2
votes
2answers
64 views

Find $\int_\Gamma\frac{2z+j}{z^3(z^2+1)}\mathrm{d}z$ where $Γ:|z-1-i| = 2$

pls, some ideas for integral solution (residue theory)? $$\int_\Gamma\dfrac{2z+j}{z^3(z^2+1)}\mathrm{d}z$$ Where $Γ:|z-1-i| = 2$ is positively oriented circle. Thx, for help!
1
vote
1answer
58 views

Cauchy's integral formula used on circle

If $\gamma$ is a piecewise, smooth, positively oriented simple closed curve in $D$, then Cauchy's formula states that $f(z)=1/2\pi i\int_\gamma {f(a)\over {a-z}}$. My textbook also stated that for ...
2
votes
1answer
54 views

Let C be a circle. Show that the only subset of C homeomorphic to a circle is C itself

I am trying to answer the question stated in the title. The hint in my book says to realize that for any z on the circle C{z} is still connected. I believe I can deal with case that shows that C{z} ...
0
votes
2answers
93 views

Paramtrizing a counterclockwise circle vs. a clockwise one

Does it make a different when you parametrize a counterclockwise full circle and a clockwise circle in the complex plane? For example, I am looking at computing an integral $\int_\gamma ...
0
votes
1answer
243 views

determine if pole is inside unit circle

i would like to know how to determine if pole of given function is inside unit circle contour? for example let us take this function $f(z)=(i-1)/(z+i)$ and we have contour ...
0
votes
1answer
95 views

Möbius map of two circles to half planes

I am very new to complex analysis and am having some trouble with finding a Möbius map that will take two unit discs to half-planes. I don't have enough reputation to post images, but here is a link ...
1
vote
3answers
79 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
100 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
2answers
308 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
134 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
451 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
284 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 ...
3
votes
2answers
88 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 ...