Questions tagged [complex-analysis]
For questions mainly about 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 variables.
52,388
questions
0
votes
1
answer
22
views
What are the functions obtained by complex polynomials evaluated at complex numbers
Given a complex polynomial $f \in \mathbb{C}[x]$, we can induce a function $ev_{\mathbb{C}}f: \mathbb{C} \to \mathbb{C}, x \mapsto ev(c)(f)$, where ev stands fpr evaluation. Denote the set of all such ...
- 1,706
0
votes
0
answers
41
views
How is real and imaginary part of a complex polynomial defined
I have been reading a survey on polynomial optimization, where on page 15, Lemma 2.5, the author used notations such as $Re(p_{z})$ and $Im(p_{z})$, where $p_{z} \in \mathbb{C}[\mathbf{x}]$, for $\...
- 1,706
0
votes
0
answers
32
views
Using Residue Theorem for functions with removable singularities
So for an Integral of the form $\int_{-\infty}^{\infty} \frac{e^{ixa} -1}{x(x^2 + 1)} dx$. My intuition is to use complex contour integration and use a contour that is a semi-circle on the upper-half ...
- 1
-1
votes
1
answer
39
views
Closed sets definition in complex analysis
Here's the translation:
Are limit points and boundary points the same? If not, why are there two versions of the definition for a closed set? Which one is correct?
A set $A \subset \mathbb{C}$ is ...
1
vote
0
answers
29
views
How to understand exponential in infinite Weierstrass product of $\sin(\pi z)$
Based on the discussion in Understanding infinite product of $sin(\pi z)$, I have a very rudimentary question: How do we derive the equation $\sin(\pi z) = \pi z \prod_{n\neq 0} (1-z/n)e^{z/n}$ in the ...
- 11
-2
votes
0
answers
19
views
Harmonic Conjugate of $\log|z|$ on Annulus? [closed]
Does $\log|z|$ possesses any harmonic conjugate in the annulus $B(0,r,1)=\{z:r<|z|<1\}$? Or equivalently I want to know if there is any holomorphic branch of logarithm that exists on the ...
- 1
1
vote
0
answers
42
views
Poincare residue and trivializing section of canonical bundle of plane cubic.
I am trying to get my hands dirty and do the following computation, but I don't feel like I'm doing it right. Help would be very much appreciated! I will tell you the setup of the calculation, and ...
- 43
1
vote
2
answers
64
views
Complex integrals that look like they agree, differ by sign (according to Mathematica)
Consider the integral
$$\int_0^\infty \frac{dz}{1-z^2 +i0^+},$$
I would assume it to agree with the integral
$$\int_0^\infty \frac{dz}{(1-z+i0^+)(1+z+i0^+)}. $$
However, according to Mathematica the ...
3
votes
0
answers
42
views
How to prove a property of the product of Eisenstein Series
I want to prove that, if $$ E_i(\tau) = \frac{1}{2\zeta(2i)}\sum_{(m,n)\in\mathbb{Z}^2-\{(0,0)\}}(m\tau+n)^{-k}$$
is the normalized Eisenstein Series of weight $i$, then $E_i(\tau)E_j(\tau)\neq E_{i+j}...
- 39
2
votes
1
answer
71
views
holomorphic function on unit disc that maps boundary of $\mathbb{D}$ into itself
Let $\mathbb{D} = \{|z| < 1\}$. Let $f : \mathbb{D} \to \mathbb{C}$ be non constant holomorphic map that extends continuously to $\overline{\mathbb{D}}$. Show that if $f(\partial \mathbb{D}) \...
- 149
-1
votes
1
answer
32
views
Does the Gauss-Lucas theorem generalize to arbitrary entire functions with a finite number of zeros?
The Gauss-Lucas theorem states that if $P(z)$ is a non-constant polynomial with complex coefficients, then all the zeros of $P'(z)$ lie within the convex hull of the set of zeros of $P(z)$.
Does this ...
- 6,280
0
votes
1
answer
47
views
Inconsistency in evaluating complex integral using two different approaches
I am trying to evaluate the following integral $$\int_0^\infty\frac{\cos x}{1+x^2}$$. I know that the following formula for integrals $$\int_{-\infty}^\infty f(x)dx=\pi\iota R(X-axis)+2\pi\iota R(\...
- 45
0
votes
0
answers
39
views
complex analysis - Prove $f(z) = \sum_{n=0}^{\infty} z^{2^n}$ is not regular on the boundary of the unit cycle
I have been trying to solve this exercise for days now, I would really appreciate some help:
Let $f : D \to \mathbb{C}$ be defined in the open unit disc. A point $w \in \partial D$ is called a regular ...
0
votes
0
answers
31
views
Application of Hurwitz's Theorem
Let $f_n$ be a sequence of entire functions converging uniformly on compact subsets to $f$. Suppose that for every $n$ the zeros of $f_n$ lie on the real axis. Show that $f$ is either identically zero ...
- 149
3
votes
0
answers
43
views
sum of entire nonvanishing functions is constant implies functions are constant
Let $f,g$ be nonvanishing entire functions such that $f + g = 1$ for every $z\in \mathbb{C}$. Do $f$ and $g$ have to be constants themselves.
My attempt: $$f = 1-g \implies \frac{1}{f} = \frac{1}{1-g}$...
- 149