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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5
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1answer
237 views

Sheaf of meromorphic functions on non-compact Riemann surfaces

Why does the first cohomology group $H^1(X, K)$ of the sheaf of meromorphic functions on a non-compact Riemann surface $X$ vanish?
7
votes
2answers
694 views

True/False Questions for Complex Analysis

I am studying for my (introductory) complex analysis final exam tomorrow. I am practicing an old final exam, which unfortunately has no answer key. Here is a link: ...
1
vote
1answer
225 views

asymptotic behavior of the real part of the Riemann zeta function for $0<\sigma<1$

consider the zeta function $\zeta(\sigma+it)$ for $\sigma>1$ : $$\zeta(\sigma+it)=\sum_{n=1}^{\infty}\frac{1}{n^{\sigma+it}}$$ And: $$\zeta(\sigma-it)=\sum_{n=1}^{\infty}\frac{1}{n^{\sigma-it}}$$ ...
4
votes
1answer
254 views

Number of roots of a complex equation/ Rouche's theorem

For $n\geq2$ consider the equation $z^n+z+n=0$ for $z\in \mathbb C$. Show that if $k$ is an integer with $1\leq k \leq n$ then inside the sector $$ S_k=\left\{z\in \mathbb C: 0< Arg(z) < ...
3
votes
2answers
221 views

convergence of complex power series - infinite convergence radius

My books states that if the convergence radius of a complex power series is $+\infty$, then the power series is uniformly convergent over every 'disk' of the complex plane, although not necessarily ...
2
votes
0answers
287 views

why $\pi \cot \pi z = \sum _{n=-\infty}^{\infty} \frac{1}{z+n}$

In prove of the above claim I need prove: $\cot (\pi\cdot z)$ is bounded for $\{y \geq 1 , 0.5 \geq x \geq -0.5\}$ In stein book a function has defined: $$\Delta ( z ) = \pi \cot \pi z - \sum ...
3
votes
2answers
348 views

uniform convergence of sequences of analytic functions

Suppose you have a sequence $f_n$ of analytic functions on an open set $\Omega$, which converges uniformly on compact subsets of $\Omega$. Can you conclude that $f_n$ converges uniformly on the whole ...
3
votes
2answers
177 views

Divergence of an infinite product

How can I prove that the infinite product $$\displaystyle\prod_{n=1}^{+\infty}(1+z^{2n})$$ diverges for $|z|>1$?
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2answers
71 views

Does $f(s)\not= 0 \Rightarrow g(s)=0$ imply that $g(s)=0\Rightarrow f(s)\not= 0$?

I have the following implication: $$f(s)\not= 0\Rightarrow g(s)=0$$ Then we can deduce that its converse is also true. $$g(s)\not= 0\Rightarrow f(s)=0$$ where ...
4
votes
1answer
159 views

complex analysis/ Taylor expansion question

Let $f:\mathbb C \rightarrow \mathbb C$ be holomorphic and $f(z)=f(-z)$ for all $z\in \mathbb C$. Show that there exists a holomorphic function $g$ such that $g(z^2)=f(z)$. If I take ...
4
votes
1answer
2k views

identity theorem in complex analysis explanation

As Conway states it the theorem is as follows: Let $G$ be an open connected set and let $f:G\rightarrow \mathbb{C}$ is analytic on $G$. Then the TFAE: $f\equiv0$ $\{z\in G: f(z)=0 \}$ has a limit ...
0
votes
1answer
342 views

How to calculate the residue of the pole of this function?

I'm learning modular form and run into such function: $$ j(\tau)=\frac{(2\pi)^{12}+\ldots}{(2\pi)^{12}q+\ldots}=\frac{1}{q}+\ldots $$ where $q=e^{2\pi i\tau}$. (I omitted the details of definition of ...
3
votes
1answer
153 views

Why can we differentiate this infinite series?

We have the formula: $$ \frac{1}{z}+\sum_{d=1}^{\infty}\left(\frac{1}{z-d}+\frac{1}{z+d}\right)=\pi\cot\pi z $$ Then in the book I'm reading, the author differentiate it $k-1$ times to get a formula ...
3
votes
2answers
965 views

Branch cut question

I have a function $$f(z)=(z-1)^{3/5}(z+1)^{2/5}$$ and I have the branch of this function chosen such that $$-\pi<\arg(z\pm1)\leq\pi$$ How do I show that a branch cut is not required on the section ...
2
votes
2answers
132 views

calculate the integral

Compute $$I=\int_C\frac{e^{zt}}{1+z^2}dz$$ where $t>0$, a real number, and $C$ is the line $\{z\mid \mathrm{Re}(z)=1\}$ with direction of increasing imaginary part. I tried to integral along the ...
0
votes
1answer
83 views

question on analytic extension

Suppose $f$ is analytic in the annulus $1<|z|<2$ and there exists a sequence of polynomials $p_n$ converging to $f$ uniformly on every compact subset of this annulus. Show that $f$ has an ...
1
vote
1answer
98 views

Rotation and fixed points (2)

The motivation to this question can be seen in Rotation and fixed points My qeustion is: The vector $(u,v)$ (also $(x,y)$) depend on the variable $s$. When $\theta (s)\ne 0\,\pmod{2\pi}$ we get ...
2
votes
0answers
135 views

winding number question

This is part of a proof from Banach algebra techniques in Operator theory by Ronald Douglas on page 170. Let $\epsilon>0$. Let $T$ be the unit circle and $\phi\in H^\infty+C(T)$. Choose $\psi\in ...
1
vote
2answers
118 views

how to evaluate this complex integral

I need to evaluate $$\int_{|z|=2}\frac{1}{(z-1)^3}dz.$$ At $z=1$, it has a pole of order $3$. I can not remember how to find the residue when there are poles with multiplicity, could any one tell me?
3
votes
1answer
200 views

About the growth of entire functions

Define $M(r)=\sup_{|z|=r}|f(z)|$. Given an increasing function $\phi(r)$ as $r\to\infty$, how to construct an entire function $f(z)$ to satisfy the inequality $M(r)>1+\phi(r)$? Attempt to solve ...
1
vote
1answer
959 views

Complex Analysis - Argument Principle vs. Rouche's Theorem

The Argument Principle Suppose a function $f$ is meromorphic on an open set that contains a circle $C$ and its interior. Further assume that $f$ has no zeroes on $C$ (but may have zeroes in the ...
0
votes
1answer
112 views

What are the hypothetical zeros of the Dirichlet eta function

My question is: What are the hypothetical zeros of the Dirichlet eta function (the alternating zeta function) in the critical strip. This notion is far from my understanding.
5
votes
1answer
603 views

Fourier transform of $ \frac{\sinh(kx)}{\sinh(x)}$ [duplicate]

Possible Duplicate: Calculating the Fourier transform of $\frac{\sinh(kx)}{\sinh(x)}$ Im trying to compute the integral of $$I = \int_{-\infty}^\infty ...
4
votes
0answers
90 views

Invertibility of a Toeplitz operator

Let $\phi$ be a real-valued function. I am trying to show that the Toeplitz operator $T_\phi$ is invertible if the function 1 is in the range of $T_\phi$. Here is what I got so far: There exists a ...
3
votes
1answer
294 views

How can the graph of a complex function be embedded in three dimensional space?

In his book Visual Complex Analysis (an awesome book, by the way), Needham, on the topic of graphing complex functions, says that Actually, the situation is not quite as hopeless as it seems. ...
4
votes
1answer
421 views

non constant bounded holomorphic function on some open set

this is an exercise I came across in Rudin's "Real and complex analysis" Chapter 16. Suppose $\Omega$ is the complement set of $E$ in $\mathbb{C}$, where $E$ is a compact set with positive Lebesgue ...
3
votes
2answers
293 views

a problem relate to analytic function

Suppose $f$ is an analytic function on $|z|\leq 1$ with $f(0)=0$, and let $|f(z)|$ have a maximum for $|z|\leq 1$ at 1, show that $f'(1)\neq 0$ unless $f$ is a constant. Remarks: 1, At first ...
2
votes
0answers
97 views

eigenvalue question for a Toeplitz operator

Let $\phi$ be a nonzero function in $L^\infty(T)$ where $T$ is the unit circle. Let $M_\phi$ be the multiplication operator and $T_\phi$ be the Toeplitz operator. Show $T_\phi$ and $M_\phi$ have no ...
1
vote
1answer
83 views

Solving two coupled functional equations in the same time

Let $ϕ(s)=f(s)\exp(ig(s))$, $s$ is a complex number. The expressions of $f$ and $g$ are not known since they depend on the gamma function. It is well known that we can solve a functional equation ...
15
votes
7answers
2k views

Calculating the integral $\int_0^{\infty}{\frac{\ln x}{1+x^n}}$ using complex analysis

I need to calculate $\int_0^{\infty}{\frac{\ln x}{1+x^n}}$ $,n\geq2$ using complex analysis. I probably need to use the Residue Theorem. I use the function $f(z)={\frac{\ln z}{1+z^n}}$ in the normal ...
4
votes
1answer
288 views

Upper bound of an analytic function (application of Pick's Lemma)

Let $f$ be holomorphic and non zero in the disk $ |z|<1$ with $|f(z)|\leqslant 1$ in $|z|<1$ and $f(0)= e^{-1}$ . What is the best possible bound upper bound for $|f'(0)|?$ Is this bound ...
2
votes
1answer
140 views

A conformal mapping question.

I am not sure about the following question, please help, thank you. Find a conformal mapping $f$ of $D=\{|z|<1\}$ onto the domain $H=\{|\textrm{Im}(w)|<\pi/2\}$ such that $f(0)=0,\ f'(0)=2$. ...
7
votes
1answer
575 views

Complex Analysis - Location of roots of a polynomial

How many roots does the polynomial $z^4 + 3z^2 + z + 1$ have in the right-half complex plane (i.e. $Re(z) \gt 0$)? I honestly can't think of how to approach the problem as it seems different from the ...
2
votes
2answers
109 views

Holomorphic function bounded by $|z|^\frac{1}{2}$ or $|z|^\frac{3}{2}$?

My question is that: On $C-\{0\}$ the function is holomorphic and satisfies $|f(z)|\le C|z|^{3/2}$ (or $|f(z)|\le C|z|^{1/2}$). $C$ here is the constant. Show that $f=0$?
0
votes
5answers
326 views

Show $\sum_{n=1}^N e^{i n\theta}.$ is bounded for $ 0< \theta < 2 \pi$, $\forall N \in \{1,2,…\}$ [duplicate]

Possible Duplicate: Prove that partial sums of $\sum_{n=1}^{\infty}{z^n}, z \in \mathbb{C}, |z|=1$ are bounded Show $$\sum_{n=1}^N e^{i n\theta}.$$ is bounded for $ 0< \theta < 2 ...
4
votes
1answer
636 views

Dirichlet Problem on an annulus.

Having found the solution for the Dirichlet problem in the region $A=\{x+iy: 0\leq y\leq 1\}$ such that $u(x,0)=0$ and $u(x,1)=1$ to be $u(x,y)=y$, I am asked to find, using conformal maps, the ...
1
vote
3answers
214 views

Find a conformal map of the given domain

Find a complex function that maps the region $$D =\left\{|z|<1, \left|z- \frac{1}{2}\right|>\frac {1}{2}\right\}$$ conformally on to the upper half plane. Can somebody help me to find the ...
0
votes
1answer
145 views

Must an odd function be defined on the whole complex plane?

Let $f:D\subsetℂ\to \mathbb{R}$ be a non-analytic function. I know that $f$ is an odd function on $D$. The function $f$ is the argument (angle) of an analytic function. $f$ has no known form. Be odd ...
2
votes
2answers
88 views

different forms of complex sine function

Pick out the true statements: (a) $|\sin z|≤1 ∀z∈\mathbb{C}$. (b) $\sin^2z+\cos^2z=1 ∀z∈\mathbb{C}$. (c) $\sin z =(e^{iz}-e^{-iz})/2 ∀z∈\mathbb{C}$. (a) is not true for large z. (b) true. (c) ...
1
vote
0answers
113 views

Symmetric points

Let $f:ℂ→ℝ$ be a non-analytic function. Let $h(s)=f(a+s)$, where $a∈ℝ$. If $h$ is an odd function, then $(0,0)$ is a symmetric point for $h$ (the graph of $f$ is symmetric with respect to the origin). ...
1
vote
3answers
97 views

A complex equation

I want to solve the following equation: $g(s)f(s)=0$ where $f$ and $g$ are defined in the complex plane with real values and they are not analytic. My question is: If I assume that $f(s)≠0$, ...
2
votes
1answer
150 views

Algebraic integral involving complex numbers

I need some help analytically proving the following with elementary tools: $$\int_1^{+\infty} \frac{z^i + z^{-i}}{z^2 + 1} ~ \mathrm{d} z = \frac{\pi}{2} \mathrm{sech} \left ( \frac{\pi}{2} \right ...
0
votes
1answer
86 views

Solving problem without referring to three circle theorem

Let $f$ be a holomorphic in the annulus $1\leqslant|z|\leqslant 3$, and let $\sup_{|z|=1} |f(z)|=4$ $\sup_{|z|=3} |f(z)|=324$ $\sup_{|z|=2} |f(z)|=V$ What is the largest possible value for V? ...
0
votes
2answers
161 views

Proving $f$ is constant How I can answer this by considering the distance between $f(z)$ and $i$. [duplicate]

Possible Duplicate: $|\operatorname{Im}f(z)|\leq |\operatorname{Re}f(z)|$ then $f$ is constant Let $f\colon\mathbb C \to \mathbb C$ be entire. Show that if ...
6
votes
1answer
149 views

invariant subspace of a Hardy space

Let $T$ be the unit circle and $H^1=\{f\in L^1(T): \int_0^{2\pi} f(e^{it})\chi_n(e^{it})dt=0 \text{ for } n>0\}$ where $\chi_n(e^{it})=e^{int}$. Let $M$ be a closed subspace of $H^1$. Then ...
6
votes
2answers
194 views

$P(z)$ defines a polynomial

Suppose that $f$ is analytic in a simply connected domain $D$ containing distinct points $z_1, z_2 ,\ldots,z_n $ and that $\gamma$ is simple closed curve enclosing $z_1, z_2 ,\ldots,z_n $. Set ...
5
votes
2answers
164 views

two problems on complex analysis

1.Pick out the true statements: a. Let $f$ and $g$ be analytic in the disc $|z| < 2$ and let $f = g$ on the interval [$−1, 1$]. Then$ f ≡g$. b. If $f$ is a non-constant polynomial with complex ...
1
vote
1answer
163 views

Schröder functional equation

I have the following Schröder functional equation: $f(h(s))=c.f(s)$ where $f,h: ℂ→ℂ$, here $f$ is not analytic and $h$ is analytic and $c∈ℝ$. My question is: How we can solve this equation (the ...
3
votes
1answer
193 views

Integral of involving Airy function without using its antiderivative

Inspired by my answer to this question, I am interested in evaluating the following definite integral $$ \frac{1}{2\pi i} \int_{c-i \infty}^{c+i\infty} \frac{dz}{\mathop{\rm Ai}^2(z)} =1 $$ without ...
4
votes
2answers
696 views

explicit formula for coefficients of Laurent series

Give an explicit formula for the coefficients of the Laurent series on $A:=\{z: |z|>1\}$ for the function $g(z)=\frac{e^z}{z-1}$. I know how to go about finding the Laurent series, using the ...