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Math Help - Residues - complex analysis confusion.

  1. #1
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    Residues - complex analysis confusion.

    Hi! I am having a hard time with residues. I understand the formal definition of a residue, but anything past that and I am struggling. My course lecturer has a very confusing way of organising the course material and it is very hard to comprehend so was wondering if anyone could help with these questions or direct me to another source which would help explain what to do?

    part i: Calculate res(f, 0) of an even meromorphic function f which has a (unspecified) pole at 0.

    part ii: Can anything be deduced about res(f,0) for an odd meromorphic function f with an unspecified pole at 0? Justify your answer.

    Part iii:
    Find \int^{\infty}_{-\infty}\frac{cosx}{x^2 + 2x + 4}\,dx and  \int^{\infty}_{-\infty}\frac{sinx}{x^2 + 2x + 4}\,dx

    Any help would be much appreciated.
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  2. #2
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    Re: Residues - complex analysis confusion.

    Just an update, I have figured out part i and ii but still wondering what to do with part iii. I'm assuming I have to use the results from part i and ii somehow.
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  3. #3
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    Re: Residues - complex analysis confusion.

    Τhe two integrals are equal with the real and imaginary parts of \lim_{R\rightarrow+\infty}\int_{c(R)}\frac{e^{iz}}  {z^2-2z+4}dz, where c(R)
    is a suitable curve you can consider, such that it contains the poles (so you can use the remainder theorem) and diminishes into the real axis for R\rightarrow+\infty.
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