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Mar
21
comment Is $\sum_{n=1}^\infty \sin\left(\frac{n\pi}{4}\right)$ convergent?
It is the first result that is proved to you in any calculus course just after defining what is a series of real numbers. If $s_n$ is the sequence of partial sums ($s_n=a_0+..+a_n$), the sum converges to $S$ if and only if $s_n\to S$, hence $a_n=s_n-s_{n-1}\to (S-S)=0$.
Mar
21
revised How to evaluate $\frac{1}{2\pi i}\int_C \frac{f(z) dz}{(z-z_1)^{m_1}(z-z_2)^{m_2}}$ and the other related contour?
added 4 characters in body
Mar
20
answered How to evaluate $\frac{1}{2\pi i}\int_C \frac{f(z) dz}{(z-z_1)^{m_1}(z-z_2)^{m_2}}$ and the other related contour?
Mar
19
comment Unique continuous complex log of a function nowhere zero
Probably in this book springer.com/birkhauser/mathematics/book/978-0-8176-4164-1 there is enough stuff for your needs.
Mar
16
comment Unique continuous complex log of a function nowhere zero
You don't select any branch. You just go on along the path defining the logarithm by analytic continuation...
Mar
15
answered Let $f$ be a meromorphic function on $\mathbb C$ such that $|f(z)|\ge |z|$
Mar
15
comment Relation between being holomorphic in $\Delta\times\Delta$ and in every relatively compact polydisk.
If I guess correctly, this is a step in the proof of Hartogs theorem about separate analyticity. There you take a domain $U$ and prove that $f$ is holomorphic on every relatively compact polydisc. If this is the situation, you have that the hypotheses hold in a neighbourhood of the polydisc, so that you can enlarge the strip a bit.
Mar
15
comment Unique continuous complex log of a function nowhere zero
I added an equivalent approach, I hope more likely to suit your needs.
Mar
15
revised Unique continuous complex log of a function nowhere zero
Added an alternative approach
Mar
15
comment Unique continuous complex log of a function nowhere zero
sorry, in my undergrad course we had fundamental groups and covering spaces before complex analysis...
Mar
15
answered Unique continuous complex log of a function nowhere zero
Mar
14
comment Examples of complex manifolds satisfying the Kodaira-Spencer condition for unobstructed deformations
Calabi Yau manifolds should do, but I don't have a reference right now ... I will look it up
Mar
12
comment Differential equation of the form to find
so, if, for instance $f(z)=z^2$, then $f(z\frac{\partial}{\partial z})=z\frac{\partial}{\partial z} + z^2\frac{\partial^2}{\partial z^2}$ ?
Mar
11
comment Is there a name for these sets of functions of several complex variables (most not analytic)?
uhm, homogeneous polynomials of degree $n$ in $2n$ variables with complex coefficients, with monomial containing every variable with power at most $1$?
Mar
11
comment Differential equation of the form to find
sorry, what is the meaning of $f(z\frac{\partial}{\partial z})$?
Mar
9
awarded  Organizer
Mar
9
revised Constructing a complex function without limits along certain curves.
edited tags
Mar
8
answered If $A^TA$ is invertible, then $A$ has linearly independent column vectors
Mar
7
answered Generalisation of Schwarz lemma
Mar
7
comment Generalisation of Schwarz lemma
Sorry, your statement of the Schwarz Lemma is somewhat strange. Shouldn't it be "if $f:U\to U$ is holomorphic and $f(0)=0$, then $|f(z)|\leq |z|$ for every $z\in U$ and $|f'(0)|\leq 1$"?