Number of Zeros of a Complex Function

Question

I need help solving this problem. I think it involves Rouche's Theorem but I am not sure.

Determine the number of zeros of the function in the upper half plane. $f(z)=z^4+3iz^2+z-2+i$

half upper plane means that imaginary part $y$ should be positive,so in your case $y=(3*z^2+1)$ — dato datuashvili, Apr 03 '13 at 05:37
here one problem is that how can we determine or how can we use $i$ here?for example we may write $f(z)=z^4+i(3*z^2+1)+z-2$ — dato datuashvili, Apr 03 '13 at 05:50
may also this help http://math.stackexchange.com/questions/81254/number-of-zeros-of-a-complex-polynomial — dato datuashvili, Apr 03 '13 at 06:25

score 0 · Answer 1 · answered Apr 03 '13 at 05:56

so in you case we have

$f(z)=z^4+i(3*z^2+1)+z-2$

upper half plane as i said is that iamginary part is positive,but first let say that we can conclude we may have two function like first $f(z)=z^4+z-2$ and $g(z)=i(3*z^2+1)$

or we could say that we have tree function

$f(z)=z^4$

$f_1(z)=i(3*z^2+1)$

$f_2(z)=z-2$

if we measure this inside simple controur for example when $|z|=1$ then we may have

$|f(z)|=1$

$f_1(z)=4$

$|f_2(z)|=-1$

so we are taking positive and minimum right because of this pdf file http://fbhs.snnu.edu.cn/kcwz/all/dianzijiaoan/chn7/7.9.pdf

so it should be $1$

or in case of two function if we concatenate first two function we will get $5$,so i think it should be for two function $5$

so please if something is incorrect,tell me and i will update,i want just to help him — dato datuashvili, Apr 03 '13 at 06:02

Number of Zeros of a Complex Function

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