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.
49,401
questions
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Shouldn't $i ^ {-1}$ = $i$? (Complex numbers) [duplicate]
I was just going through Grant's (3b1b) Lockdown Math and he says that $i^{-1)}$ is actually $-i$ but I think it should be just $i$.
This website provides the following solution:
$$i^{-1} = \frac{1}{i}...
4
votes
1
answer
69
views
What does the equation $x^2+y^2=r^2$ represent when $x, y, r$ are complex numbers?
I know this question is vague or maybe broad and subjective.
But, I am interested in studying the equation $x^2+y^2=r^2$ when $x,y,r$ are complex numbers.
What are a few directions that I can follow ...
- 73
1
vote
0
answers
24
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False proof that uniform convergence of Fourier series implies smoothness
I found the following (obviously mistaken) proof that any continuous function $f\in C[\mathbb{T}]$ for which $|{f-\mathcal{S}_{N}f}|_{\infty}\to0$ must be smooth.
Can you please help me find the flaw?
...
- 147
0
votes
0
answers
23
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Riemann-Roch theorem for general divisor
$\newcommand{\u}{\mathfrak{U}}$I have a question about the proof of Riemann-Roch theorem in Farkas&Kra Riemann surfaces.
Definition. For a divisor $\u$ on $M$, we set a $\Bbb C$-vector space
$$L(\...
- 4,318
1
vote
0
answers
27
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A question related to the complex derivative of a function
I am working on the following problem.
Consider a continuous function $\phi:[-1,1]\to \mathbb{C}$ on
$[-1,1]$. Let $$g(z):=\int_{-1}^{1}\frac{\phi(t)}{t-z}\:dt.$$
I am interested in finding the ...
- 418
0
votes
0
answers
16
views
A question related to an inequality involving two entire functions
I am currently trying to solve the following problem from a previous qualifying exam.
Let $\alpha,\beta: \mathbb{C}\to \mathbb{C}$ be two non-constant
entire functions with exactly the same zeroes of ...
- 418
0
votes
0
answers
23
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Degree 2 Holomorphic Map Between Annuli
Consider the annulus $A(D) = \{D < |z| < 1\}$ where $0 \leq D < 1$. I need to find all $D$ such that there is a holomorphic map $f: A(D) \rightarrow A(D)$ of degree 2. By degree 2, I mean ...
- 425
3
votes
1
answer
30
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Use of Fubini's Theorem in Papa Rudin's Holomorphic Fourier Transforms
I am starting to read on chapter 19, Holomorphic Fourier Transforms from Real and Complex Analysis by Walter Rudin. In the first page of that chapter I came across the function $$f(z) = \int_0^\infty ...
- 199
0
votes
1
answer
28
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Hardy Spaces - outer function is holomorphic
I am studying Banach Spaces of Analytic Functions by Hoffman. In Chapter 5 Page 61, the textbook claims that
If $u \in L^1 (\mathbb T)$ then the function $F: \mathbb D \to \mathbb C$
\begin{align*}
F(...
- 3,605
3
votes
0
answers
43
views
Limit of the sum of cosine
I want to find the value of $$\sum_{k=0}^\infty \cos\left[\left(k+\frac{1}{2}\right)\pi x\right]\cos\left[\left(k+\frac{1}{2}\right)\pi t\right].$$
I think it should relate to delta function because ...
- 41
0
votes
0
answers
19
views
Equality of power series around two different points
Let's say I have a real function, $f(x) = \sum_{n=0}^{\infty} a_nx^n$ with a positive convergence radius $r_1>0$, then we know $f$ converges at $|x|<r_1$ and diverges at $|x|>r_1$, can we ...
- 113
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0
answers
33
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Is the function $f\circ h$ holomorphic?
Let $f:\mathbb{C}^{n}\to\mathbb{C}$ be a holomorphic function, and if $a,b\in\mathbb{C}^{n}$, let $h:\mathbb{C}\to\mathbb{C}^{n}$ be the function defined by:
$$
h(z)=a+bz,
$$
for all $z\in\mathbb{C}$....
- 753
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0
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17
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Show that holomorphic bijections between open sets in the complex plane preserve the local orientation.
Show that holomorphic bijections between open sets in the complex plane preserve the local orientation.
I'm trying to find how "local orientation" is defined, but I can't find a definition ...
- 17
1
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0
answers
21
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Maximum Value Principle: if $u$ is harmonic on $\{x+iy:0 <x<1\}, u(x+iy) \leq A cosh(ky)$, then $u \leq (1-x)A_0 + xA_1$ where $A_t = \sup\{u(t+iy)\}$
Let $\Omega = \{x+iy : 0 < x < 1 \text{ and } y \in \mathbb R\}$. Suppose $u: \bar\Omega \to \mathbb R$ is continuous, harmonic on $\Omega$ and satisfies $u(z) \leq A \cosh(k \text{ Im}(z))$ ...
- 2,981
0
votes
1
answer
31
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Cauchy Theorem on Analytic Theorem (on Newman's Proof of Prime Number Theorem)
Newman's Short Proof of the Prime Number Theorem, P. 707
https://people.mpim-bonn.mpg.de/zagier/files/doi/10.2307/2975232/fulltext.pdf
For $g(z) = \int_0^{\infty} f(t) e^{-zt} dt$ ($Re(z) = 0$), $g_T (...
- 311
0
votes
0
answers
23
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Laurent series of $\frac{1+z}{1-\sin z}$ around $\frac\pi2$
I need to do residue calculations for $\frac{1+z}{1-\sin z}$ but I'm running into problems with making the Laurent series.
The residue should be 2 [wolframalpha].
$\frac{1+(z-\frac\pi2)}{1-\sin(z-\...
- 21
1
vote
1
answer
54
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There is $R > 0$ such that there are no holomorphic functions $f: \{z: |z| < R \} \to \mathbb C \backslash \{0,1\}$ with $f(0) = a$ and $f'(0)= b$
Suppose that $a,b \in \mathbb C \backslash \{0\}, a \neq 1$. Prove that there is some $R > 0$ such that there are no holomorphic functions $f: \{z: |z| < R \} \to \mathbb C \backslash \{0,1\}$ ...
- 2,981
0
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0
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14
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Relation between diameter of a holomorphic function defined on the unit disc and it's derivative at 0.
following question has been discussed before, for example here and here.
For 2 reasons I ask this again:
$1.$I have solved this question in an other way but the solution doesn't seem to be correct ...
2
votes
1
answer
32
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$f$ is holomorphic in $U(z_0,\delta)$ $\implies\displaystyle{\lim_{n\to\infty}\frac{f(z_n)-f(w_n)}{z_n-w_n}}=f'(z_0)$.
Suppose $f$ is holomorphic in $U(z_0,\delta)$ and $z_n\to z_0,w_n\to z_0\ (n\to\infty), z_n\neq w_n$,
then$$\lim_{n\to\infty}\frac{f(z_n)-f(w_n)}{z_n-w_n}=f'(z_0).$$
Unlike in real analysis, we can't ...
- 6,090
8
votes
1
answer
142
views
Evaluating the sum $\sum_{k=1}^\infty (-1)^k / k^2$ via a contour integral
I'm evaluating the sum:
\begin{align*}
\sum_{k=1}^\infty (-1)^k \frac{1}{k^2}
\end{align*}
I expressed the sum via a complex contour integration. But I'm not getting out the right answer.
My method: ...
- 91
0
votes
0
answers
34
views
Stein and Shakarchi: "positive counterclockwise orientation"
I am having trouble understanding a passage in Stein and Shakarchi's complex analysis textbook. Here is the passage for reference. (It earlier wrote $z = re^{i \theta}$ and then wrote Euler's identity....
0
votes
0
answers
30
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prove $\frac{1}{2\pi i}dz/z$ is the generator of $H^1(\Bbb{P}^1\setminus \{0,\infty\},\Bbb{Z})$
I was trying to prove that the singular cohomology $H^1(\Bbb{P}^1\setminus \{0,\infty\},\Bbb{Z})$ has a deRham representative $dz/z$. That is $\frac{1}{2\pi i}dz/z \in H^1(\Bbb{P}^1\setminus \{0,\...
- 4,284
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0
answers
34
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Duality of $H^1$ and BMO.
While proving that the dual of $H^1$ is $BMO$ in Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, page 143, Stein says that we have $\left\Vert g \right\Vert_{H^1} \...
- 11
1
vote
0
answers
45
views
$\mathbb{C}$ is complete
Can
$\mathbb{R}^2 \simeq \mathbb{C}$ and because $\mathbb{R}^2$ is complete $\implies \mathbb{C} $ is complete
be an argument to show thaf $\mathbb{C}$ is complete? Or can you give me a proof please?...
- 397
1
vote
1
answer
97
views
I derived a formula for $[x!]^\prime$. Is it correct?
The starting point was that $ \Gamma'(x+1)=\Gamma(x+1)\psi(x+1)$ where $\psi(x+1)=-\gamma+H_{x}$ . Hence $$ [x!]' = x!\biggl[-\gamma+\sum_{k=1}^{x}\frac{1}{k}\biggl]$$ For example $ [4!]' = 24[-\gamma+...
- 29
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0
answers
37
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Cauchys integral formula for the nth derivative [closed]
Let $\gamma$ be a positively oriented, simple, closed contour such that $2023i$ lies in the interior of $\gamma$. Let $f : \mathbb{C} \rightarrow \mathbb{C}$ be a holomorphic function on $\mathbb{C}$ ...
2
votes
1
answer
65
views
How is a Hankel contour different from a keyhole contour?
From what I'm guessing a keyhole contour is one that looks like this
and because it can be shown that the contribution from $C_R$ and $C_\epsilon$ vanishes as $R\to \infty$, a Hankel contour looks ...
- 151
1
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0
answers
37
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Proof of $\limsup_{n\to\infty} \|a^n\|^{1/n} \le r(a)$
Theorem $2.2.6$ in Principles of Harmonic Analysis by Anton Deitmar and Siegfried Echterhoff proves the well-known formula for the spectral radius in a unital Banach algebra, namely $r(a) = \lim_{n\to\...
- 11.4k
0
votes
0
answers
30
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Saddle Point analysis of integral
I want to make saddle point approximation of the integral $$ I = \frac{1}{2 \pi i } \int_{\gamma - i \infty} ^{\gamma + i \infty} \frac{1}{z^6} e^{zt} dz $$ but as I see the function in the exponent i....
0
votes
0
answers
40
views
Definition of meromorphic $q$-differential
In Farkas Kra riemann surfaces textbook, it defines a meromorphic $q$-differential as follows:
Let $q$ be an integer. By a (meromorphic) $q$-differential $\omega$ on $M$ we mean an assignment of a ...
- 4,318
0
votes
1
answer
43
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Compute the $\int_{|z|=2} \frac{cos(z)}{z^2-2z-3} dz$
I am trying to compute $$\int_{|z|=2} \frac{cos(z)}{z^2-2z-3} dz $$
I tried to work it out with the Cauchy–Goursat theorem but the function has a singularity at $z=-1 $ which is an interior point of ...
- 23
-1
votes
0
answers
40
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The image of $\frac{\sin(\sqrt z)}{\sqrt z}$ [closed]
How to show that the entire function $f(z) = \frac{\sin(\sqrt z)}{\sqrt z}$ could attain any value at infinity times?
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answers
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Find fixed points of the following transformation: (i) w=3iz+1/z+i (ii) w=3z+4/z-1 [closed]
Ans :
(i)w=3iz+1/z+i : i,i
(ii) w=3z+4/z-1 z=2
0
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0
answers
61
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What is the symbol for a function that is complex differentiable?
Suppose $f:D \subset \mathbb{R}^2 \to \mathbb{R}^2$.
If $f$ is differentiable (or $\mathbb{R}$-differentiable) we say $f \in C^1(D;\mathbb{R}^2)$ or $f \in C^1(D)$ (meaning the partial derivatives ...
- 407
1
vote
1
answer
64
views
$(f(z+h) - f(z))/h$ converges uniformly as $h \to 0$
Let's denote $D_x = \{ z \in \mathbb{C} \mid |z| < x \} $.
Suppose $f(z)$ is holomorphic in $D_R$ then I want to prove that for any $0 < r < R$ a function $\displaystyle{\frac{f(z+h) - f(z)}{...
- 2,357
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votes
0
answers
14
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show that Ĝ is non empty and an abelian group when equipped with pointwise operations. [closed]
Assuming G is a locally compact group ( not abelian). A character of G is a continuous group homomorphism x: G → 𝕋 , where 𝕋 is the circle group . Denote by Ĝ the set of characters of G.
Here how ...
3
votes
0
answers
46
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Prove that Kahler form induced by Fubini-Study metric on $\Bbb{P}^n$ is the generator of the $\Bbb{Z}$ coefficient cohomology
I try to prove that the Kahler metric induced from the Fubini-Study metric on $\Bbb{P}^n$ is a generator of the cohomology $H^2(\Bbb{P}^n,\Bbb{Z})$
My attempt first by the Poincare duality we know ...
- 4,284
0
votes
1
answer
22
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Product of trigonometric functions as a trigonometric polynomials
We are given a non-negative trigonometric function $f = (\cos^2(\theta))^n(\sin^2(\theta))^m$,
where $m=N-n$ and $n,m<N$. I would like to understand if such a function can be reshaped to look like ...
- 2,149
-2
votes
0
answers
26
views
RZH Inequalities, 1/2 confusion. Complex analysis reading recommendation.
I'm trying to understand why the requirement for 1/2 for convergence of the series for $\zeta(s)$. Any good recommended reading on this would be great.
I have the below. What exactly is incorrect ...
1
vote
0
answers
64
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Expand $\cos^a x$ in terms of $\cos kx$, $\sin mx$
If $a\geq 0$, expand $\cos^a x$ in terms of $\cos kx$, $\sin mx$
$$\cos^a x=\left(\frac{e^{ix}+e^{-ix}}2\right)^a$$
Since $a$ is a non negative real number, so by General Binomial theorem
$$\cos^a x=...
- 95
-3
votes
1
answer
86
views
Calculating ζ(2) [closed]
I'm looking for a way to calculate $\zeta(2)$ via the integral
$$\zeta(2) = \int_{0}^{\infty} \dfrac{x}{e^x - 1}\, dx$$
without using a contour integral or appealing to the sum of inverse squared ...
0
votes
1
answer
50
views
Can we decompose complex differential equation into two real?
there is a simple system:
$$
-y''(x) = f(x)
\\
y(0) = 0
\\
y(1) = 0
$$
where $y$ and $f$ is complex function of real variable $x$.
Is it legal to decompose this functions into two real (four real for ...
- 11
1
vote
0
answers
21
views
How to apply the second mean value theorem on $ \int_{Y^2}^{{Y'}^2} z^{-1/2} e^{2 \pi i N z} dz $?
$Y' > Y > 0$ is given. $N$ is a positive integer. How we can apply the second mean value theorem on $\int_{Y^2}^{{Y'}^2} z^{-1/2} e^{2 \pi i N z} dz $ to conclude that this integral has absolute ...
- 167
-3
votes
1
answer
34
views
A complex function which is complex differentiable only at 0 but not even continuous elsewhere
Can you give an example of a function $f:\mathbb{C} \to \mathbb{C}$ which is complex differentiable at $0$ but not even continuous at all $z\neq 0$?
- 107
1
vote
2
answers
113
views
Contour integration with $\int_{-\infty}^{\infty}{\frac{1-\cos(2z)}{z^2(z^2+1)}}\, dz$
$$\int_{-\infty}^{\infty}{\frac{1-\cos(2z)}{z^2(z^2+1)}}\, dz$$
The contour is shaped like this (image from this answer)
With $\epsilon$ being the radius of the smaller semi-circle. The integral over ...
- 451
3
votes
3
answers
105
views
How to compute the area of the given region (NBHM $2022$)?
This is a question from NBHM $2022$ exam. It asks to find the area of the region $\{z+\frac{z^2}{2} \mid z\in \mathbb{C},|z| \leq 1\}$
Now $z+\frac{z^2}{2}$ = $\frac{(z+1)^2}{2}-\frac{1}{2}$.
The $-\...
- 33
1
vote
1
answer
46
views
f is holomorphic on the unit disk. Show that there exist an sequence ${z_n}$ such that $|z_n|$ converges to 1 and ${f(z_n)}$ is bounded [duplicate]
I'm trying to prove this by contradiction, which comes out that every limit of $|f(z)|$ on boundary tends to infinity. But I have no further idea, any help?
- 33
2
votes
2
answers
60
views
Laurent Series given some annuli
The question is to represent $g(z) := \frac{1}{(z-3)(z+1)}$ by a Laurent series, with regards to the annuli $A_1:= \{z \in \mathbb{C} : 1 < |z| < 3\}$ and $A_2:= \{z \in \mathbb{C} : 3 < |z|\}...
1
vote
1
answer
81
views
Inequalities via Number of Zeros & Jensen's Formula
I am working on a problem in which we take some entire function $F$ which has zeros at every integer square root, i.e. $F(\sqrt{n}) = 0$ for every $n \in \mathbb{Z}^+$.
I need to show that, for any ...
- 174
-1
votes
0
answers
34
views
How to solve the integral of x cot(x) using the polylogarithmic function step by step [closed]
Photo Of the integral
The Taylor series can be used to express this structure as a geometric series or directly the sum that defines the polylogarithm.
A recommendation is to know the different ...
