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.
53,648 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{...
- 3,745
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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 ...
- 1,247
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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 ...
- 21
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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 ...
- 331
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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 $v(x,0)$ ...
- 573
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answers
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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
$$
\tilde f(u) := f(u +...
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Does the closest root of a n-th partial sum of a power series to a root of the power series converge?
Let
$$
f(z)=\sum_{k\ge \text{0}} b_{k}\, z^{k}
$$
be a power series with complex coefficients, and suppose $a\in\mathbb {C}$ satisfies $f(a)=0$.
For each index $n$, consider
$$
f_{n}(z)=\sum_{k=\text{...
- 9,329
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1
answer
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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 $...
- 2,354
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answer
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How do you think about uniform continuity?
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 ...
- 2,354
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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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votes
1
answer
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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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2
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The intuition behind defining trigonometric functions in the complex plane as special combinations of exponential functions [closed]
I was studying Complex Analysis from "A First Course of Complex Analysis" and the authors stated directly that sine and cosine are defined as follows (without any intuition):
$$ \sin\left(z\...
- 31
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votes
1
answer
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Formal Justification of Alternating Harmonic Series: $1-\frac{1}{2}+\frac{1}{3} - \cdots = \ln(2)$ [duplicate]
It's well known that $\sum_{n=1}^\infty \frac{(-1)^n}{n}=\ln(2)$, which can most easily be seen by the following derivation:
$$\frac{1}{1-x} = \sum_{n=0}^\infty x^n \tag{Geometric Series}$$
$$\frac{1}{...
- 5,547
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0
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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 $\...
- 31