The theory of functions of one complex variable with an emphasis on the theory of complex analytic (or holomorphic) functions of one complex variable. Typical topics include: Cauchy's integral formula, singularities, poles, meromorphic functions, Laurent and Taylor series, maximum modulus principle, ...

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8 views

Elliptic functions $f(z+\lambda_1)=af(z) \; , \; f(z+\lambda_2)=bf(z) $

Let $\lambda_1$ and $\lambda_2$ be complex numbers with nonreal ratio. Let $f(z)$ be an entire function and assume there are constants $a$ and $b$ such that $$f(z+\lambda_1)=af(z) \;\;\;\;,\;\;\;\; ...
5
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2answers
47 views

Evaluating sums using residues $(-1)^n/n^2$

I am an alien towards compelx analysis, with very little know I am posing a question, who someone may want to help with. Evaluate: $$\frac{1}{4}\cdot \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$$ In ...
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2answers
52 views

Why is it so difficult to use geometric methods to construct weight one modular forms?

I am trying to understand why it is that geometric methods of constructing modular forms of weight one for $SL(2,\mathbb Z)$ fail. I have a relatively rudimentary understanding of it, but it would be ...
3
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1answer
30 views

Simply connected and connected in complex analysis

Can some one please help me with this, why is third set in the picture not simply connected. The definition of simply connected (in space of complex numbers) is: A set is said to be simply ...
2
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1answer
69 views

Show that $4\frac{\partial}{\partial z}\frac{\partial}{\partial\bar{z}}=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$

I am supposed to prove the following: $$4\frac{\partial}{\partial z}\frac{\partial}{\partial\bar{z}}=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\,\,\,$$ Using the definitons ...
7
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2answers
119 views

How can I prove $\sum_{n=1}^{\infty }\frac{1}{n^3(n+1)^3}=10-\pi ^2$

Can the residue theorem prove this? $$\sum_{n=1}^{\infty }\frac{1}{n^3(n+1)^3}=10-\pi ^2$$
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2answers
23 views

How is $ \lim_{z \to z_o} (z-z_o)\frac{f(z)}{g(z)} = \lim_{z \to z_o} \frac{f(z)}{g(z)-g(z_o)/(z-z_o)}= \frac{f(z_o)}{g'(z_o)}$?

I was reading this proof in Gamelin Complex Analysis (page 196): If $ f(z) $ and $ g(z) $ are analytic at $ z_o $ and if $ g(z) $ has a simple zero at $ z_o $ $$ Res[ \frac{f(z)}{g(z)},z_o ] = ...
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6answers
662 views

Evaluating $\int_0^\infty \frac{\log (1+x)}{1+x^2}dx$

Can this integral be solved with contour integral or by some application of residue theorem? $$\int_0^\infty \frac{\log (1+x)}{1+x^2}dx = \frac{\pi}{4}\log 2 + \text{Catalan constant}$$ It has two ...
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2answers
25 views

Comparison of the consequences of uniform convergence between the real and complex variable cases,

In the real variable case, I think that uniform convergence preserves continuity and integrability, i.e., for an integral of a sequence of continuous (or integrable) functions, which converge ...
3
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2answers
177 views

Prove that an analytic function on complex plane to its proper open and simply connected set is constant

Suppose that $V$ is an open, simply connected, proper subset of $\mathbb{C}$. Suppose that $f\colon\mathbb{C}\rightarrow V$ is holomorphic. Prove that $f$ is constant function. Give counter example ...
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0answers
18 views

Convergence implies Abel summability, and we only need to consider when $s=0$?

Suppose $\displaystyle c_n\in\mathbb{C}\textrm{ and}\sum_{n=1}^{\infty}c_n=s$. Then, prove $\displaystyle\lim_{r\to 1^{-}}\sum_{n=1}^{\infty}r^{n}c_n=s$. In my text, the author hinted that: we only ...
7
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0answers
70 views

Function's analytic continuation is its own derivative

This is the question we were asked at the university by our professor for complex analysis. Not as an exam, but as a challenge. I don't think he knew the answer himself. Find a nontrivial example of ...
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0answers
26 views

inequalities related to linear factional transformation and schwarz`s lemma

In both questions, it is said that I should use schwarzs lemma and linear factional transformation. But I don`t have any ideas how to use it. please give me some more hint
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0answers
40 views

What does complex square root as defined on Wikipedia look like: two questions

If you look at the third picture here, the surface representing the complex square root intersects the negative real axis at $0$. Later in the article the definition of the complex square root is ...
3
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1answer
75 views

how to compute this complex integral with high order polynomial?

Compute $$ \lim_{R \to \infty} \frac{1}{2\pi i} \int_{|z|=R} \frac{(2z^2+z-1)P'(z)}{P(z)+3}dz $$ where $P(z)=z^{10}+2z^9+z^5+1$. It seems like I may use residue but it contains too high ...
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1answer
34 views

Chain rule for the Wirtinger derivative $\frac{\partial}{\partial \overline{z}}$

I came over a calculation which used the identity $$ \frac{\partial(u \circ f)}{\partial \overline{z}} = \left( \frac{\partial u}{\partial \overline{z}} \circ f \right) \overline{f'}. $$ I tried to ...
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1answer
19 views

Bounded functions composed with Möbius maps

Hopefully easy question here: What is the most succinct method/technique to prove the following statement?: Let $u \in L^{\infty}(\Bbb D)$. Show that $||u(\varphi_{z})||_{\infty}$ is ...
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0answers
24 views

How to prove that $f$ can be expressed as a ratio of polynomials, given that $|f(z)|=1$ when $|z|=1$? [duplicate]

Given: $f$ is analytic in $| z|\leq1$ and $|f(z)|=1$ when $|z|=1$. Prove that $f(z)=P(z)/Q(z)$ where $P$, $Q$ are polynomials.
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1answer
23 views

Help with simplification of an expression

I was solving the residues of $f(z)e^{zt} = e^{zt}\frac{\ln(z)}{z^2+1}$ as follows: $$\operatorname{Res}(f(z)e^{zt}, i) = \lim_{z\to i} (z-i)\frac{e^{zt}\ln(z)}{(z-i)(z+i)} = ...
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0answers
17 views

how to prove that holomorphic function mapping complex onto complex is linear? [duplicate]

How to prove that any one to one holomorphic function mapping complex plane onto itself is linear?
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1answer
44 views

Can two analytic functions that agree on the boundary of a domain, each from a different direction, can be extending into one function?

Let $D=\{z:|z|\leq 1\}$ be the unit disc in $\mathbb{C}$. Say $f$ is analytic on $D$ and $g$ is analytic on $\overline{D^c}$, and that $f|_{\partial D}=g|_{\partial D}$. Is there necessarily an ...
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2answers
124 views

Laurent-series expansion of $1/(e^z-1)$

Find the Laurent series for the given function about the indicated point. Also, give the residue of the function at the point. $$ \frac{1}{e^z - 1} $$ at $z_0=0$(four terms of laurent series). I ...
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1answer
30 views

Interpolation sequences and open mapping theorem

I'm using Garnett's "Bounded Analytic Functions" as a course text and looking at interpolation sequences. $z_n$ is a sequence of interpolation if for each sequence $a_n \in l^{\infty}$ there exists $f ...
3
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1answer
64 views

Holomorphic function with a unique fixed point

Let $\omega \subset \mathbb C$ be a simple connected set and $\,f:\omega \to A$ is an analytic function where $A \subset \omega$ is compact. Show that $f$ has an unique fixed point. I think we can ...
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2answers
42 views

Taylor expansion of the Error function

The error function $\operatorname{erf}(z)$ is defined by the integral $$ \operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2}\,dt,\quad t\in\mathbb R$$ Find the Taylor expansion of ...
6
votes
1answer
100 views

Show that $\{e^{in}: n\in\Bbb N\}$ is Dense in the Unit Circle

This problem gave me fits when I was in grad school. Looking back at it now, it still escapes me. The problem is from Conway's Functions of One Complex Variable. I'm looking for a proof from basic ...
25
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1answer
732 views

Which sets are removable for holomorphic functions?

Let $\Omega$ be a domain in $\mathbb C$, and let $\mathscr X$ be some class of functions from $\Omega$ to $\mathbb C$. A set $E\subset \Omega$ is called removable for holomorphic functions of class ...
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2answers
26 views

In dual numbers, what number is represented by the following matrix?

In dual numbers, what number is represented by the following matrix? \begin{pmatrix}0 & 0 \\1 & 0 \end{pmatrix}
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1answer
24 views

Fourier transform, quadratic function

I'm trying to compute this convolution: $\frac{2 \alpha}{\alpha ^2 + 4 \pi ^2 x^2} * \frac{2 \beta}{\beta ^2 + 4 \pi ^2 x^2}$ I know that the Fourier transform of a convolution of two functions is ...
2
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0answers
33 views

Need help with holomorphic functions on a domain interval removed.

I want to prove that for a region $\Omega$ with interval $I=[a,b]\subset\Omega$, if $f$ is continuous in $\Omega$ and $f\in H(\Omega-I)$, then actually $f\in H(\Omega)$. Is this problem related to ...
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3answers
28 views

How to show that a complex function have a branch in a domain

I've given as homework to show that the function $$f(z)=\sqrt{\frac{z+1}{z-1}} $$ has a branch on $G = \mathbb C \backslash [-1,1] $. I'm having a hard time in finding the way to approach this kind ...
3
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3answers
330 views

Inverse fourier transform of $ 1/(1+s^2)$

Hoi, I want to have the inverse fourier transform $\mathcal{F}^{-1}(\frac{1}{1+s^2})$. So I thought about using some properties of fourier-transform. But knowing the answer I must make some sort of ...
0
votes
1answer
34 views

$|p(z)| \leq M$ for $|z| \leq 1$ Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$

Let $p(z)= \sum_{k=0}^n a_k z^k$ , $a_n \neq 0$ , be a polynomial of degree $n$ such that $|p(z)| \leq M$ for $|z| \leq 1$. Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$ This was an exam ...
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1answer
28 views

Two different results with contour integration

This is probably going to be a stupid question ( I don't feel great today) but I can't get around this problem. $$I = \int_\mathbb R \frac 1 {(3x-2i)^2} dx $$ I thought that using contour ...
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1answer
45 views

$ \int_\gamma \frac{1}{z\sin z}dz$ where $\gamma$ is the circle $|z| = 5$

My understanding is that if this integral exists in the real sense, i.e. real Riemann-wise, then I can apply the residue theorem. If not, I may use the Cauchy Principal Value, to obtain a value. To ...
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1answer
117 views

Help to solve complex equation related to the Gamma function

I would need some help to solve the next complex equation for $y\in\mathbb {R}$, which I already know to be real-valued: $$ \frac {1} {2i}\left ((2\pi)^{\text {iy}}\text {}\text {Sin}\left (\frac ...
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1answer
35 views

How can I prove that the zeroes of $f(z)=1+1/2^z$ have no real part?

I want to prove that the zeroes of the function $f(z)=1+1/2^{z}$ have no real part. Is the following correct? $f(z)=0$ so $2^{z} = -1$ and $-1=e^{i\pi}$ so $e^{i\pi} = e^{z\ln2}$ therefore $z= ...
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0answers
21 views

an example of functions which is essentialy bounded but not continuous in circle

Can you give me an example of a function which is essentially bounded but not continuous in the unit circle and bounded in the open unit ball?
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0answers
39 views

Removable singularity of a harmonic function

Assume that $h$ is harmonic in the punctured unit disk $\mathbb D\backslash\{0\}$ such that $$ \lim_{r\to0}h(re^{it})=0 $$ for all $t\in\mathbb R$. Can $h$ be extended to a function harmonic in ...
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1answer
84 views

Prove that an analytic function, real-valued on radii $[0, 1)$ and $[0, e^{i\pi\sqrt 2})$, is constant on the open unit disk

Suppose $f$ is analytic in the open unit disc and real valued on the radii $[0, 1)$ and $[0, e^{i \pi \sqrt 2})$. Prove that $f$ is constant. I'm not sure how to solve this.
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1answer
331 views

Hausdorff Dimension of Arbitrary Julia Set

I am looking to find an exact solution to the Hausdorff dimension of a Julia set $J(f)$ for a polynomial $f: z \mapsto z^2 +c$ given an arbitrary $c$. I know this question is known for a number of ...
4
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1answer
685 views

Complex Variable vs Real Analysis 1

I took Real Analysis 1 last semester, and it was challenging, but not as bad as I thought it would be. I am considering taking Function of a Complex Variable this semester, but I am torn. I don't know ...
3
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1answer
1k views

Proof that an analytic function that takes on real values on the boundary of a circle is a real [duplicate]

Possible Duplicate: Let f(z) be entire function. Show that if $f(z)$ is real when $|z| = 1$, then $f(z)$ must be a constant function using Maximum Modulus theorem I'm having trouble proving ...
2
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1answer
76 views

An entire and one-to-one function must be of the form AZ+B, A non-zero. How to rule out higher degree polynomials in z? [duplicate]

Show that if f is entire and one-to-one, then it must be of the form AZ+B, with A not equal to zero. I am editing my question, since there are duplicates on this forum to the question of why an ...
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1answer
307 views

“Complex analysis,” by Elias M. Stein: Answers.

Does this book have answers to its problems in the back? I can't seem to view the back of the book properly on amazon preview. Thanks.
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1answer
31 views

Holomorphic function and nth derivative.

Let $K$ be a open connected subset of complex numbers and $f$ holomorphic on $K$. If $f=0$ on some open disc $D$ in $K$, then is it true that $n$th derivative of $f$ is $0$ for all points in $D$ ...
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0answers
24 views

Searching for a constant transformation in $ \mathbb C$

I am having a continous transformation: $f: \mathbb C \to \mathbb C $ with a set $B \subseteq \mathbb C $, which is bounded. Now I want to proove that $ A = f^{-1} (B)$ is NOT bounded! I know it ...
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2answers
58 views

How to find the residues of $\frac{1}{(z^4+4)^2}$?

How to find the residues of this function? $$\frac{1}{(z^4+4)^2}$$ So far, I found the poles: $z_1=-1-i$, $z_2 = -1+i$, $z_3=1-i$, $z_4=1+i$. I know they are of the second order. But I have ...
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0answers
28 views

Is restriction of $exp:\mathbb{C} \to \mathbb{C}^*$ to $A = \{ x+iy : x \in R, y \in ]1, 1+2\pi]\} \subset \mathbb{C}$ a bijection?

I have this question: Is the restriction of exp function $exp:\mathbb{C} \to \mathbb{C}^*$ to $A = \{ x+iy : x \in R, y \in ]1, 1+2\pi]\} \subset \mathbb{C}$ a bijection? Here's what I tried: ...
0
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1answer
17 views

Biholomorphic mapping of $\tan(z)$

I'm supposed to solve this question: Show that the function $\tan$ maps the vertical strip $-\frac{\pi}{4}<x<\frac{\pi}{4}$ biholomorphically to $\dot B(0,1)$ It is obvious that ...