mrf
Reputation
31,595
99/100 score
 1d reviewed Looks OK Showing epimorphism without using the Freyd-Mitchell Embedding Theorem 2d awarded Nice Answer 2d comment show the complete Proof of the proposition 4.3.6 from the book “Basic Complex Analysis” What part(s) are you having trouble with? 2d answered Using the Inverse Function Theorem for complex functions 2d revised Using the Inverse Function Theorem for complex functions deleted 24 characters in body May 3 comment Is $f'(z)$ defined on $A\subseteq \mathbb{C}$, if $f_n'\to f'$ uniformly? @Hopeless Uniformly on $A$ is stronger than uniformly on every compact subset $K$ of $A$. Example: take $A$ as the unit disc, and $f_n(z) = z^n$, then $f_n$ converges uniformly (to $0$) on every compact subset of the disc, but not on the disc itself May 3 revised Applications of Rouché's theorem edited body May 3 revised Applications of Rouché's theorem edited body; edited title May 3 comment Complex integration problem via Cauchy's integral formula @Ant Probably because (the usual formulation of) Cauchy's integral formula requires the integrand to be of the form $$\frac{f(z)}{z-z_0}$$ where $f$ is holomorphic. In particular, the integrand can only have one pole inside the curve. To use Cauchy's integral formula here, we need to split the curve or the function. (Of course, once we have residue calculus there are quicker approaches.) May 3 answered Is $f'(z)$ defined on $A\subseteq \mathbb{C}$, if $f_n'\to f'$ uniformly? May 3 revised Complex integration problem via Cauchy's integral formula added 5 characters in body; edited title May 3 comment Compute $\int_{\Gamma} z^{-2} dz$ where $\Gamma=C_1(0)$ with its usual (counterclockwise) parametrization. You don't need the existence of an anti-derivative everywhere inside $\Gamma$. May 3 answered Compute $\int_{\Gamma} z^{-2} dz$ where $\Gamma=C_1(0)$ with its usual (counterclockwise) parametrization. May 2 comment Verifying the result of this complex integral (not integrable analytically) The poles can be expressed in terms of Lambert's W-function. I asked Maple to give me a numerical approximation. May 2 comment Verifying the result of this complex integral (not integrable analytically) It's not entirely trivial. Some estimation is required. (There are certainly other poles. For example close to $2.0888 \pm 7.4615i$.) May 2 comment Verifying the result of this complex integral (not integrable analytically) You should probably also argue why $z=0$ is the only pole inside the unit circle. Apr 28 comment Investigate complex integral on the circle There are two singularities inside $|z|=2$. Either start with a partial fractions decomposition or split the curve into two pieces, each enclosing just one of the singularities. (Once you know residues, there are quicker approaches.) Apr 28 comment Complex Analysis with absolute integral Then the inequality is not true. The value of the integral is $\approx 1.63$. (The integrand is positive and $> 1$ on $(0,1]$.) Apr 28 comment Complex Analysis with absolute integral What does $f$ have to do with the integral? Apr 28 comment Expanding a complex function in Taylor series Start with doing partial fractions decomposition.