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4
votes
2answers
61 views
Calculating integral with branch cut.
I'm learning how to calculate integrals with branch points using branch cut. For example:
$$I=a\int_{\xi_{1}}^{\xi_{2}}\frac{d\xi}{(1+\xi^{2})\sqrt{\frac{2}{m}\left(E-U_{0}\xi^{2}\right)}}$$
where ...
2
votes
2answers
118 views
difficulty understanding branch of the logarithm
Here is one past qual question,
Prove that the function $\log(z+ \sqrt{z^2-1})$ can be defined to be analytic on the domain $\mathbb{C} \setminus (-\infty,1]$
(Hint: start by defining an appropriate ...
0
votes
1answer
63 views
Square root principle value convention
Why is the principal square root of a complex number defined as
$\sqrt z = \sqrt r e^{-i \varphi / 2}$
for $\varphi \in (-\pi, \pi]$ ?
Wouldn't it be more natural to let $\varphi \in [0, 2\pi)$ as it ...
0
votes
0answers
180 views
Complex analysis, branch cut question
1) evaluate $F(z) = -i(z-iR)/(z+iR)$, $R$ positive and real, then find $f(z)=1/\pi\ln{F(z)}=u(x,y) + iv(x, y)$;
2) Make branch cuts of $f(z)$ s.t. $f(z)$ i single value inside $|z|=R$;
3) show $v(x, ...
1
vote
4answers
225 views
Proving $\sqrt{2z-2\log(z)-2}$ is analytic near $z=1$.
I am trying to prove $f(z)=\sqrt{2z-2\log(z)-2}$ is analytic near $z=1$. The issue is proving there is no branch point.
If I try the approach of taking the path $z=1+r\exp(i\theta)$ with $r=\epsilon$ ...
4
votes
1answer
78 views
How do we know how many branches the inverse function of an elementary function has?
How do we know how many branches the inverse function of an elementary function has ?
For instance Lambert W function. How do we know how many branches it has at e.g. $z=-0.5$ , $z=0$ , $z=0.5$ or ...
4
votes
2answers
577 views
Branch cut for $\sqrt{1-z^2}$ - Can I use the branch cut of $\sqrt{z}$?
I was trying to clarify some questions I had about elliptic integrals using
http://websites.math.leidenuniv.nl/algebra/ellcurves.pdf
There they define the map
$$\phi\colon w\mapsto ...
0
votes
3answers
93 views
Improper integration using complex methods
Sorry for my English if there is any mistake. The exercice for which I need help is the following:
Compute using complex methods:
$I=\int_1 ^\infty \frac{\mathrm{d}x}{x^2+1}$
i) Choose the complex ...
2
votes
1answer
538 views
Complex Analysis - Question about branch cuts
I am having trouble understanding how branch cuts work. For example, the function $f(z)=
\sqrt{z}$ has a branch cut where you reject the negative real axis. But how do you define the output so that ...
2
votes
1answer
161 views
Analytic branches of $z^{-i}$.
How to describe all the branches of the function $z^{-i}$, analytic in the whole complex plane except the positive real axis?
I consider $z^{-i}=e^{-i \log z}$ and the branch becomes whole complex ...
3
votes
1answer
275 views
summation of an infinite series involving arctan
I'm having problems with the following calculation.
Let $a >0$
$$
\begin{align}
& \sum_{n=1}^\infty \arctan \left(\frac{2a^2}{n^2}\right) = \text{Im} \sum_{n=1}^\infty \log \left( 1 + ...
1
vote
1answer
131 views
How to calculate $\int_{|z|=r}\ln(1-z)\,dz$ in dependence of $r\neq1$?
With the integration I mean one counter-clockwise turn around the origin, i.e.
$$\int_{\phi=0}^{2\pi}\ln(1-re^{i\phi})ire^{i\phi}d\phi$$
For $r<1$, this is simply a contour integration on a ...
