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Question 1

2votes

0answers

110views

Contour integral with four branch points on the unit circle

I want to compute the contour integral$$\oint_{|z|=2} z \sqrt{z^4-1}\text{d}z,$$where the path is positively oriented (it is the blue one below).It is non-zero thanks to the four branch-points $\...

94thomas

31

askedNov 20 at 13:57

Question 2

2votes

1answer

155views

What is the Riemann surface of $\sqrt{z^2}$? How do you prove that the origin is not a branch point?

Is $\sqrt{z}×\sqrt{z}=\sqrt{z^2}$ a two-valued function that takes values $\pm z$? However, I read that in this case the origin is not a branch point but a "non-unique point." Is there a ...

nakamura

45

askedSep 23 at 3:59

Question 3

1vote

1answer

109views

Problem in choosing value of phases when evaluating a complex integral with branch cut

I'm trying to evaluate the following integral using the residue theorem:$$ \int_0^1 \frac{\sqrt{1-x^2}}{x^2 - 2} dx $$First of all, we need to identify the singularities of the function, once ...

Luke__

492

askedSep 7 at 0:34

Question 4

1vote

1answer

75views

Value of a function when approaching a branch cut from different directions

I'm trying to understand how to compute real integrals using residue theorem when dealing with complex multi-valued functions. For example. Let's consider the following real function, extended to the ...

Luke__

492

askedAug 7 at 15:13

Question 5

4votes

1answer

196views

Integrating along a contour containing branch points

Let $f(z)=\sqrt{z^2-1}$. It suffices to cut $[-1,1]$ to make $f$ single-valued. Now let $C$ be a closed contour “containing” $[-1,1]$ “inside” it. For example, take $C$ to be the circle centered at ...

a.e

53

askedJul 12 at 23:43

Question 6

0votes

0answers

42views

Confusion regarding choosing multiple $\arg(z-z_0)$ restrictions for functions with several branch cuts

I thought of a very simple example to explain my confusion.Let's say I want to describe the function $y=\sqrt{z+1}\sqrt{z-2}$. I want to make sure it is single valued, so I take $-\pi < \arg(z+1)&...

sillyQsman

109

askedJul 7 at 15:03

Question 7

0votes

0answers

34views

Help in methodical understanding of the full Riemann surface and all possible local monodromies of $y(x)=\sqrt{x\sqrt{5} +\sqrt{2-x^2}}$

I think I understand the structure of this Riemann surface, but a) I'd like to be sure, and b) I am trying to rephrase it in a more systematic way, which will in principle allow me to solve any ...

sillyQsman

109

askedJul 6 at 14:34

Question 8

1vote

1answer

48views

Contour for integrating a function which also has branch points at infinity

I was wondering what contour to take to integrate a function $f(z)$ along a real line from $x=0$ ($x = Re[z])$ to $x=\infty,$ where the $f(z)$ has branch points at $x=+1$, $x=-1$, and $z=\infty$ (in ...

AD-Phys

11

askedJul 3 at 0:10

Question 9

1vote

1answer

69views

Does a function always have an even number of algebraic branch points?

Suppose we have a multivalued complex function on the Riemann sphere with a finite number of branch points, all of them algebraic. (A branch point $p$ is algebraic if there is a finite $n$ such that ...

Carmeister

4,273

askedJun 21 at 17:59

Question 10

2votes

0answers

97views

When is $\sqrt{(z-a)(z-b)} = \sqrt{z-a}\sqrt{z-b}$ true?

This seems a quite easy concept, but I feel like I am missing something. The function $f(z) = \sqrt{(z-a)(z-b)}$, with $a, b \in \mathbb{C}$, is a multi-valued function, and as such, it's true that$$...

sconsolato

31

askedJun 10 at 20:33

Question 11

3votes

1answer

214views

On the behaviour of a function around branch cuts

I'm a physics student. I'm currently taking a complex analysis course and I'm struggling to understand how the determination of a function is given on its branch cut. Since it is hard for me to even ...

Luke__

492

askedApr 21 at 12:39

Question 12

1vote

1answer

92views

2-Fold Branched Cover of 2-sphere with Euler Characteristic 3

The Reimann-Hurwitz formula for a branched cover $f:M \rightarrow N$ is$$\chi(M)=\operatorname{deg}(f)\chi(N)-b,$$where $b$ is the total branching index of $f$ (I'm reading Teleman 2003). I want to ...

levitopher

2,705

askedApr 13 at 16:40

Question 13

0votes

1answer

150views

Understanding the branches of $\sqrt{z(z-1)}$

I have just begun studying complex analysis seriously and the topic of branch cuts causes a lot of confusion to me. In one of the answers on this website, it is shown that there is a branch of $\sqrt{...

Максим Неважно

63

askedJan 25 at 23:58

Question 14

2votes

0answers

41views

How to compute density at branch point for multi-valued function?

The complex line $\{(z, w) \in \mathbb{C}^2: z^2=w^3 \}$ is a two-dimensional minimal surface living in $\mathbb{R}^4$ and it has a singular point at the origin, which is also a branch point. How to ...

IceAgeV

45

askedJan 16 at 8:38

Question 15

4votes

1answer

93views

Puiseux series for solution of polynomial equations

I want to study the behavior of a curve $\mathcal{C}$ implicitly defined as the zero of function $\mathbf{F}: \mathbb{R}^n\rightarrow\mathbb{R}^{(n-1)}$, $\mathcal{C}:\{\mathbf{x}\in \mathbb{R}^n \...

Franco

81

askedJan 15 at 11:35

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Questions tagged [branch-points]

Contour integral with four branch points on the unit circle

What is the Riemann surface of $\sqrt{z^2}$? How do you prove that the origin is not a branch point?

Problem in choosing value of phases when evaluating a complex integral with branch cut

Value of a function when approaching a branch cut from different directions

Integrating along a contour containing branch points

Confusion regarding choosing multiple $\arg(z-z_0)$ restrictions for functions with several branch cuts

Help in methodical understanding of the full Riemann surface and all possible local monodromies of $y(x)=\sqrt{x\sqrt{5} +\sqrt{2-x^2}}$

Contour for integrating a function which also has branch points at infinity

Does a function always have an even number of algebraic branch points?

When is $\sqrt{(z-a)(z-b)} = \sqrt{z-a}\sqrt{z-b}$ true?

On the behaviour of a function around branch cuts

2-Fold Branched Cover of 2-sphere with Euler Characteristic 3

Understanding the branches of $\sqrt{z(z-1)}$

How to compute density at branch point for multi-valued function?

Puiseux series for solution of polynomial equations

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