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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.