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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293 views

isomorphism between torus

I am working on Lectures On Riemann Surfaces by Forster. I am having trouble figuring out the following question. 1.5 a) Let $\Gamma,\Gamma'\subset\mathbb{C}$ be two lattices. Suppose ...
7
votes
1answer
204 views

Determine all the values of $1^{\sqrt{2}}$

I can't seem to understand how to solve this. I mean, if we weren't dealing with complex numbers, then I suppose it is clearly 1, but I don't know how to approach this. Apparently the answer is ...
0
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0answers
198 views

Prove that all the roots of $\cos z=a$, where $−1 \leq a \leq 1$, are real.

I'm trying to apply how I solved this question for $\sin z = a$, but I ran into a small issue. $\cos z = \frac{e^{iz} + e^{-iz}}{2} = a$. Then, if we state $w = e^{iz}$, then we obtain $w + ...
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1answer
323 views

Prove that all the roots of $\sin z = a$, where $-1 \leq a \leq 1$, are real.

I'm having a hard time to understand this. Is this saying, that $z$ is real whenever $a$ is between -1 and 1? If so, would I go about tackling it like so: $\sin z = \frac{e^{iz} - e^{-iz}}{2i} = a$. ...
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1answer
165 views

integral involving square root using complex methods: what choice of path?

I'm asked to compute using complex methods the following integral: $$ I(a)= \int_0^1 \mathrm{d}x \frac{\sqrt{1-x^2}}{x^2-a^2},$$ where $a>1.$ What I know is the following: for $|z|<1,$ the ...
3
votes
1answer
171 views

Is there a flaw in this proof of Marty's theorem (normal families)?

In Ahlfors' Complex Analysis text, page 227 the author claims that if the expression $$\rho(f)=\frac{2 |f'(z)|}{1+|f(z)|^2}$$ is locally bounded (here $f \in \mathfrak F$ is a family of meromorphic ...
2
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1answer
264 views

Homework Problem: Complex Analysis Chain Rule

My classmates and I were given that we had to verify, \begin{eqnarray} \frac{\partial}{\partial z} (f \circ g) = (\frac{\partial f}{\partial z} \circ g)(\frac{\partial g}{\partial z}) + ...
2
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1answer
258 views

Complex Differentiable Functions Mapping Upper Half Plane to Itself?

Let $H$ denote the upper half plane of $\mathbb{C}$ (not including the real line). I have a theorem in my book that states: "Every automorphism of $H$ is a fractional linear transformation $f(z) = ...
2
votes
1answer
74 views

Show in between steps in this Riemann zeta function equivalence/reduciton

In the answer chosen by the OP in this question I had trouble understanding the steps taken to get the equivalences/reduce the zeta function into another one. Can somebody show me the steps to go from ...
4
votes
3answers
124 views

Analytical closed form for a definite integral

I am doing a computation in quantum field theory and the following integral occurred to me $$ I(a)=\int_{-\infty}^{+\infty}\frac{e^{-a\sqrt{x^2+1}}dx}{x^2+1} \qquad a\ge 0. $$ I would like to know ...
2
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1answer
92 views

Questions about $\ln(z)$ recurrence and fixed points.

Define property $A_R$ for an analytic function $f(z)$ as $1)$ $f(z)=0$ has exactly one solution being $z=0$ for $|z|<R$ where $R$ is a radius. And $f(z)$ is analytic within the radius $R$ ...
2
votes
1answer
148 views

Show that $\sigma^{-\lambda(x) - 1}$ is continuous on $(0,1)$.

Let $V$ be an open connected subset of $\mathbb{C}$ and $A(V)$ be the set of all (complex-valued) analytic functions on $V$. If $\lambda \in A(V)$ with $\Re \lambda(x) < 0$ for all $x \in V$ , ...
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0answers
164 views

Definite integral with a parameter

Suppose we assumed that the parameter $\alpha$ is real, and then evaluated some definite integral $I(\alpha)$, which depends on $\alpha$. Can we then claim that $I(\alpha)$ also holds for $\alpha \in ...
3
votes
1answer
113 views

Properties of plurisubharmonic functions

In book: (Oxford science publications._ London Mathematical Society monographs, new ser., no. 6) Maciej Klimek -Pluripotential theory -OUP (1992) Theorem 2.9.14 (ii): Let $\Omega$ be an open ...
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0answers
112 views

Are non-constant, analytic functions topologically conjugate to $z\rightarrow z^m$ when $f'(0)=0$?

Given a non-constant, analytic function $f$ of the form $f(z) = z^m + a_{m+1} z^{m+1} + a_{m+2} z^{m+2} + \cdots$ one can show that $f$ can be written in the form $f(z) = f_1(z)^m$ where $f_1$ is ...
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0answers
157 views

Finding holomorphic functions such that $z=(f(z))^n$

I'm having trouble in this homework question. Let's suppose $U=\mathbb{C}-\{z\in\mathbb{C}:Re(z)\leq 0\}$ and $n\in\mathbb{Z}^*_+$. I need to find all holomorphic functions $f$ that satisfies the ...
2
votes
1answer
46 views

Question about fixpoints and zero's on the complex plane.

Define property $A$ for an entire function $f(z)$ as $1)$ $f(z)=0$ has exactly one solution being $z=0$ $2)$ $f(z)=z$ has exactly one solution $=>z=0$ (follows from $1)$ ) $3)$ $f(z)$ is not a ...
0
votes
1answer
49 views

Complex Analysis

Determine the derivative of the following functions and state where they are analytic. $$f(z) = \log(z^3) \quad \Rightarrow f'(z) = \frac{3z^2}{z^3} = \frac{3}{z}$$ Hence, this function is analytic ...
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2answers
197 views

Complex Analysis Question About Analytic Functions

I have some questions about knowing where and where not functions are analytic. Here's a function, f(z)= $\frac{Log(z+4)}{z^2+i}$ -I know that this function is not defined for ...
4
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1answer
485 views

Implicit function theorem for several complex variables.

This is the statement, in case you're not familiar with it. Let $ f_j(w,z), \; j=1, \ldots, m $ be analytic functions of $ (w,z) = (w_1, \ldots, w_m,z_1,\ldots,z_n) $ in a neighborhood of $w^0,z^0$ ...
-1
votes
1answer
59 views

Square RootComplex Number and which branch can be used for integration contour?

What is the definition of the square root of a complex number and which branch of the square root one can use for the integration contour. For example if we have the following square root. ...
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0answers
18 views

$\binom{z}{n}$ converges to $0$ if and only if $\Re z >-1$

$\binom{z}{n} = \frac{z\cdot (z-1) \cdot \ldots\cdot(z-n+1)}{n!}$ and $z\in\mathbb{C}$ is a complex number. Please, can you help me to solve this problem.
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1answer
145 views

Measure & integration vs Complex analysis

If I want to go down a statistics (masters degree) track that's a bit heavy on the math side, and I had to choose between complex analysis and measure theory as a course which one should I take and ...
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1answer
49 views

How to show $z$ choose $n$ is bounded sequence, where $z$ is complex number

Given a complex number $z$ and a positive integer $n$, we define "$z$ choose $n$" by $$\frac{z(z-1)\cdots(z-n+1)}{n!}.$$ How can we show that the sequence of all $z$ choose $n$ ($n\ge 1$) is a ...
3
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1answer
88 views

Bounded Holomorphic Function - Banach Space?

Can someone, help me in this question, please? Let $U\subset\mathbb{C}$ be open set and $H_\infty(U)=\{f:U\to\mathbb{C}:f\text{ is bounded and homolorphic}\}$. Show that $H_\infty(U)$ is a closed ...
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2answers
83 views

$\det (A^{-1})$ from eigenvalues of $A$

Suppose I have invertible square matrix $A$ in the complex field and I know all of its eigenvalues and they may be assumed to be non zero. Is there a way to write $\det(A)$ and $\det (A^{-1})$? PS. ...
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2answers
117 views

Eigenvalues of $(A+B)^{-1}$

Suppose I know the eigenvalues of $A$ and $B$. Is there a way to write eigenvalues of the following? (1). $(A+B)$ (2). $(I+A)$ (3). $(I+A)^{-1}$ (4). $(A+B)^{-1}$ where $A, B$ are matrices in ...
2
votes
1answer
63 views

Trigonometric Polynomial Coefficients

Suppose $a(z)=\sum_{j=-n}^n a_j z^j \geq 0$ on the unit circle $|z|=1$. I would like to prove the seemingly simple fact that $a_j=\overline{ a_{-j}}$. My attempt: \begin{align} a(e^{i\theta}) = ...
2
votes
1answer
110 views

Example 1.7 Girondo's Introduction to compact riemann surfaces

Consider first the algebraic equation $$y^{2}=\prod_{k=1}^{2g+1}(x-a_k)$$ where $\{a_k\}_{k=1}^{2g+1}$ is a collection of $2g+1$ distinct complex numbers, and let $$ S^\circ =\{(x,y) \in ...
3
votes
1answer
101 views

Analyticity of a real function on $[0,\infty)$

I'm struggling to understand the difference of the analyticity of a real and a complex functions. Consider the following real valued function which is a minimal example of a somewhat more involved ...
2
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2answers
164 views
5
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1answer
634 views

Uniformly bounded sequence of holomorphic functions converges uniformly

Consider an open connected set $\Omega\subset \mathbb{C}$, and $f_n\subset H(\Omega)$. Suppose $f(z)=\lim_{n\to\infty}f_n(z)$ exists and $|f_n(z)|\leq M$ for all $z\in \Omega$. Show that ...
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3answers
217 views

If the critical values of a complex polynomial lie in the unit disc then the preimage of the unit disc is connected.

Let $p(z)$ be a complex polynomial such that $p'(z) = 0 \implies p(z) \in D,$ where $D$ is the open unit disc: $D= \{z\in \mathbb{C}\:\: |\:\:|z| < 1 \}.$ I want to prove that $p^{-1}(D)$ is ...
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1answer
220 views

Monotonic log det function?

I want to claim that the follwoing function is monotonically increasing in $d_j$. ...
0
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1answer
87 views

Bound of $\log \det$

I want to find a bound to the function $$R(d_i, ...
9
votes
3answers
761 views

Rigorous derivation/explanation of delta function representation?

I am interested in a derivation of the following representation for the Dirac delta function: $$\delta(x-a)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i p (x-a)}dp$$ It is clear to me how the property ...
4
votes
3answers
477 views

Which book on complex analysis is good for self study?

Which book on complex analysis is good for self study? I am an average student and have just a very basic knowledge of this subject.I want to cover up to Runge's Theorem. I heard about few books- ...
4
votes
1answer
105 views

Codomain of holomorphic function always all of $\mathbb{C}$?

If a holomorphic function $f:\mathbb{C}\to\mathbb{C}$ is bounded, i.e. $|f| \lt A$ for some constant $A$ for the entire domain of $f$, then $f$ is constant according to Liouville's theorem. Does ...
2
votes
3answers
423 views

Given $\sin z=5$. Find $e^{iz}$ (Complex Trigonometric Function)

Given $sin$ $z=5$. Find $e^{iz}$. Here is what I have done: \begin{align} \sin z &= \frac{e^{iz}-e^{-iz}}{2i}=\frac{e^{i(x+iy)}-e^{-i(x+iy)}}{2i}\\ &=\frac{e^{-y}(\cos x+i\sin x)-e^{y}(\cos ...
1
vote
1answer
63 views

Evaluate the integral $\int_{r\leq R}\frac{\mathrm dr}{r^2-k^2}$

I try to evaluate: $$\int_{0\leq r\leq R}\frac{\mathrm dr}{r^2-k^2}$$ Where $k$ is a complex number with positive imaginary part and non zero real part. $R$ is a positive number. By Cauchy's theorem ...
5
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1answer
233 views

Why does the amoeba shrink to its skeleton when we go to infinity?

Let $f\in\mathbb{C}[X_1^{\pm1},\ldots,X_n^{\pm1}]$ a Laurent polynomial. Let $\mathrm{Log}:(\mathbb{C}\setminus\{0\})^n\to\mathbb{R}^n$ defined by ...
4
votes
1answer
64 views

Is this expression well-defined: $\int_{-i}^{i} \frac{dz}{z}$? How to evaluate it?

I'm learning some basic complex analysis and came across this integral $$\int_{-i}^{i} \frac{dz}{z}.$$ First of all, Wolfram can't calculate it, but it might be because he treats $i$ like a real ...
2
votes
1answer
39 views

Analyticity of Products

Assume we have two functions $f,g:\Omega\rightarrow\mathbb{C}$ that are analytic and a third function $h:\Omega\rightarrow\mathbb{C}$ with $f=g\cdot h$. Can one now show that $h$ is analytic as well? ...
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1answer
95 views

Problem computing a complex line integral

I am asked to evaluate the function $f: \mathbb{C} \rightarrow \mathbb{C}: z \mapsto \overline{z}$ over the curve $C$ which is the union of the line from 0 to 1 and the line from 1 to 1+$i$. So, ...
3
votes
1answer
247 views

Univalent functions normal family.

Prove that: The family of $S$ of univalent functions on the unit disc with f(0)=0, f'(0)=1 is a normal family. I'm pretty sure i have to do it with Zalcmans Lemma: a family of analytic functions on ...
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2answers
101 views

Inverse Fourier transform of $(z-i/2)^{-N/2}$

How can I compute the integral corresponding to the inverse Fourier transform of $(z-i/2)^{-N/2}$: \begin{equation} I \equiv ...
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0answers
416 views

Branch Cut for $\ln(1 - z^2)$

"Given that $g(z) = \ln(1-z^2)$, defined on $\mathbb{C}\backslash \left(-\infty, 1\right]$, i.e. the branch cut is from $-\infty$ to $1$ along the real axis. Find $g(-i)$ given $g(i) = \ln(2)$" I ...
8
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1answer
76 views

Let $M_1,M_2,M_3,M_4$ be the suprema of $|f|$ on the edges of a square. Show that $|f(0)|\le \sqrt[4]{M_1M_2M_3M_4}$

Let $G$ denote the interior of the square with vertices $1,i,-1,-i$. Suppose $f$ is holomorphic on $G$ extends continuously to $\overline{G}$, and $M_1,M_2,M_3,M_4$ are the suprema of $|f|$ on the ...
2
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0answers
299 views

Using the theorem of Rouché in order to show the fundamental theorem of algebra

Infer from the theorem of Rouché that every non-constant polynomial does have a zero point in $\mathbb{C}$ (Fundamental Theorem of Algebra). Good day, consider the polynomial $$ ...
2
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1answer
39 views

A sequence on a line through the origin in $\mathbb{C}$ must converge on that line?

Does this make sense? Lemma: Consider a sequence $(z_n) \subset \mathbb{C}$. If $z_n \to z$ and $Arg(z_n) = Arg(z_m)$ for all $n,m \in \mathbb{N}$, then $Arg(z) = Arg(z_n)$. Proof: Fix $n \in ...