Questions tagged [complex-analysis]

Question 1

I posted this same question over 2 years ago on MO but didn't get any comments or answers. The question is admittedly quite narrow, but I am still very interested in its resolution. Let $U$ be the ...

Question 2

Let $\mathbb{D}^*$ denote the set $\{z \in \mathbb{C} : |z| < 1, z\neq 0\}$. Let $f : \mathbb{D}^* \longrightarrow \mathbb{C} \setminus \{\pm10\}$ be a holomorphic map. Which of the following is/...

Question 3

I'm dealing with the following integral $$\int_{-\infty}^\infty \frac{ke^{ikx}}{\sqrt{k^2+m^2}}dk$$ where $m,x$ are some real positive fixed constants. I asked a question about the calculation of this ...

Question 4

I know when finding the order of a zero, $z_0$, of an analytic function, you can just differentiate the function until substituting $z_0$ in does not give zero. Instead, to save time on ...

Question 5

In my short note on a Dirichlet generating function for the Euler pentagonal number coefficients, I use a Hankel contour in Section 2.4. I'd like to check whether the text correctly expresses what I ...

Question 6

In a QFT class, we were calculating the following integral $$\int_{-\infty}^{\infty}dk \frac{ke^{ikx}}{\sqrt{k^2+m^2}}$$ and we decided to do a contour calculation. We chose a branch cut at negative $...

Question 7

Prove that if $f$ is holomorphic at some point and also $f(z)=f(\overline{z})$ then $f$ must be constant. Letting $f=u+iv$, and using Cauchy-Riemann, I was able to prove that $u(x,0)$ and $v(x,0)$ ...

Question 8

In my work, an integral of the following type arose: $$\int_0^\infty dx J_l(a x) J_l(b x) \frac{1}{x - c + i 0}.$$ Here $J$ is the Bessel function of the first kind. Assume that $a$, $b$, and $c$ are ...

Question 9

I've been trying to solve this problem, but I don't know how to begin solving it. I know that the residue theorem must be used at some point, but not much else. Any help is welcome. Let be the ...

Question 10

Let $G\subset \mathbb{C}$ be a connected open set, and $H(G)$ the set of all holomorphic functions defined on $G$. Since $G$ admits an exhaustion of compact sets $G=\bigcup_n K_n$ s.t. $K_n\subset int(...

Question 11

It's well known that $\sum_{n=1}^\infty \frac{(-1)^n}{n}=\ln(2)$, which can most easily be seen by the following derivation: $$\frac{1}{1-x} = \sum_{n=0}^\infty x^n \tag{Geometric Series}$$ $$\frac{1}{...

Question 12

Let $ C $ be a compact connected Riemann surface of genus $ g > 1 $. I used the definition of a Kuranishi family of $ C $ as in [Teichmüller space via Kuranishi families] 1. Using these Kuranishi ...

Question 13

My background is in physics, so I never had a proper course in either real or complex analysis; topics like uniform convergence weren't touched upon. I really like analysis though, so for the last few ...

Question 14

I have a physics related problem where I need to calculate integrals of following type $\displaystyle\int_{-\infty}^{\infty} d\epsilon\, A(\epsilon) g^R(\epsilon + \omega/2)g^A(\epsilon - \omega/2)$, ...

Question 15

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 $\...

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Questions tagged [complex-analysis]

Is there a dense set of Lipschitz functions on the unit ball which all peak at the same point on the boundary?

singularity type of a holomorphic function on punchured unit disk at $z=0$

Is Abel summability the same as contour integration?

Finding zeros of the product of two analytic functions

Hankel Contour: Are Residues Counted or Excluded?

Help in filling in the gaps from a contour integration in QFT [duplicate]

if $f$ is holomorphic and $f(z)=f(\overline{z})$ then $f$ is constant

Definite integral of the two Bessel functions / (x-a)

Gaussian Integral solved with complex integration [closed]

Why doesn't Montel's Theorem imply the space of analytic functions is locally compact?

Formal Justification of Alternating Harmonic Series: $1-\frac{1}{2}+\frac{1}{3} - \cdots = \ln(2)$ [duplicate]

How to prove that the moduli space $\mathcal{M}_g$ is Hausdorff using Kuranishi families?

How do you think about uniform continuity?

How to deform contour in the complex plane correctly

Contour integral with four branch points on the unit circle

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