Questions tagged [complex-analysis]
For questions mainly about the theory of complex analytic/holomorphic functions of one complex variable. Use [tag:complex-numbers] instead for questions about complex numbers. Use [tag:several-complex-variables] instead for questions about holomorphic functions of more than one complex variable.
54,267 questions
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Complex logarithm base 1
Is a logarithm with base 1 defined in the field of complex numbers? I have not found any information about this. In real numbers, this is uncertain because $ \ln(1) = 0 $ and
$ \log_a(b)= \frac {\ln(...
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Is the set of finite Blaschke products a graded ring?
I have heard tell that there are many analogies between Blaschke products and polynomials.
A finite Blaschke product is a rational function on the complex unit disk $B(z)=\mu\prod_{i=1}^{\deg B}\frac{...
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Picard-Lefshetz method for computing integrals: a simple example
I am trying to use Picard-Lefshetz theory to turn a conditionally convergent integral into absolutely convergent and compute it using the saddle point approximation. The integral is a toy model which ...
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Strict monotone differences of analytic functions [closed]
I am working with two analytic functions $f$ and $g$ and want to study their behavior along a horizontal line in the complex plane.
Let $v_0 \in \mathbb{R}$ be fixed, and define
$$
\tilde f(u) := f(u +...
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Is there a dense set of Lipschitz functions on the unit ball which all peak at the same point on the boundary?
I posted this same question over 2 years ago on MO but didn't get any comments or answers. The question is admittedly quite narrow, but I am still very interested in its resolution.
Let $U$ be the ...
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singularity type of a holomorphic function on punchured unit disk at $z=0$
Let $\mathbb{D}^*$ denote the set $\{z \in \mathbb{C} : |z| < 1, z\neq 0\}$. Let $f : \mathbb{D}^* \longrightarrow \mathbb{C} \setminus \{\pm10\}$ be a holomorphic map. Which
of the following is/...
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Is Abel summability the same as contour integration?
I'm dealing with the following integral $$\int_{-\infty}^\infty \frac{ke^{ikx}}{\sqrt{k^2+m^2}}dk$$
where $m,x$ are some real positive fixed constants. I asked a question about the calculation of this ...
- 2,354
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Finding zeros of the product of two analytic functions
I know when finding the order of a zero, $z_0$, of an analytic function, you can just differentiate the function until substituting $z_0$ in does not give zero. Instead, to save time on ...
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Help in filling in the gaps from a contour integration in QFT [duplicate]
In a QFT class, we were calculating the following integral $$\int_{-\infty}^{\infty}dk \frac{ke^{ikx}}{\sqrt{k^2+m^2}}$$
and we decided to do a contour calculation. We chose a branch cut at negative $...
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if $f$ is holomorphic and $f(z)=f(\overline{z})$ then $f$ is constant
Prove that if $f$ is holomorphic at some point and also $f(z)=f(\overline{z})$ then $f$ must be constant.
Letting $f=u+iv$, and using Cauchy-Riemann, I was able to prove that $u(x,0)$ and $v(x,0)$ ...
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Definite integral of the two Bessel functions / (x-a)
In my work, an integral of the following type arose:
$$\int_0^\infty dx J_l(a x) J_l(b x) \frac{1}{x - c + i 0}.$$
Here $J$ is the Bessel function of the first kind. Assume that $a$, $b$, and $c$ are ...
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Gaussian Integral solved with complex integration [closed]
I've been trying to solve this problem, but I don't know how to begin solving it. I know that the residue theorem must be used at some point, but not much else. Any help is welcome.
Let be the ...
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Why doesn't Montel's Theorem imply the space of analytic functions is locally compact?
Let $G\subset \mathbb{C}$ be a connected open set, and $H(G)$ the set of all holomorphic functions defined on $G$. Since $G$ admits an exhaustion of compact sets $G=\bigcup_n K_n$ s.t. $K_n\subset int(...
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Formal Justification of Alternating Harmonic Series: $1-\frac{1}{2}+\frac{1}{3} - \cdots = \ln(2)$ [duplicate]
It's well known that $\sum_{n=1}^\infty \frac{(-1)^n}{n}=\ln(2)$, which can most easily be seen by the following derivation:
$$\frac{1}{1-x} = \sum_{n=0}^\infty x^n \tag{Geometric Series}$$
$$\frac{1}{...
- 5,547
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How to prove that the moduli space $\mathcal{M}_g$ is Hausdorff using Kuranishi families?
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 ...
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