Tagged Questions
1
vote
3answers
70 views
Another complex analysis question
I am going to have an analysis exam soon and I found the following question in a past paper:
Evaluate the integral counterclockwise
$$\int |z| \overline{z} \, dz$$
where y is the closed curve ...
1
vote
1answer
90 views
Analysis Exam Questions
I am going to have an analysis exam soon and I found the following question in a past paper:
Evaluate
$$\int \frac{-y \, dx + x \, dy}{x^2+y^2}$$
a) Once counterclockwise around the circle $$x^2 + ...
1
vote
3answers
194 views
Circle in the complex plane
Show analytically (finding the centre and radius) that $z(t)=\frac{1}{(1-i)^{-1}-t}=\frac{2}{1+i-2t}$ where $z(t)\in C $, that $z(t)$ traces out a circle in the complex plane as $t$ is varied.
2
votes
1answer
85 views
What are the subsets of the unit circle that can be the points in which a power series is convergent?
Let $A\subset\Bbb C$ be a subset of the unit circle. Consider the following condition on $A$.
Cond. There exists a sequence $\{a_i\}_{i=1}^\infty$ of complex numbers such that $$\sum_{n=1}^\infty ...
2
votes
2answers
198 views
circles and linear fractional transformations
I'm realizing how little (in some respects) I know about circles. Here's something that emerged out of something I was fiddling with. My question is whether this is
"well known" in the way that ...
5
votes
1answer
232 views
Extension of real analytic map on the unit circle
Given a real-analytic map $f : \mathbb{S}^1 \rightarrow \mathbb{S}^1$, where
$$\mathbb{S}^1 = \{z \in \mathbb{C} : |z| = 1\},$$
does it admit a complex-analytic extension $\tilde{f} : U \rightarrow ...
2
votes
2answers
61 views
every point on boundary of region of convergence is singular
I am given the following function:
$$f(z)=1+z^2+z^4+z^8+z^{16}+ \cdots$$
and shall show that it is holomorphic in the unit disc, that $f\to\infty$ as $z\to e^{2i\pi/2^n}$, and that every point on ...
