Questions tagged [branch-cuts]

Question 1

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 $\...

Question 2

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 ...

Question 3

I want to prove that $$\kappa(s)=\frac{d\theta}{ds}\tag{1}$$ to verify that $$\int_{s_1}^{s_2}\kappa(s)\,ds=\int_{\theta_1}^{\theta_2}=\theta_2-\theta_1.$$ (Motivation for this is my previous question,...

Question 4

I am looking at the Jacobi Theta function $\theta_1(z)$ with $\tau\in i\mathbb R$ $$\theta_1(z+1)=-\theta_1(z)\quad \theta_(z+\tau)=-e^{-2\pi iz}e^{-\pi i \tau}\theta_1(z)$$ I take the principal ...

Question 5

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 ...

Question 6

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)&...

Question 7

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 ...

Question 8

Consider the integral $$I = \int_{-1}^{1}{(x+1)^{\frac{1}{3}}(1-x)^{\frac{2}{3}} dx}$$ A common method of evaluating the integral is to consider $f(z) = {(z+1)^{\frac{1}{3}}(z-1)^{\frac{2}{3}}}$ with ...

Question 9

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 $$ ...

Question 10

A question of vocabulary I am not sure about. Let's take the complex function (with values in $\mathbb{R} \in \mathbb{C}$ in that case...): $$z \mapsto \arg(z)$$ Let's take $z$ and $\theta$ ...

Question 11

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 ...

Question 12

An integral representation of the modified Bessel function is $$I_\nu(t) = \frac1{2\pi i} \oint_C \exp(t (z+1/z))\frac{\mathrm dz}{z^{1+\nu}}$$ where the contour $C$ starts in $\infty-i\epsilon$ and ...

Question 13

I am considering the $\phi^3$ theory in $d = 3$ dimensions. The propagator is given by the following formula $$ G(p^2) = \frac{-i}{p^2 + m_0^2 + M^2(p^2) - i\epsilon} $$ where $$ M^2(p^2) = -\frac{g_0^...

Question 14

Let's consider the function \begin{equation} f(x) = \frac{1}{x - iy_0} \frac{1}{x - iy_1} \left( \frac{ia - x}{ia + x} \right)^{b - i\frac{c}{x}} . \end{equation} Here, $\{a,c,y_0,y_1\}\in\mathbb{R}$ ...

Question 15

Consider the following contour in the complex plane $$\Gamma_m = \{e^{(2m+1)\pi i} r \in \mathbb{C} : r>0\},\quad m\in \mathbb{Z}$$ For any value of $m$ this is exactly the same locus in $\mathbb{C}...

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Questions tagged [branch-cuts]

Contour integral with four branch points on the unit circle

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

Rigorous relation between pointing angle $\theta$ and arc length $s$ in total curvature in $\mathbb R^2$

Argument of the theta function

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

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 $y(x)=\sqrt{x\sqrt{5} +\sqrt{2-x^2}}$

Tracking the 'phase' of complex contour integrals

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

Is there a specific way to name the branches of a multivalued complex function?

On the behaviour of a function around branch cuts

Apparent paradox from deforming a Hankel contour on the Riemann sphere

Determining Discontinuity Across Branch Cuts for $\phi^3$ propagator

Complicated contour integral with branch cut

Is there some lift of the complex plane on which these contours are different?

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