Thread Subject:

what's the root of this equation and in Matlab and Maple?

Hi all,

In learning Maple and Matlab and math,

I did the following experiment:

How to find all roots(in complex domain) of the equation:

y/b=tan(y), where "b" is a constant, and b is a negative real number.

First of all, are there closed form solutions for this equation in complex
plane? I can find one which is y=0.

But there are many more, in fact, I guess there are infinitely many roots.

Are there good ways to represent them out in closed form?

Second, are there ways to find out all the roots in complex plane using
Matlab and Maple?

Thanks!

In article <f86bp3$a8o$1@news.Stanford.EDU>, Gino Victor
<gino_victor@hotmail.com> wrote:

> Hi all,
>
> In learning Maple and Matlab and math,
>
> I did the following experiment:
>
> How to find all roots(in complex domain) of the equation:
>
> y/b=tan(y), where "b" is a constant, and b is a negative real number.
>
> First of all, are there closed form solutions for this equation in complex
> plane? I can find one which is y=0.
>
> But there are many more, in fact, I guess there are infinitely many roots.
>
  there are infinitely real roots , in least one in [n*pi,(n+1)*pi] for
all integers n.
> Are there good ways to represent them out in closed form?
>
   not worth while ones:) All the real roots != 0 are near (n+1/2)*pi
, based on the graph of tan(x). for all integers n. |x[n]- (n+1/2)pi|
gets smaller and smaller as n->oo if the first root y = 0 is not
counted in sequence of roots {x[n].n an integer}

"Gino Victor" <gino_victor@hotmail.com> wrote:
> Hi all,
>
> In learning Maple and Matlab and math,
>
> I did the following experiment:
>
> How to find all roots(in complex domain) of the equation:
>
> y/b=tan(y), where "b" is a constant, and b is a negative real number.
>
> First of all, are there closed form solutions for this equation in
> complex plane?

No.

> I can find one which is y=0.

Can't we all? :-)

> But there are many more, in fact, I guess there are infinitely many
> roots.

Of course there are. (No guessing involved.)

> Are there good ways to represent them out in closed form?

No.

> Second, are there ways to find out all the roots in complex plane using
> Matlab and Maple?

You again refer to roots "in complex plane", but unless I'm overlooking
something, all the roots are _real_.

David W. Cantrell

In article <20070725002958.753$86@newsreader.com>, David W. Cantrell
<DWCantrell@sigmaxi.net> wrote:
> .........
> You again refer to roots "in complex plane", but unless I'm overlooking
> something, all the roots are _real_.
> David W. Cantrell
-----------------------
  Given the fact that your b constant is negative, David is right. Only
real values can be roots. Letting x and y be the real and imaginary parts
of z = x + i*y, just write out the imaginary part of tan(z) and try to
equate it to the imaginary part of z/b. You'll soon see that no matter
what x is, the two quantities cannot be equal unless y is set to zero.

Roger Stafford

"Gino Victor" <gino_victor@hotmail.com> writes:

> Hi all,
>
> In learning Maple and Matlab and math,
>
> I did the following experiment:
>
> How to find all roots(in complex domain) of the equation:
>
> y/b=tan(y), where "b" is a constant, and b is a negative real number.
>
> First of all, are there closed form solutions for this equation in complex
> plane? I can find one which is y=0.

In general the others don't have closed-form solutions.

> But there are many more, in fact, I guess there are infinitely many roots.
>
> Are there good ways to represent them out in closed form?
>
> Second, are there ways to find out all the roots in complex plane using
> Matlab and Maple?


If y = s + i t with s and t real, the real and imaginary parts of your
equation become

2 s/b = sin(2 s)/(cos(s)^2 + sinh(t)^2)
2 t/b = sinh(2 t)/(cos(s)^2 + sinh(t)^2)

If s and t are both nonzero, this implies

sin(2 s)/(2 s) = sinh(2 t)/(2 t)

However, for real nonzero s and t we have
|sin(2 s)/(2 s)| < 1 < |sinh(2 t)/(2 t)|
so that can't happen. We conclude that the roots are all on the
real and imaginary axes. If b < 0, it's not hard to show there
are no imaginary roots (other than 0), so you're left with the
real roots, of which there is one in each interval
((n-1/2) pi, (n+1/2) pi).
--
Robert Israel israel@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

"Robert Israel" <israel@math.MyUniversitysInitials.ca> wrote in message
news:rbisrael.20070725054405$6bb7@news.ks.uiuc.edu...
> "Gino Victor" <gino_victor@hotmail.com> writes:
>
>> Hi all,
>>
>> In learning Maple and Matlab and math,
>>
>> I did the following experiment:
>>
>> How to find all roots(in complex domain) of the equation:
>>
>> y/b=tan(y), where "b" is a constant, and b is a negative real number.
>>
>> First of all, are there closed form solutions for this equation in
>> complex
>> plane? I can find one which is y=0.
>
> In general the others don't have closed-form solutions.
>
>> But there are many more, in fact, I guess there are infinitely many
>> roots.
>>
>> Are there good ways to represent them out in closed form?
>>
>> Second, are there ways to find out all the roots in complex plane using
>> Matlab and Maple?
>
>
> If y = s + i t with s and t real, the real and imaginary parts of your
> equation become
>
> 2 s/b = sin(2 s)/(cos(s)^2 + sinh(t)^2)
> 2 t/b = sinh(2 t)/(cos(s)^2 + sinh(t)^2)
>
> If s and t are both nonzero, this implies
>
> sin(2 s)/(2 s) = sinh(2 t)/(2 t)
>
> However, for real nonzero s and t we have
> |sin(2 s)/(2 s)| < 1 < |sinh(2 t)/(2 t)|
> so that can't happen. We conclude that the roots are all on the
> real and imaginary axes. If b < 0, it's not hard to show there
> are no imaginary roots (other than 0), so you're left with the
> real roots, of which there is one in each interval
> ((n-1/2) pi, (n+1/2) pi).
> --
> Robert Israel israel@math.MyUniversitysInitials.ca
> Department of Mathematics http://www.math.ubc.ca/~israel
> University of British Columbia Vancouver, BC, Canada

Thanks Robert! No way to represent the roots out in closed-form?

"Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid> wrote in
message
news:ellieandrogerxyzzy-2407072240480001@dialup-4.232.0.126.dial1.losangeles1.level3.net...
> In article <20070725002958.753$86@newsreader.com>, David W. Cantrell
> <DWCantrell@sigmaxi.net> wrote:
>> .........
>> You again refer to roots "in complex plane", but unless I'm overlooking
>> something, all the roots are _real_.
>> David W. Cantrell
> -----------------------
> Given the fact that your b constant is negative, David is right. Only
> real values can be roots. Letting x and y be the real and imaginary parts
> of z = x + i*y, just write out the imaginary part of tan(z) and try to
> equate it to the imaginary part of z/b. You'll soon see that no matter
> what x is, the two quantities cannot be equal unless y is set to zero.
>
> Roger Stafford

Thanks Roger! No way to write the roots all out as closed-form?

On Jul 25, 4:10 am, "Gino Victor" <gino_vic...@hotmail.com> wrote:
[...]

> Thanks Robert! No way to represent the roots out in closed-form?

How many times do you need to be told "no" ?

"Gino Victor" <gino_victor@hotmail.com> wrote:
> "Robert Israel" <israel@math.MyUniversitysInitials.ca> wrote in message
> news:rbisrael.20070725054405$6bb7@news.ks.uiuc.edu...
> > "Gino Victor" <gino_victor@hotmail.com> writes:
> >
> >> Hi all,
> >>
> >> In learning Maple and Matlab and math,
> >>
> >> I did the following experiment:
> >>
> >> How to find all roots(in complex domain) of the equation:
> >>
> >> y/b=tan(y), where "b" is a constant, and b is a negative real number.
> >>
> >> First of all, are there closed form solutions for this equation in
> >> complex plane? I can find one which is y=0.
> >
> > In general the others don't have closed-form solutions.
> >
> >> But there are many more, in fact, I guess there are infinitely many
> >> roots.
> >>
> >> Are there good ways to represent them out in closed form?
> >>
> >> Second, are there ways to find out all the roots in complex plane
> >> using Matlab and Maple?
> >
> >
> > If y = s + i t with s and t real, the real and imaginary parts of your
> > equation become
> >
> > 2 s/b = sin(2 s)/(cos(s)^2 + sinh(t)^2)
> > 2 t/b = sinh(2 t)/(cos(s)^2 + sinh(t)^2)
> >
> > If s and t are both nonzero, this implies
> >
> > sin(2 s)/(2 s) = sinh(2 t)/(2 t)
> >
> > However, for real nonzero s and t we have
> > |sin(2 s)/(2 s)| < 1 < |sinh(2 t)/(2 t)|
> > so that can't happen. We conclude that the roots are all on the
> > real and imaginary axes. If b < 0, it's not hard to show there
> > are no imaginary roots (other than 0), so you're left with the
> > real roots, of which there is one in each interval
> > ((n-1/2) pi, (n+1/2) pi).
>
> Thanks Robert! No way to represent the roots out in closed-form?

As I said before: No.

However, the nonzero roots can be expressed in series form:

For tan(x) = mx with m < 0 and x <> 0, letting q = (2n + 1)pi/2,

x = q - 1/(mq) + (1 - 3m)/(3(mq)^3) + (3 -20m + 30m^2)/(15(mq)^5) +...

for integer n.

Example: Suppose m = -1 and we're interested in the root of tan(x) = -x
  which is near 21pi/2. Using just the first three terms of the series to
  approximate that root, we get 33.01700094. For comparison, that root is
  actually 33.01700103...

I wanted to show that example to demonstrate that, even using very few
terms of the series, we can get good accuracy in some cases. But be warned
that such accuracy with few terms cannot be expected when n is small or
when |m| is small.

BTW, you might be interested in comparing the above series with my items
(2) and (3) at <http://mathworld.wolfram.com/TancFunction.html>. But of
course that concerns only the case m = +1; you were asking about m < 0,
which I think has been less well studied.

David W. Cantrell