intend to add some more explanation in the near future.  However,
>this whole j thing is difficult because getting that physical
>intuition has not really come to me yet.  I wonder if you must simply
>accept, as an article of faith that Eulers identity works and that
>(e^jwt+e^-jwt)/2 = cos(wt).  That is, the staring point is accepting
>the identity.  The equivelent of an axiom in geometry, then build off
>of the axiom.

Hello bulegoge,

    I like your "complex numbers" demos.

I'm not sure if it would benefit you but 
you have a look at

http://www.dspguru.com/info/tutor/QuadSignals.pdf

Nice work bulegoge.

[-Rick-]

b...@columbus.rr.com wrote:
> On Jan 21, 2:03 pm, Darol Klawetter <darol.klawet...@l-3com.com>
> wrote:
>> On Jan 21, 6:48 am, buleg...@columbus.rr.com wrote:
>>
>>> Hi,
>>> I have begun a complex number tutorial which shows how e^jwt works. It
>>> is located:
>>> http://fourier-series.com/fourierseries2/complex_tutorial.html
>>> I have more work to do on this, but come take a look.  I created these
>>> programs for complex numbers in order to explain the complex
>>> representaion of the fourier series.
>>> Brent
>> Cute, but you should explain why Euler needed "j" in his identity.
>> Explain why it's used as the vertical axis of the phasor plane. Many
>> engineers want a physical intuition that is defied by a reference to
>> the square root of -1. Most introductory texts just state Euler's
>> identity and leave it at that.
> 
> I intend to add some more explanation in the near future.  However,
> this whole j thing is difficult because getting that physical
> intuition has not really come to me yet.  I wonder if you must simply
> accept, as an article of faith that Eulers identity works and that
> (e^jwt+e^-jwt)/2 = cos(wt).  That is, the staring point is accepting
> the identity.  The equivelent of an axiom in geometry, then build off
> of the axiom.

No. At the very least, you can add the Taylor series of cos(x) to the 
Taylor series of i*sin(x) and observe that the sum is identical to the 
Taylor series of exp(i*x). A technicality makes that fall short of being 
a proof, but it sure is a good indication.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

On Jan 21, 9:37 pm, Jerry Avins <j...@ieee.org> wrote:
> buleg...@columbus.rr.com wrote:
> > On Jan 21, 2:03 pm, Darol Klawetter <darol.klawet...@l-3com.com>
> > wrote:
> >> On Jan 21, 6:48 am, buleg...@columbus.rr.com wrote:
>
> >>> Hi,
> >>> I have begun a complex number tutorial which shows how e^jwt works. It
> >>> is located:
> >>>http://fourier-series.com/fourierseries2/complex_tutorial.html
> >>> I have more work to do on this, but come take a look.  I created these
> >>> programs for complex numbers in order to explain the complex
> >>> representaion of the fourier series.
> >>> Brent
> >> Cute, but you should explain why Euler needed "j" in his identity.
> >> Explain why it's used as the vertical axis of the phasor plane. Many
> >> engineers want a physical intuition that is defied by a reference to
> >> the square root of -1. Most introductory texts just state Euler's
> >> identity and leave it at that.
>
> > I intend to add some more explanation in the near future.  However,
> > this whole j thing is difficult because getting that physical
> > intuition has not really come to me yet.  I wonder if you must simply
> > accept, as an article of faith that Eulers identity works and that
> > (e^jwt+e^-jwt)/2 = cos(wt).  That is, the staring point is accepting
> > the identity.  The equivelent of an axiom in geometry, then build off
> > of the axiom.
>
> No. At the very least, you can add the Taylor series of cos(x) to the
> Taylor series of i*sin(x) and observe that the sum is identical to the
> Taylor series of exp(i*x). A technicality makes that fall short of being
> a proof, but it sure is a good indication.
>
> Jerry
> --
> Engineering is the art of making what you want from things you can get.
> ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯- Hide quoted text
-
>
> - Show quoted text -

That relationship is cool, It points to complex numbers as having more
meaning than might otherwise have been thought.  It made complex
numbers a real "head scratcher", but, for me, this relationship does
not really bring forth any more insight into what imaginary numbers
are.  To me, imaginary numbers are like a hidden world that works
behind the scene , but always turns up invisible to us.

For me the real beauty is that the e^jwt is more intuitive as a
solution to a differential equation than say, sinx and cos x is.
Sure, it is observed that the fourth derivative of a sine wave is a
sine wave, but the e^jwt really brings the soultion to a differential
equation to life.

We can intuitively understand solutions to differential equations that
have x, x^2, x^3 as solutions.
We can intuitively understand solutions that have e^-x or e^-t as
solutions.

 To understand rotational/sinusoidal solutions, however, we need to
understand complex numbers, which (perhaps) nobody really intuitively
understands.  I guess when it all boils down, the only way to get a
complete solution to a differential equation (one that includes not
just the correct frequeny, but the correct phasing), you need the
hidden magic of complex numbers going on "behind the scenes".

I know this sounds like rambling jibberish, and probably blows my
credibility, but, oh well.

Brent