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Find the domain of convergence of :
$\displaystyle\sum_{n=1}^{\infty} \frac{e^{in}}{(z+1)^n} +\sum_{n=0}^{\infty} \frac{(z+1)^n}{e^{\frac{1}{2}+n}}\ \ \ \ (z\in\mathbb{C})$
I've found that it diverges for any complex number $z$, is this correct ?
I'm new in the complex analysis and I'm stuck with this integral :
$I=\displaystyle \int_{|z|=4} \frac{\mathrm{d}z}{(z^2+9)(z+9)} $
the exercise is about Cauchy integral, I don't want the whole solution, just give me a hint (Please don't post fully worked solutions)
what I have done : using partial fraction we get :
$I=\displaystyle \frac{1}{90} \int_{|z|=4} \frac{1}{9+z} + \frac{(9-z)}{9+z^2} \mathrm{d}z$
I'm trying to do this : $\displaystyle \int_{|z|=4} \frac{\mathrm{d}z}{9+z} $.
but $|9|>4$, how to do the integration in this case ?
Studying some Newtonian mechanics, I've encountered this differential equation :
$y'+a y^2=b$
where $a,b$ are constants.
how could we solve it ? (I trying to get an algebraic solution)
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