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Help with a partial fraction please. watch

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    Can somebody partial fractionate the following for me please?

    \dfrac{z^2}{z^4+1}

    I think I'm being really stupid here. I've got to this point  A(z^2 - i) + B(z^2 + i) = z^2 and my brain has gone a bit fuzzy because I can't see what to sub in for z to deduce A and B. Help.
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    (Original post by Preeka)
    Can somebody partial fractionate the following for me please?

    \dfrac{z^2}{z^4+1}

    I think I'm being really stupid here. I've got to this point  A(z^2 - i) + B(z^2 + i) = z^2 and my brain has gone a bit fuzzy because I can't see what to sub in for z to deduce A and B. Help.
    I wouldn't factorise the denominator that way. Note that z^4+1 = (z^2 - \sqrt2 z + 1)(z^2 + \sqrt2 z +1). Then attempt partial fractions.

    You've made a mistake in breaking your fraction into the general partial fractions. I'm not even sure if you can do partial fractions with complex denominators using the standard method? Correct me if I'm wrong.

    If it is allowed then you should have found that:

    \dfrac{z^2}{x^4+1} \equiv \dfrac{Az+B}{z^2+i} + \dfrac{Cz+D}{z^2-i}.

    Rather than just numerators of A and B, as you seem to have used.
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    (Original post by Farhan.Hanif93)
    I wouldn't factorise the denominator that way. Note that z^4+1 = (z^2 - \sqrt2 z + 1)(z^2 + \sqrt2 z +1). Then attempt partial fractions.

    You've made a mistake in breaking your fraction into the general partial fractions. I'm not even sure if you can do partial fractions with complex denominators using the standard method? Correct me if I'm wrong.

    If it is allowed then you should have found that:

    \dfrac{z^2}{x^4+1} \equiv \dfrac{Az+B}{z^2+i} + \dfrac{Cz+D}{z^2-i}.

    Rather than just numerators of A and B, as you seem to have used.
    Thanks
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    (Original post by Preeka)
    Thanks
    I know that the factorisation I proposed works fine but I'm not too sure about having denominators of z^2+i and z^2-i. I would wait for someone more knowledgable to confirm that.
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    If you're trying to find the partial fraction of that then you're probably doing something wrong. Partial fractions are only really useful when they can be done in the real domain.

    What is the full question? Or did you just make up this question?
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    (Original post by Farhan.Hanif93)
    I wouldn't factorise the denominator that way. Note that z^4+1 = (z^2 - \sqrt2 z + 1)(z^2 + \sqrt2 z +1). Then attempt partial fractions.

    You've made a mistake in breaking your fraction into the general partial fractions. I'm not even sure if you can do partial fractions with complex denominators using the standard method? Correct me if I'm wrong.

    If it is allowed then you should have found that:

    \dfrac{z^2}{x^4+1} \equiv \dfrac{Az+B}{z^2+i} + \dfrac{Cz+D}{z^2-i}.

    Rather than just numerators of A and B, as you seem to have used.
    I wouldn't have originally broken it down into complex denominators but that's how my lecturer seems to have done it so it is valid. She's genius so I trust her enough on that account. Maybe I did mess up the numerators parts though.
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    (Original post by noobynoo)
    If you're trying to find the partial fraction of that then you're probably doing something wrong. Partial fractions are only really useful when they can be done in the real domain.

    What is the full question? Or did you just make up this question?
    The whole question is to do with integration around a contour so it makes use of Cauchy's Integral formula as at one point, partial fractioning is required because both points lie inside the contour in question and in her answer sheet, my lecturer has broken it down like that but obviously assumed I'd be able to deduce the middle steps by myself (which I'm just failing at for this question )
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    (Original post by Preeka)
    The whole question is to do with integration around a contour so it makes use of Cauchy's Integral formula as at one point, partial fractioning is required because both points lie inside the contour in question and in her answer sheet, my lecturer has broken it down like that but obviously assumed I'd be able to deduce the middle steps by myself (which I'm just failing at for this question )
    fair enough.

    In your question have you tried
    

A=B= 1/2
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    (Original post by noobynoo)
    fair enough.

    In your question have you tried
    

A=B= 1/2
    yeah A and B do equal 1/2 according to the answers. I just had a hard time seeing why and how. I asked a friend and she said, just substitute z^2 = i and it works out to give the right values for A and B which I suppose seems obvious now that she told me. I still don't know how mathematically sound that is but if it works, it works. I'm not in the mood to investigate anymore after the realisation that I can't even do partial fractions at times . Stupid partial fractions. Grr.
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    (Original post by Preeka)
    yeah A and B do equal 1/2 according to the answers. I just had a hard time seeing why and how. I asked a friend and she said, just substitute z^2 = i and it works out to give the right values for A and B which I suppose seems obvious now that she told me. I still don't know how mathematically sound that is but if it works, it works. I'm not in the mood to investigate anymore after the realisation that I can't even do partial fractions at times . Stupid partial fractions. Grr.
    First see how many z^2 there are:

    

A+B=1

    then do the same with the constants:

    

Ai-Bi=0

    solve the simultaneous equations.
 
 
 
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