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.
52,391
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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
35
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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
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1
answer
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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
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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
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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 $$\int_{-\infty}^\infty f(x)dx=\pi\iota R(X-axis)+2\pi\iota R(\...
- 45
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complex analysis - Prove $f(z) = \sum_{n=0}^{\infty} z^{2^n}$ is not regular on the boundary of the unit cycle
I have been trying to solve this exercise for days now, I would really appreciate some help:
Let $f : D \to \mathbb{C}$ be defined in the open unit disc. A point $w \in \partial D$ is called a regular ...
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30
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Application of Hurwitz's Theorem
Let $f_n$ be a sequence of entire functions converging uniformly on compact subsets to $f$. Suppose that for every $n$ the zeros of $f_n$ lie on the real axis. Show that $f$ is either identically zero ...
- 149
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votes
0
answers
41
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sum of entire nonvanishing functions is constant implies functions are constant
Let $f,g$ be nonvanishing entire functions such that $f + g = 1$ for every $z\in \mathbb{C}$. Do $f$ and $g$ have to be constants themselves.
My attempt: $$f = 1-g \implies \frac{1}{f} = \frac{1}{1-g}$...
- 149
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Completely monotone function is analytic
I want to prove the following.
Let $I:=[a,b)$ be finite interval, $f$ is completely monotone on $I$.
Then it can be extended analytically into the complex z-plane
($z=x+iy$), and the function $f(z)$ ...
- 3,501
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How to get the following estimate of integral invoving Airy function
$$
\mbox{Define}\quad
G(x,y)=\frac{Ci\left(\gamma(x-y_c)\right)Ai\left(\gamma(y-y_c)\right)}{\epsilon\gamma},
$$
where $y>x,$ $y_c$ is a complex number such that $\Re(y_c)>0,$
$\epsilon$ is a ...
- 57
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0
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64
views
Sign of a complex integral
If one consider the complex value function
$$
f(z)=\frac{1}{(z-1)^{3/2}(z-2)^{3/2}}
$$
with branch cut chosen to be between $z=1$ and $z=2$.
Consider
$$
\oint f(z) dz,
$$
where the contour is taken to ...
- 1,129
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1
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Complex integral with fractional singularities
If one consider the complex value function
$$
f(z)=\frac{1}{\sqrt{z-1}\sqrt{z-2}}
$$
with branch cut chosen to be between $z=1$ and $z=2$. Could someone please explain why
$$
2\int_1^2 f(x)dx=\oint f(...
- 1,129
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$\int_0^{+\infty}\frac{\cos (ax)-\cos(bx)}{x^2}\mathrm dx$ using complex analysis [duplicate]
$$
\mbox{I defined the complex function}\quad
\operatorname{f}\left(z\right) =
{{\rm e}^{{\rm i}z} \over z^{2}}
$$
and in the end i just take the real part of my answer. I want to use Residues Theorem....
- 17
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2
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73
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Definite Integral of $\int_{0}^{2\pi}\frac{d\phi}{k-(a+bcos(\phi))^2+(a+bcos(\phi))}$ with $a,b \in \mathbb{R}$ and $k\in \mathbb{C}$
I tried solving this integral introducing the unit circle parametrization on the complex plane
$$
z = {\rm e}^{{\rm i}\phi}\quad\mbox{and substituting}\quad \cos\...
- 1
1
vote
0
answers
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Proof of Paley-Wiener Theorem
I'm trying to understand the proof of the following version of Paley-Wiener theorem under the additional assumption $f \in L^2$:
I understood the part $(2) \Rightarrow (1)$ but I couldn't follow a ...
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