Math Is Fun Forum

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Tricky integral of a rational function

Hi gAr;

I do not know who put that integral together. I do know that there is a paper out there on doing that integral. I was unable to get it because it was in a journal that was not free.
Thanks for providing that page!

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Tricky integral of a rational function

Hi bobbym,

It's solved in a journal??
A.M.M?

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Tricky integral of a rational function

Hi gAr;

I saw it when this problem was first posed here. Unfortunately I was not following my signature strictly at the time. I wrote nothing down and soon forgot when and where.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Tricky integral of a rational function

Oh, okay!

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Tricky integral of a rational function

Hi;

All I can remember from the abstract was that this one could be done by hand but only by a few.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Tricky integral of a rational function

How can they say it can be done only by a few?!

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Tricky integral of a rational function

Hi gAr;

I guess they mean if the person does not know 10000 theorems in topology, he/she does not have a chance.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Tricky integral of a rational function

Hi bobbym,

Maybe.
Anyway, I'll check with what I know for now.

Looks like we need to use "landen transformations", and I do not understand it yet.

Last edited by gAr (2011-07-17 03:07:11)

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Tricky integral of a rational function

Hi gAr;

I have been working on reducing the order of the polynomials but keeping the area under the curve invariant.

Have not had much luck.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Tricky integral of a rational function

Hi bobbym,

Okay.
Do you know "landen transformations"? I do not understand it yet. It's full of similar integrals, but I can't find a good example.

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Tricky integral of a rational function

hi bobbym and gAr

The factorisation of

is

where the values are approximately
a = 0.709546081
b = 0.339510641
c = 0.562755113
d = 0.111761451   
e = 2.14679097
f = 0.548727908

That’s as close as I can get.

Now, can someone explain Cauchy to me please?

Bob
994

Last edited by Bob (2011-07-17 22:05:23)

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Tricky integral of a rational function

Hi Bob;

I am getting a residue of zero for all 3 singularities. So something is wrong!

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Tricky integral of a rational function

hi bobbym,

Nice to hear from you today.  It looks like this morning's posts need  a 'spring clean'.

This bit of complex theory is not yet in my brain.  I'd like you to explain what you are doing please.

Now, can someone explain Cauchy to me please?

Also are you able to check the factorisation by multiplying it out in Mathematica for me?

Thanks, 

Bob
996

Last edited by Bob (2011-07-17 23:04:34)

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Tricky integral of a rational function

Hi Bob;

I have a migraine so I am slower than usual.

Also are you able to check the factorisation by multiplying it out in Mathematica for me?

That is the easy part.

This the formula I am trying to use.

But if each residue is 0 then the sum of all of them will be 0.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Tricky integral of a rational function

Hi bobbym,

Since they are approximate values, can it yield a residue?

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Tricky integral of a rational function

Hi gAr;

But only an approximate one? We do not have to worry about it because all the residues are equal to 0. The sum of them is 0 and that means the integral is 0, which is false.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Tricky integral of a rational function

Hi,

Okay.
Did you try NResidue[] or Residue[] ?

I thought since finding residue is also numerical, we can settle for numerical integration!

Anyway, did you check "landen's transformation"?
http://library.msri.org/books/Book55/files/13landen.pdf

There are many other similar files.

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Tricky integral of a rational function

Hi gAr;

I did not check out landens transformations yet. I have a whole book here of transformations to handle rational integrands. I was unable to get any of them to work.

I used Residue[]. I do not think I am using the correct formula.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Tricky integral of a rational function

Hi bobbym,

Okay.

By some substitutions, I found that these three are also equivalent:

Can't proceed, though!

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Tricky integral of a rational function

Hi gAr;

I read the PDF you found. I can not get any of that to work unless I can see an example done from start to finish. They do not provide one that I can follow.

I started with this partial fraction and worked with the 6th degree denominators.

I never got anywhere.

What did you do to get yours?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Tricky integral of a rational function

I can not get any of that to work unless I can see an example done from start to finish.

That's my problem too, no complete example...

I substituted 1/x for x.

The other two, used the partial fractions you wrote.
Subs. y for x+1, and y for x-1 in the other, and recombined.

After deriving that, substitute y by its reciprocal.

Last edited by gAr (2011-07-18 00:32:41)

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Tricky integral of a rational function

That's my problem too, no complete example...

I can not stand that. It is so frustrating. The book I have is full of incomplete examples. I can get very little from it. I never mentioned it but I consider that worse than the odd numbered exercises.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Tricky integral of a rational function

And one day even the authors cannot understand it, when they forget.

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Tricky integral of a rational function

True! But at least they did know at one time. Thanks to the way they do the example their readers will never know!

I can get the denominator down to a degree 10. Are these transformations trying to reduce the integrand to a quadratic one. Or are they trying to produce an integrand that splits into partial fractions?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Tricky integral of a rational function

How would people feel if the journals they buy also contain stuff like that?!

I'm not sure what the transformations do. It just looks like the coefficients are changed!

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."