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Thread: complex values and residuee theorem

  1. #1
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    complex values and residuee theorem

    I have two questions, I have tried to find stuff on them but failed
    (1) find all complex values z such that e^(z+2)=1+i

    I tried to put z=log(1+i)-2 but dont think thats not right

    (2)use the residue theorem to evaluate
    integralc(tanz)dz where c is the positively orientated circle |z|=2

    i thought with the residue theorem you had to have a fraction so you could set the bottom to zero and get a singularity. I thought you could change tanz into sinz/cosz and set cosz to zero. but thats wrong

    any help appreciated on either question
    thanks
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by eleahy View Post
    I have two questions, I have tried to find stuff on them but failed
    (1) find all complex values z such that e^(z+2)=1+i

    I tried to put z=log(1+i)-2 but dont think thats not right
    It's right, assuming you realize that $\displaystyle \log(1+i)$ is a set, and not a number.


    (2)use the residue theorem to evaluate
    integralc(tanz)dz where c is the positively orientated circle |z|=2

    i thought with the residue theorem you had to have a fraction so you could set the bottom to zero and get a singularity. I thought you could change tanz into sinz/cosz and set cosz to zero. but thats wrong

    any help appreciated on either question
    thanks
    Note that $\displaystyle \tan(z)$ has simple poles at $\displaystyle \pi\left(k+\frac{1}{2}\right)$ for all $\displaystyle k\in\mathbb{Z}$. Thus, find the number of them lying inside the disc $\displaystyle D(0,2)$ and use the residue theorem which says then that, if $\displaystyle p_1,\cdots,p_n$ are those finitely many poles then $\displaystyle \displaystyle \oint_{|z|=2}\tan(z)=2\pi\sum_j \text{Res}(\tan(z),p_j)$ and since each pole is simple you need only calculate the residue by $\displaystyle \text{Res}(\tan(z),p_j)=\lim_{z\to p_j}(z-p_j)\tan(z)$
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  3. #3
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    Calculating the residues can be done like this:


    To get the answer from the final part you can use the Maclaurin series for sin.
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