Skip to main content

Questions tagged [branch-points]

A branch point is a point in the complex that can map from a single point to multiple points in the range.

Filter by
Sorted by
Tagged with
1 vote
0 answers
23 views

Branch Points and Branch Line of the Function $f(z) = \cos^{-1}z$

I know the branch line and branch points of common function like $\log{z}$ and aware of the definitions. But what's the general approach here to find branch line and branch points of any multivalued ...
mat09's user avatar
  • 157
1 vote
0 answers
40 views

Integral involving a simple pole and 4 branch points, pairwise in-and-out of the contour

For some time lately, I am facing difficulty in evaluating the following integral coming from a physics context: $$I = \frac{1}{2 \pi}\int^{\pi}_{-\pi} \frac{h-\cos k}{\sqrt{(h-\cos k)^2+\gamma^2 \sin^...
prikarsartam's user avatar
0 votes
1 answer
60 views

Complex Integral and Residue involving multiple Branch Points inside the contour

I encountered the integral: $$\oint_{|z|=1} \frac{f(z) \ dz}{\sqrt{(z-a)(z-b)}} \ \ \ \ \text{with} \ \ \ |a|,|b| < 1$$ So that the branch points are inside the contour. I am not adding the ...
prikarsartam's user avatar
0 votes
1 answer
59 views

Branch points of $\sqrt{z-1}$

Let, $f(z)=\sqrt{z-1}$ I am reading the book "Visual complex analysis" by Tristan Needham and after reading the section of branch points and branch cuts, I gave myself this definition of a ...
RAHUL 's user avatar
  • 1,521
2 votes
0 answers
26 views

totally ramified covering of $\mathbb{P}^1$ with 3 branch points.

I'm trying to construct a totally ramified covering (of order $d$) of the complex projective line $\mathbb{P}^1$ with exactly 3 branch points $0,1,\infty$. Here is what I'm trying: consider the smooth ...
S.Gau at Math's user avatar
1 vote
1 answer
37 views

Question About Existence Of Branch For $(z^2 -1)^{ \frac12}$.

Consider the function $f :z\mapsto (z^2 -1)^{ \frac12}$. Now Here proved that it has branch such that $f$ is analytic in $|z|>1$.Where branch cut is $[-1,1]$. Now f is composition of two function ...
Meet Patel's user avatar
0 votes
1 answer
77 views

Rigorous proof that $\sqrt{z}$ has a branch point at zero.

Here is the definitions I am working with Define the map $$ (z-1)^{\frac{1}{2}}$$ defined as the inverse of $$\begin{align} f: \mathbb{C} & \rightarrow \mathbb{C} \\ z &\mapsto z^{2}+1 \end{...
Maths Wizzard's user avatar
0 votes
0 answers
40 views

When working with multiple branch cuts, is there a way to chose the arguments so that log of a product can be opened as sum of individual logarithms?

Suppose a function $\eta (z)=log(\psi (z))$ where $$\psi (z)=\prod_{k=1}^{n} \left(z-z_k\right)$$ We know that $log(z)=log|z|+i(argz)$, this implies that $$log(\prod_{k=1}^{n} \left(z-z_k\right))=\...
Madhav Asthana's user avatar
0 votes
1 answer
44 views

An improper integral of an inverse of a square root of a higher degree polynomial.

Let $1 \le n_1 < n $ and $ n\ge 3$ be integers and let ${\bf \lambda}= \left( \lambda_j \right)_{j=1}^n \in {\mathbb R}$ such that $\lambda_j > 0 $ for $j=1,\cdots, n_1$ and $\lambda_j <0 $...
Przemo's user avatar
  • 11.5k
0 votes
1 answer
45 views

What is the branch cut of composite of multivalued complex function

I have the following function where I want to identify the Riemann surface. $$ f(z)=\log\left(\sqrt{z^2+1}\right). \quad\quad\quad (1) $$ The square root function has a Riemann surface $R_{SR}$ with ...
user239970's user avatar
1 vote
0 answers
31 views

Radius of convergence at $2+i$ of $f_i(z)=\frac{1}{\phi_i-2^{1/4}}$ with $2^{1/4}$ being the positive real root

Let $\phi_k(z), k=0,1,2,3$ the branch cuts of $z^{1/4}$. Consider $$f_k(z)=\frac{1}{\phi_k-2^{1/4}}, \quad 2^{1/4}=|2|^{1/4}e^{i0}>0$$ Find the radius of convergence of the series expansion at $2+i$...
Mateo's user avatar
  • 63
3 votes
1 answer
113 views

Branch cut in integral function

I'm not very versed in complex analysis and I'm trying to understand some concepts on branch cuts and contour integration. Consider a function $$ I(s)=\int_0^1 d\alpha\ \frac{1}{f(s,\alpha)}, $$ such ...
Marcosko's user avatar
  • 175
1 vote
0 answers
35 views

Riemann surface of the hypergeometric function

The hypergeometric function $_2F_1(a,b,c,z)$ has a branch cut extending from $z=1$ to $z=\infty$. Does this define an infinite-sheeted Riemann surface (like that for $\log{z}$) or one with a finite ...
mp62442's user avatar
  • 35
0 votes
0 answers
30 views

Branch cut of $\log(z^2-1)$ if $\arg(z^2-1)\in\left(-\frac{7\pi}{4}, \frac{\pi}{4}\right]$

Using the principal branch $\arg(\xi(z))\in\left(-\dfrac{7\pi}{4}, \dfrac{\pi}{4}\right]$ for $$f(z)=\log(z^2-1)=\log(\xi(z)),$$ what's its branch cut's equation/how does it look like? I was doing the ...
Conreu's user avatar
  • 2,578
1 vote
0 answers
40 views

Branch cuts of $\log(z^2-1)$ for unusual principal values

Say we got a multivalued function $$\log(z^2-1)\equiv\log(\xi(z)).$$ Usually, we would choose principal values such as $\arg\xi(z)\in[0,2\pi)$ or $\arg\xi(z)\in[-\pi,\pi)$. In order to find the branch ...
Conreu's user avatar
  • 2,578
1 vote
1 answer
92 views

Local form of holomorphic functions near a zero.

My question is one from complex analysis.Suppose $f(z)$ is a holomorphic function in a neighborhood of $z_0$ and that $f(z_0)=f'(z_0)=0$ and $f''(z_0)\neq 0$ that is,$f$ has a zero of order $2$ i.e. $...
Kishalay Sarkar's user avatar
1 vote
1 answer
83 views

why including the origin in a closed path causes discontinuity in complex functions like the square root?

I am taking a complex analysis course and the professor confused me on the following matter: On one hand, given a complex function like $f(z)=z^{1/2}$, when we complete a closed circuit around the ...
R24698's user avatar
  • 389
1 vote
1 answer
104 views

show that $z = 0$ is a branch point of a function

I have the following function: $$f(z) = \left( \frac{z}{z+x_0} \right)^q,$$ where $q \in \mathbb Q, \ q = \frac{p}{r}$ with $p \neq r$ and $r \neq 0$ and $x_0 \in \mathbb R$, $x_0 > 0$. I want to ...
syphracos's user avatar
  • 486
1 vote
1 answer
217 views

Branch cuts for $\sqrt{z(z-1)}$

I am studying for quals and found this question in a previous exam: Find a branch cut for $\sqrt{z(z-1)}$ that is analytic in $\mathbb{C} \> \setminus [0, 1]$ and takes the value of $-\sqrt{2}$ at ...
Jabbath's user avatar
  • 274
0 votes
0 answers
23 views

Conjugating Branch Points $f=ghg^{-1}$

Consider the function $f:=ghg^{-1}$ on $\widetilde{\mathbb{C}}$ where $g$ is a homeomorphism and $h$ is a rational map. Why is it true that branch points of $h$ are transformed by $g$ to removeable ...
OllyT777's user avatar
  • 125
4 votes
1 answer
103 views

Writing $(-z)^\alpha$ in terms of $z^\alpha$

I have a very silly confusion in complex analysis. Let $z\in \mathbb{C}$ and $\alpha\in \mathbb{C}$. Now consider $(-z)^\alpha$ and suppose that we want to relate this to $z^\alpha$. One thing that I ...
user avatar
0 votes
0 answers
23 views

How to find vanishing order of coefficients of Newton polynomial?

I'm given the curve $y^3=x^3-1$ and I want to find the genus. I have a ramified cover $\pi: Y \to \mathbb{P}^1$, where $Y$ is given by the curve. I know that $1, \omega, \omega^2$ are the $x$ values ...
cheeseboardqueen's user avatar
0 votes
0 answers
30 views

Find the image of an analytic branch $f\left(2i\right)=2+i\cdot \left(\frac{9}{2}\pi \right)$, means, find $I_m\left(f\right)=?$

had a test a few days ago and was not sure about my solution. Please check it out and tell me if I was wrong, none I asked knew what to do \ others did like me. EDIT: ( I really hope I am not lying ...
LearningToCode's user avatar
3 votes
2 answers
440 views

Is the textbook WRONG on the branch cut?

As defined in the textbook, it takes the branch cut along positive real axis. So the argument angle is $[0,2\pi)$. This is no problem. But in the example, it sets $f(z)=\frac1{\sqrt{2-z}}$ and claims ...
user avatar
1 vote
1 answer
85 views

Approaching the branch cut and the limit of $f(x+iy)$ when $y\to0+$

Consider complex function $f(z)=\sqrt{z(z-1)}$, here $z\neq0,1$. I referred to similar questions, found branch cut and branch points. The branch points are $z=0,1$. According to the definition of ...
Cunyi Nan's user avatar
  • 742
0 votes
0 answers
45 views

Evaluating roots of complex-valued functions [duplicate]

I have seen a lot of arguments recently where, for instance: $$ \sqrt{z^2-1} = \sqrt{z-1} \cdot \sqrt{z+1} \hspace{5mm} (*) $$ without ever specifying the chosen branch or the chosen branch cut (these ...
Matteo Menghini's user avatar
0 votes
2 answers
195 views

what are the branch points and branches of $g(z)=(z+ \sqrt{z})^{1/3}$?

And what if we for example shifted one of the roots, eg $f(z)=(z+ \sqrt{z-3})^{1/3}$? I already asked a more extensive version of this question here Branch cut/ points for square roots inside cubic ...
Noam's user avatar
  • 67
0 votes
0 answers
92 views

Branch cut/ points for square roots inside cubic roots- incorrect branching by mathematica or my mistake?

There's a lot of great information here about understanding the branch cuts and branch points of functions of the form ( for example ) $(z^3+1)^{1/2}$, sums of simple roots and products thereof. ...
Noam's user avatar
  • 67
0 votes
0 answers
26 views

How to find the branch points of this logarithmic function on a complex domain

Now that I know that the branch points of a function $\sqrt[n]{P(z)}$ can be determined by the integral division of the zero points of $P(z)$ by n. Then how to find the branch points of this ...
aoi chiyuki's user avatar
1 vote
0 answers
74 views

Branch points of $\sqrt{\cos(z)}$.

I’m asking this question because I’m trying to write the series expansion of $\sqrt{\cos(z)}$ about $0$, and it seems problematic because its series expansions are not unique (if I’m doing the ...
Dfgvjighgdrg's user avatar
0 votes
1 answer
89 views

How do you show that branch cuts result in a single valued function for logarithmic branch points?

I know the way that basically everywhere says to do it, but I'm having trouble putting it into equations that make sense for logarithmic branch points. Everywhere says to trace a path around the ...
Sidereus's user avatar
0 votes
1 answer
114 views

Branch Cut related confusion

Suppose I have some function like $f(z)=(z^2-1)^{1/3}$ and I know that it has branch cuts at $\pm1$. Suppose, I'm integrating, and I care about the value of the function on the contour, that I'm ...
Nakshatra Gangopadhay's user avatar
0 votes
0 answers
63 views

If f(z) has a branch point at z=0, what does 1/f(z) have at z=0?

If f(z) has a pole of order n at z=0 then 1/f(z) has a zero of order n at z=0. Is there an equivalent for if f(z) has a branch point at z=0 instead? If f(z) has a branch point at z=0, what does 1/f(z) ...
MedusaOblongata's user avatar
0 votes
0 answers
25 views

Determine complex exponent to make multi-valued function negative

I have the following complex function and I would like to make it so that $g_k(z=20) = -\frac{1}{75}$ where $z+z_0=\rho \exp(j(\theta+2 k \pi))$. $$g(z)=\frac{1}{\sqrt{z+5} (z-5)}$$ I know that $z_0=-...
Eind997's user avatar
  • 113
0 votes
1 answer
60 views

Finding branch cut of special function

I am looking for the branch cut of this function \begin{equation} f(z) = \sqrt{3-\sin^2{z}} \end{equation} I have found the branch points of this equation as \begin{equation} z_{b1} = \frac{\pi}{2} + ...
Ta Quang Ngoc's user avatar
5 votes
0 answers
147 views

Compute $\int_{0}^{2\pi}\frac{e^{ikt}}{|e^{it}-e^{it_0}|^m}~\text{d}t$, where $k\in\mathbb{Z}$, $t_0\in\mathbb{C}$, and $m=1,3,5,\dots$

I want to compute $$ \int_{0}^{2\pi}\frac{{\rm e}^{{\rm i}kt}}{\,\left\vert\,{{\rm e}^{{\rm i}t} - {\rm e}^{{\rm i}t_{0}}}\,\,\right\vert^{\, m}\,}\,{\rm d}t, $$ where $k\in\mathbb{Z}$, $t_{0}\in\...
David's user avatar
  • 51
1 vote
0 answers
33 views

Neighborhood of infinity containing a Jordan Curve

I just recently started a chapter on elementary multi-valued functions of a complex variable, and one part of the book text a bit confuses me. To give a little context, the segment covers branch ...
LoLLipop's user avatar
2 votes
0 answers
99 views

Branch point "Visual Complex Analysis" visualization confusion

I have a question regarding a visualization in Tristan Needham's Visual Complex Analysis. Now let us return to [29]. By arbitrarily picking one of the three values of $\sqrt[3]{p}$ at $z=p$, and then ...
random0620's user avatar
  • 2,971
3 votes
0 answers
87 views

Do branch cuts always connect branch points?

While doing some exercises on complex functions, I noticed that every time I find a branch cut in the complex plane it is always in between two branch points, such as in the function \begin{equation} ...
M_Vezz's user avatar
  • 31
1 vote
1 answer
564 views

Branch cuts and continuity of $f(z) = \sqrt{z^2-1}$ while visualizing it

The branch cuts of $f(z) = \sqrt{z^2-1}$ have been discussed on this site before. However, there are certain things I still do not understand. I'm following the discussion in Morse and Feshbach's ...
B215826's user avatar
  • 325
0 votes
1 answer
63 views

Confusion about branches in exponents

I understand that the function $f(z)=\sqrt{z}$ hast two branches and two branch points, ($z=0,z=\infty$). I also get that how more generally, $f(z)=\sqrt[n]{z}$ has the same branch points but has $n$ ...
Nobody's user avatar
  • 343
0 votes
1 answer
979 views

Understanding branch cut of the following function $(z^3+1)^{1/3}$

I'm trying to understand branch cuts in complex analysis. Let's say I have the following function $$f(z)=(z^3+1)^{1/3}.$$ I see that the branch points in this case are the solutions to the equation $z^...
NabbKitha's user avatar
  • 540
1 vote
1 answer
126 views

Branch points of $\sqrt{z^2}$

When I was solving a problem in complex analysis, I encountered the function $\sqrt{z^2}$. I am wondering if this function has a branch point. My attempt The definition of a branch point given by our ...
Kaira's user avatar
  • 1,565
1 vote
0 answers
56 views

Verify whether point is branch point

I need to check whether $z=\infty$ is a branch point of the function $$f(z)=\sqrt[p]{(z-a_1)(z-a_2)}$$ where $p \in \mathbb{N}$ and $a_i \in \mathbb{C}$ for $i=1,2$. I know that we could make the ...
Barreto's user avatar
  • 866
1 vote
0 answers
26 views

Integration along a contour containing a branch cut

Consider the following integral $I = \int_{-\sqrt{b^{2} + a^{2}}}^{\sqrt{b^{2}+a^{2}}} (a^{2} - z^{2})^{\lambda}e^{- i \omega z}\mathrm{d}z$ In this integral, $a$, $b$ and $\omega$ are real numbers. ...
BB_'s user avatar
  • 87
0 votes
1 answer
252 views

Determine a branch of $f(z)=\log(2iz−z^2)$

Determine a branch of $f(z)=\log(2iz-z^2)$ that is analytic at $z=1$. Then find $f(1)$ and $f'(1)$. First we note that $g(z)=2iz-z^2$ and recall $\mathcal{L}_\tau:=\log|z|+i\arg_\tau z$. So a branch ...
help's user avatar
  • 607
1 vote
0 answers
67 views

How to find branch cuts and branch points of $\sqrt(z^a +1)$

I have tried to find out the branch cuts and branch points of the function $\sqrt(z^a +1)$, where $0<a<2$. The function can be written as $e^{\frac{1}{2}(\log(z^a+1))}$ and from here I have ...
Le0's user avatar
  • 61
1 vote
0 answers
74 views

Proving that : $Q_0(z)=\frac{1}{2}\ln\left(\frac{z+1}{z-1}\right)$ is single valued.

The Legendre function of second kind $Q_\nu(z)$ has branch points at $z=\pm 1$. The branch points are joined by a cut along the real axis. Show that $$Q_0(z)=\frac{1}{2}\ln\left(\frac{z+1}{z-1}\right)$...
Young Kindaichi's user avatar
2 votes
0 answers
146 views

Is there an in depth classification of branch points in complex analysis?

Disclaimer: I have posted this question on mathoverflow.net following the instructions of this topic. In complex analysis we have well know results about isolated singularities. Poles are ...
Diego Santos's user avatar
  • 1,283
0 votes
2 answers
274 views

How to determine if $z+\sqrt{z-1}$ and $\frac{\sin{\sqrt{z}}}{\sqrt{z}}$ is a multi-valued function?

What I know: I understand that for a complex function to be a multi-valued funciton they must have some branch points. I understand branch points as this definition (translated into English from my ...
Ian Hsiao's user avatar
  • 121

1
2 3 4 5