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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2
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1answer
26 views

integral calculate by complex analysis methods

Calculate using methods from comples analysis. $$ \int_0^{2\pi} \,\sin ^{2n} \phi\, d\phi$$ So this is how I started: $$\sin^{2n} \phi = \left[\frac{e^{i \phi}-e^{-i \phi}}{2i}\right]^{2n} = ...
1
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2answers
23 views

Problem on bilinear transformation

Under the transformation $\displaystyle w=\frac{z-1}{z+1}$, show that the map of the straight line x=y is a circle and find its centre and radius. My attempt: Putting z=x+ix, $\displaystyle ...
0
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2answers
41 views

How to show that $\delta_{x_n}\xrightarrow{w}\delta_{x} \iff x_n \to x$

Let $x_n$ be a sequence of reals. Show that $$\delta_{x_n}\xrightarrow{w}\delta_{x} \iff x_n \to x$$ Since the weak convergence is equivalent to pointwise convergence of characteristic functions ...
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0answers
12 views

Star-shaped slit complex plane

How would one formally prove that the slit complex plan, $\mathbb{C}^{+}=\mathbb{C}-[0,\infty)$ is star shaped with star-center, $p\in(-\infty,0)$? I know how to show it graphically, but wasn't sure ...
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0answers
19 views

is this definition of meromorphic function missing a requirement?

Looking at Definition 1.1.33 in the book Complex geometry by Huybrechts: Def. Let $U \subset \mathbb{C}^n$ be open. A meromorphic function $f$ on $U$ is a function on the complement of a nowhere ...
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0answers
15 views

Property about multiplication of finite order entire functions

Lets say $f_1$ and $f_2$ are entire functions of finite order a,b respectfully. And say $a>b$. I need to prove the order of $f_1f_2$ is $a$. I've tried many things by now.. they all failed. And I ...
2
votes
0answers
22 views

Find a real entire function $f(z)$ asymptotic to $\ln(x^2+1)$ for real $x$.

Find a real entire function $f(z)$ asymptotic to $\ln(x^2 +1)$ for real $x$. More specific I want $f(0)=0$ and $\frac{1}{2} \ln(x^2+1) < f(x) < 2 \ln(x^2+1)$. Or prove it does not exist.
3
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2answers
63 views

And another real integral to be solved by contour integration

I want to solve $$\int_0^\infty\frac{1}{x^3+x^2+x+1}dx$$ and i have really learned a lot already by failing to solve it. I want to solve it using a clever contour. It is possible to do it using ...
0
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0answers
13 views

Natural surjection from complex upper half plane into modular curve

I am considering the natural surjection $\pi : \mathcal{H} \to Y(\Gamma)$ where $\mathcal{H}$ is the complex upper half plane and $Y(\Gamma)$ the modular curve of the congruence subgroup $\Gamma$. ...
1
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1answer
13 views

Series of points in a bounded sector of a complex half-plane

The question is: consider an infinite sequence of points which lie in a bounded sector of the complex plane, whose angular width is strictly less than pi (that is, it's an open sector of a ...
1
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1answer
25 views

Show $\prod \limits_{\substack{k=0\\k\neq k_0}}^{K-1}\left(\exp\left[{\frac{2 \pi i (k-k_0)}{K}}\right]-1\right)=(-1)^{K-1} K$

I know it is true that, for any $k_0 \text{ s.t. } 0\leq k_0 < K$, $$\prod \limits_{\substack{k=0\\k\neq k_0}}^{K-1} \left(\exp\left[{\frac{2 \pi i (k-k_0)}{K}}\right]-1\right)=(-1)^{K-1} K$$ ...
0
votes
1answer
50 views

Can we only integrate over continuous functions?

Look at this definition: Why is it assumed that f is continuous? Is it just to make sure the curve we get is integrable? COuld a weaker condition make a problem?
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3answers
49 views

A complex polynomial in $z$ and $\bar z$ contains no terms with $\bar z$ if and only if its $\bar z$-derivative is zero

I am struggling with this exercise: Let $p(x,\bar{z})=\sum a_{lm}z^l\cdot\bar{z}^m$ be a polynomial in $z$ and $\bar{z}$ (so only finitely many $a_{lm}$ are non-zero). Show that p contains no ...
0
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2answers
21 views

Find angle $\alpha$ from a complex vector

I'm trying to solve this problem from a Russian book: Find the angle which is needed to rotate the vector $3\sqrt{2} + i2\sqrt{2}$ to obtain the vector $-5+i$. EDIT: $\tan\dfrac{\pi}{6} \neq ...
0
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1answer
18 views

Unclear result in characterization of subharmonic functions.

Let $\Omega$ be a region in $\mathbb{C}$ and let $\phi:\Omega \to [-\infty,+\infty)$ be an upper semicontinuous function. TFAE: \ i)$\phi$ is subharmonic in $\Omega$, ii) for any disc ...
0
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2answers
34 views

Essential singularity of a product $fg$

I came across this question: If $f:\mathbb{D}-\{0\}\longrightarrow \mathbb{C}$ is holomorphic where $z=0$ is an essential singularity of $f$ and $g:\mathbb{D}\longrightarrow \mathbb{C}$ is ...
0
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1answer
15 views

finite geometric series has a known sum. does this imply anything about the halfway point ?

Hi: I have a question about a geometric series and it has nothing to do with complex analysis because the series is real-valued. But I know people who follow this tag are really talented and I have ...
2
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0answers
30 views

Generating series - Finite groups of order $n$

I am wondering if something of interest can be said about one of the two series $$G_1(x)=\sum_{n=1}^{+\infty}{\mathcal{G}(n)z^n}$$ $$G_2(s)=\sum_{n=1}^{+\infty}{\frac{\mathcal{G}(n)}{n^s}}$$ where ...
1
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1answer
32 views

complement of zero set of holomorphic function is connected

I'm stuck with the following part of exercise 1.1.8 in Hubrechts book Complex geometry: Prove that, if $U \subset \mathbb C^n$ is open connected, then $U \setminus Z(f)$, the complement of zero set ...
2
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1answer
33 views

About the definition of complex multiplication

Some people say that the complex product is the way it is to respect the distributive law of multiplication. However, the distributive law acts in the whole number, like: $$(a+b)(c+d) = ac + ad + bc ...
5
votes
1answer
92 views

If $f(\mathbb{C})\subset \mathbb{C}-[0,1]$ then $f$ is constant

If $f:\mathbb{C}\longrightarrow\mathbb{C}$ is an entire function such that $f(z)\neq w$ for all $z\in \mathbb{C}$ and for all $w\in [0,1]\subset \mathbb{R}$, how to prove that $f$ is constant (without ...
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2answers
41 views

If $f[\mathbb{T}]\subset \mathbb{R}$ then $f$ is constant

If $f:\overline{\mathbb{D}}\longrightarrow\mathbb{C}$ is a holomorphic function over $\mathbb{D}$ and $f(\mathbb{T})\subset \mathbb{R}$ then is $f$ constant? Consider: ...
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1answer
32 views

Examples of divergent series summed by means of the analytic continuation of the corresponding

For my Bachelor's thesis, I am investigating divergent series. This is (yet another) question on this topic. Apparently, a divergent series $$ S = \sum_{n=1}^{\infty} a_{n} $$ can be summed by means ...
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1answer
18 views

Suppose that $f(z) = g(z)/h(z)$ is analytic on the annulus $\{1 < |z| < 2\}.$ Show that $f$ can be written as $f = G(z)/H(z)$

Suppose that $g, h$ are continuous, nowhere vanishing functions on $\{|z| < 2\},$ $\{{|z| > 1} ∪ ∞\}$ respectively. Suppose that $f(z) = g(z)/h(z)$ is analytic on the annulus $\{1 < |z| < ...
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0answers
26 views

If $\sum_{n=0}^{\infty}f^{(n)}(z_0)$ converges then $\sum_{n=0}^{\infty}f^{(n)}(z)$ converges [duplicate]

If $f:\mathbb{C}\longrightarrow \mathbb{C}$ is an entire function such that $\displaystyle \sum_{n=0}^{\infty}f^{(n)}(z_0)$ converges for some $z_0$, how to prove that ...
0
votes
0answers
7 views

An entire divison of two p-order entire fucntion is also p-order at most

Let $f_1$ and $f_2$ be two entire functions such $g=f_1/f_2$ is also entire. It is given that $f_1,f_2$ are of finite order $p$. I need to show show that $g$ is also of order $p$ at most. I've been ...
0
votes
1answer
80 views

Can a finite value for $\int_1^\infty \exp(x^2)\,dx$ be defined?

Why should $$\int_1^{\infty}\exp(ix^2)dx,\int_1^{\infty}\exp(-ix^2)dx,\int_1^{\infty}\exp(-x^2)dx$$ converges but not: $$\int_1^{\infty}\exp(x^2)dx$$ Is there any way that assigns a value to ...
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0answers
43 views

How many zero's does a general real entire function $f(z)$ have?

Let $f(z)$ be a real entire function. How do we find the number of solutions for $f(w)=0$ ? Can we express the number of zero's of $f$ in terms of its Taylor coëfficiënts ? Im not looking for the ...
0
votes
1answer
14 views

Principal Part of Laurent series' expansion of $f(z)=\frac{\sin(z^3)}{(1-\cos z)^3}$

I need to calculate principal part of the Laurent series expansion of $f$ at $z_0=0$ with $$ f(z)=\frac{\sin(z^3)}{(1-\cos z)^3} $$ I can see that $f$ has a pole of order 3 at $z_0=0$ , and also ...
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0answers
31 views

Jensen's Formula and the measure of $\{\theta \in [0, 2\pi]: f(e^{i\theta}) = 0\}$

Let $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$. Suppose $f$ is continuous on $\overline{\mathbb{D}}$ and analytic on $\mathbb{D}$ with $f(0) \neq 0$. Then if $r$ is such that $0 < r < 1$ ...
0
votes
0answers
24 views

Power series with complex variables inequality

I am struggling to prove the following inequality: For $z \in \mathbb{C}, r \in \mathbb{R}, n \in \mathbb{N}$, if $|z| \leq r$ and $1 \leq r < n$ then ...
0
votes
2answers
49 views

geometric description of set of complex number

A set of complex number: $$S=\{ z\in \Bbb C : |z|=\lambda |z-1|\}$$ what's the geometric description? I try to draw it ... which seems like a circle but cannot find the equation to describe it..
2
votes
3answers
27 views

Factorization of $z^4 +1 = (z^2 - \sqrt 2z+1)(z^2 + \sqrt 2 z+1)$ for complex z

How can I get this equation from LHS to RHS by using the four roots of $z^4 +1 = 0$ are $z=\pm\sqrt{\pm i}$ $$z^4 +1 = (z^2 - \sqrt2 z+1)(z^2 + \sqrt2 z+1)$$
1
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0answers
18 views

Validity of Laurent series's principal part calculation

I need to calculate the principal part of the Laurent expansion of $f$ around a given $z_0$ in an annulus of the form $\{z\in \mathbb{C}:0<|z-z_0|<r$} and then use this to find $Res(f,z_0)$ ...
0
votes
1answer
61 views

Show that an entire function that is real only on the real axis has at most one zero, without the argument principle

Could someone advise me on how to approach this problem: Suppose an entire function $f$ is real if and only if $z$ is real. Prove that $f$ has at most $1$ zero. without the use of argument principle ...
2
votes
2answers
46 views

If $f,g$ are entire functions and$\ fg\equiv 0$ then either $f \equiv 0$ or $g\equiv0. $

Let $f,g$ be entire functions such that $g \not\equiv 0.$ If $fg\equiv0$ in $\mathbb{C},$ could anyone advise me how to show $f \equiv0$ in $\mathbb{C} \ ?$ Thank you.
0
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1answer
27 views

Smooth parametrisation in the complex plane.

My book defines a complex smooth parametrisation like this. First a parametrisation is a complex funtion z of a real variable t. Where t is defined on $[a,b]$. It is smooth if $z'(t)$ exists and is ...
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1answer
25 views

Verification of Laurent Series calculation

I tried to calculate the Laurent series of these functions but I have no way to verify my answers. i) $$ \begin{align} f(z)=\frac{e^{z^2 }-1}{z^4}, \mathbb{D}=\mathbb{C} \backslash \{0\} ...
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vote
5answers
72 views

If $p(z)$ is an injective polynomial $\Longrightarrow$ $p(z)=az+b$ [on hold]

If $p(z)$ is an injective polynomial, how to prove that $p(z)=az+b$ with $a\neq 0$. $p(z)\in\mathbb{C}[z]$. Any hint would be appreciated.
0
votes
1answer
20 views

Supremum of the set $\{\operatorname{Re}(iz^3+1) : |z|<2\}$

I need to find supremum of the set of all real numbers of the form $\operatorname{Re}(iz^3+1)$ such that $|z|<2$. By the inequality $-|w|\le \operatorname{Re}(w)\le |w|$ we have ...
1
vote
2answers
36 views

If $p(z)$ is a monic polynomial then $p(z)+b=(z-z_1)(z-z_2)\cdots (z-z_n)$

I need some help with this problem: If $p(z)$ is a monic polynomial of degree $n$ then there is a $b\in\mathbb{C}$ such that $p(z)+b=(z-z_1)(z-z_2)\cdots (z-z_n)$ where $z_1,z_2,\dots,z_n$ are simple ...
2
votes
1answer
66 views

There is no nonconstant entire function $f$ such that $f(z+1)=f(z)$ and $f(z+i)=f(z)$ [duplicate]

Claim: there is no entire non-constant function $f$ such that $f(z+1)=f(z)$ and $f(z+i)=f(z), \forall z\in \mathbb{C}.$ May I verify if my proof is valid? Or is there a better way to approach this ...
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0answers
30 views

Prove: $|f(z)| ≤ \frac{2A|z|}{1−|z|}$.

Let $f$ be a holomorphic function on the unit disc $\{z : |z| < 1\}$ satisfying $f(0) = 0$ and $Ref(z) ≤ A$ for some positive number $A > 0.$ Prove: $|f(z)| ≤ \frac{2A|z|}{1−|z|}$. Not sure how ...
10
votes
6answers
383 views

Reference request for undergraduate complex analysis.

I am a second year student studying electrical engineering. I self-study pure mathematics and want to pursue a career as a mathematician. What are some prerequisites for studying complex analysis? ...
0
votes
2answers
52 views

Complex Analysis Problem and Advice

Let $f$ be an odd function that is holomorphic in $\mathbb{C}- \{0\}$ such that $|f(z)| \leq \dfrac{1}{|z|}+ |z|^2, $ where $z \neq 0.$ Could someone advise on how to show $f(z) = \dfrac{a_{-1}}{z} + ...
2
votes
1answer
14 views

Guidance / Help with Laurent series expansion in a certain annulus

I am trying to study complex analysis and I've come across this $$ \begin{align} f(z)= \frac{1}{1+z^2} \end{align} $$ I need to determine the Laurent series expansion for the annulus ...
0
votes
1answer
21 views

A question about the relation between divergence and absolute divergence.

Princeton Lectures in Complex Analysis by Stein and Shakarchi says the following: If $|z| > R$, then a similar argument proves that there exists a sequence of terms in the series whose ...
2
votes
2answers
24 views

how to calculate |exp(-ia)+exp(-ia')|^2

What is the correct way to calculate something like $|\exp(-ia)+\exp(-ia')|^2$ ? I have tried simply multiplying the term inside the absolute value by its complex conjugate, ...
3
votes
2answers
33 views

Characterization of entire functions to be a polynomial

I need some help with this proposition: If $f:\mathbb{C}\longrightarrow \mathbb{C}$ is an entire function such that $\{z\in \mathbb{C}:f(z)=w\}$ is finite for all $w\in \mathrm{Im} (f)$ then $f$ is a ...
0
votes
2answers
61 views

Direct evaluation of a series from Euler's identity.

Is there a direct way to evaluate: $$ \sum_{k=0}^{\infty} (-1)^k \dfrac{\pi^{2k}}{(2k)!}=-1 $$ Note that this follows from Euler's identity.