partial fractions

[PG1995]

Hi

Could you please help me with this query? Thank you.

Regards
PG

Useful Link(s):
1: https://www.purplemath.com/modules/factquad.htm

 

[panic mode]

you have sum of products. to find coefficient values you need to figure out values that will eliminate one of terms.
since each of the product terms has form Ki*(x-xi), you choose x such that (x-xi)=0 and do it for each of products.
for example when xi=4, you would want x=-4 etc.

 

[steveB]

PG,

You are correct that you need to be careful when you change the form of equations. There is always risk that the slight differences in form can have significant consequences to what we are trying to do. In this case, the form is slightly different, but you should view equation 3 as a method for determining the A and B constants. Then those constants are substituted in eq. 2 to make eqn. 4, and eqn. 4 is a true equivalence. So, how do you know the method will always work? In math, you have to prove things to know them. Intuitively, it's not hard to see that the method does produce the correct constants that create a real equivalence, but proving something for all cases is not always easy even when the result is obvious, without the proof.

So, you can be confident that the method of partial fractions has been proved valid by mathematicians; however, if you want to prove it to yourself and not take the word of mathematicians, you'll have to do some clever thinking, or track down an existing proof and study it until you are sure that it is correct. Personally, when i do partial fractions, I do it intuitively and then look at the final answer to verify that the new form is an exact representation of the old form. Hence, I prove it on a case by case basis as part of my process of double checking my answers. This is much easier than a generalized proof for all the common and uncommon cases of partial fraction expansion.

Steve

 

[PG1995]

Thank you, panic mode, Steve.

In this case, the form is slightly different, but you should view equation 3 as a method for determining the A and B constants. Then those constants are substituted in eq. 2 to make eqn. 4, and eqn. 4 is a true equivalence.

You are saying that eq. 4 is true equivalent of eq. 2, and eq. 3 is just a tool to reach the final answer, right? Are eq. 2 and eq. 3 true equivalents of each other? In my opinion, they are not. Here is why I think so. If you agree with me, then how can you say that eq. 4 is true equivalent of eq. 2 when eq. 4 was derived from something, i.e. eq. 3, which is not true equivalent of eq. 2. I hope you get where I'm having trouble. Thank you.

Regards
PG

 

[MrAl]

Hi,

What you should do is develop a general method for doing these expansions. You should be able to do it in discrete steps like:

1. Recognize the forms needed
2. Equate the two
3. Multiply out
4. Combine like powers of s (or whatever the variable of interest is)
5. Solve for the constants.

Something like that
But it's better if you do this yourself as you will see how this works the same for any expansion.

 

[steveB]

If you agree with me, then how can you say that eq. 4 is true equivalent of eq. 2 when eq. 4 was derived from something, i.e. eq. 3, which is not true equivalent of eq. 2. I hope you get where I'm having trouble. Thank you.

I do agree with you about 2 and 3 not being quite the same thing, but I think the point could be debated. Eq. 4 and eq. 2 are the same equation, but 4 has numbers in place of the constants. The question is not whether 4 is equivalent to 2, but whether 4 is a correct equivalence. You judge that by looking at eq. 4 only. Is it or is it not a correct equation? To prove it is not correct, you need only find one point where the equation does not hold true. Surely it is equivalent because the values and the singularities match, and since it is equivalent, we can use the right hand side in place of the left hand side, if that form is more useful.

 

[PG1995]

Thank you, MrAl, Steve.

I do agree with you about 2 and 3 not being quite the same thing, but I think the point could be debated.

Okay. Then, let's debate it! I say they are not entirely equivalent; the equation 2 is not defined for x=4 and x=-1.

Regards
PG

 

[steveB]

Thank you, MrAl, Steve.



Okay. Then, let's debate it! I say they are not entirely equivalent; the equation 2 is not defined for x=4 and x=-1.

Regards
PG

You'll have to find someone else to debate with you because I agree with you. I could play the devil's advocate, but this strikes me as one of those questions that leads to pointless debating about semantics. It is easy to understand the nature of both equations and to understand the nature of the particular singularities displayed by one equation. Once we understand their natures, there is nothing to be gained by answering this question because it becomes a semantics debate about what it means for two things to be equivalent. Notice that you said they are not "entirely equivalent". You are even struggling to phrase the question properly because the word "equivalent" is one of those absolute words like "perfect". One thing can not be more or less perfect than another, and likewise, things can not be partially equivalent. They either are or are not equivalent.

 

[MrAl]

Hello again,


You guys are talking about Equ 2 and Equ 3 now i see.

The domain of:
y=A/(x-4)

does not include the number 4 because that makes the expression infinite.

However, the domain of:
A/((x-4)*(x+1))=B/((x-4)*(x+1)

may include x=4 because then both sides are still equal so there's not really a problem.

 

[PG1995]

Hi

I think this might be useful to someone like me. Thanks.

Regards
PG

 

[MrAl]

Hi,

Yes that's a common problem. The missing point is found by changing the equation slightly. That's in pure math however which sometimes works and sometimes doesnt work in real life.