Christian Ivicevic
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 Apr 25 comment Show that a power series is analytic inside its radius of convergence I really like your last explanation - at least it is more formal than what I had in mind thus far. What do you think of the addition that we are looking for variables $z$ close to our new center $z_0$ and thus using the triangle inequality holds for those cases? Prosaic: your example would be false in general as your mentioned point $z=3$ wouldn't be in the circle as it is too small since the outer radius isn't allowing it to be bigger. Apr 24 comment $(x-x_0)^0$ in power series Whoever just downvoted me: would you be so kind as to explain why? Maybe I can improve my answer. Apr 24 comment Holomorphic, entire functions and the Cauchy-Riemann equations @Ant So ignoring $c_1$ and $c_2$ in 2. I am getting $f(z)=-\mathrm iz^2$ which seems to be correct - I assume you started with a polynomial of degree 2 since there are no higher degrees in my representation of $f(x+\mathrm iy)$, didn't you? Apr 24 comment Holomorphic, entire functions and the Cauchy-Riemann equations @Ant So it is only $\mathbb C$-differentiable at $z=0$ but there is no holomorphic restriction, then? Apr 24 comment Show that a power series is analytic inside its radius of convergence My newest result is that $f(z)$ converges for \$|z|