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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3
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1answer
161 views

Order of $\frac{f}{g}$

An entire function is of finite order $\rho$ if $$\rho = \inf \{\lambda \geq 0 \ | \ \exists A, B > 0 \ s.t. \ |f(z)|\leq Ae^{B|z|^{\lambda}} \forall z \in \mathbb{C} \}$$ Prove that if $f$ and ...
3
votes
1answer
40 views

Superior limit of integrals of entire functions

Let $f$ be an entire function on $\mathbb{C}$. If $f$ is not constant, then I want to prove \begin{equation} \limsup_{R\to\infty}\int_{\lvert z\rvert=R}\lvert f(z)\rvert\,\lvert dz\rvert=\infty. ...
1
vote
1answer
90 views

Complex Analysis Integration & Branch Cuts

Suppose that the curve $C$ is any path between $z=0$ and $z=1$ which does not go through any singularities of the function below. I'm trying to show the following: $$\int_{C} \frac{1}{1+z^{2}}\,dz = ...
1
vote
2answers
46 views

Need help with a proof concerning zero-free holomorphic functions.

Suppose $f(z)$ is holomorphic and zero-free in a simply connected domain, and that $\exists g(z)$ for which $f(z) =$ exp$(g(z))$. The question I am answering is the following: Let $t\neq 0$ be a ...
0
votes
2answers
111 views

Prove that $f(z) = \sum\limits_{k = 1}^\infty \frac{z^{2^k}}{2^k}$ is continuous in the closed unit disc and holomorphic inside it.

I have started off by assuming that there is a disc of radius $r$ for which $|z|<r$ for $r \in (0,1)$ and $z \in D_r$. This implies that $|z|^{2^k} < r^{2^k}$ And after that, I don't know ...
1
vote
1answer
76 views

In a Banach space X, its two Schauder bases have the same cardinal number?

The definition of Schauder basis is, there exist a set family F(whose cardinal number can be finite countable or uncountable), s.t. any x in X could be uniquely expressed countalbe linear combinations ...
0
votes
1answer
54 views

A proof in $\mathbb{R}^2$ regarding the Cauchy-Riemann equations

Let $u,v$ be a pair of smooth, real valued functions on $\mathbb{R}^2$. Let $(x,y)$ be a point on $\mathbb{R^2}$. Show that the mapping $(x,y)\to(u,v)$ is conformal at the points where the Jacobian ...
0
votes
1answer
56 views

Why does $\sin(\operatorname e^i)$ in complex variables have the following solution?

If possible I would like to know the definitions to look at so I can master this material. According to my professor $$ \sin(\operatorname e^i) = \sin(\cos1)\cosh(\sin1)+i\cos(\cos1)\sinh(\sin1) $$ ...
3
votes
3answers
244 views

Show, a holomorphic function with constraint on real and imaginary part is constant

Let $f: G\rightarrow \mathbb{C}$ be a holomorphic function on a domain. Let $\left[\Re{(f)}\right]⁴+\left[\Im{(f)}\right]⁴$ have a local maximum in $G$. Why is $f$ than already constant? If I could ...
0
votes
1answer
49 views

Can we deduce that the zeros of $g$ are also isolated?

Let $f:Ω→ℂ$ be a non-zero holomorphic function and $g:Ω→ℂ$ be a non-zero non-holomorphic function. We know that all the zeros of $f$ are isolated. Assume that $$f(s)=0⇒g(s)=0$$ Can we deduce that the ...
-2
votes
1answer
62 views

Solve $\cos \pi z = 0$ for $z \in \mathbb{C}$ [closed]

$\cos \pi z = 0$, so $\cos \pi x \cosh \pi y - i \sin \pi x \sinh \pi y = 0$, $\cosh \pi y$ never be $0$, so $\cos \pi x = 0, \pi x=\pm \pi/2+2k\pi, x = \pm1/2+2k.$ Is this the right way to do?
1
vote
2answers
82 views

On Cauchy-Riemann equations

Given $f:\mathbb C\to \mathbb C$ is a non-constant entire function. Then which of the following is possible? Re $f(z)=$ Im $ f(z)$, Im$\,f(z)<0$, Re$\,f(z)$ is bounded, $f(z)\neq 0,$ for all ...
2
votes
4answers
371 views

Is $f(z)=\bar{z}$ continuous?

I have $z\in \mathbb{C}$, is $f(z)=\bar{z}$ continuous on the whole complex plane? Note that $\bar{z}$ is the conjugate of $z\in \mathbb{C}$ I was thinking that if $z$ is on the real line, then ...
0
votes
1answer
79 views

Cauchy principal value of two integrals

I want to calculate $P.V. \int_{-\infty}^{\infty} \frac{e^{ix}}{x}dx$ and $P.V. \int_{-\infty}^{\infty} \frac{f(x)}{x(x-i)}dx$ I stat by using the definiton to calculate the first one: ...
6
votes
2answers
171 views

Continuity of the derivative

As we all know, all the basic properties of holomorphic functions (i.e. functions which are differentiable in the complex sense) can be deduced from Cauchy's formula. Moreover, Cauchy's formula itself ...
3
votes
1answer
114 views

complex analysis (Univalent function )

The Distortion Theorem tells us that if $f$ is a univalent function on $\mathbb{D}:=\{z:|z|<1\}$, then $|f'(z)|\leq 12\,|f'(0)|$ for $|z|\leq\frac12$. By iterating this, prove that if ...
0
votes
1answer
56 views

Closed sets and sequences in Metric spaces

Suppose $x \subset X$ is a closed set, the sequence {$ {x_j}$}${ } \subset F$ and $x \in X$. Show that if $x_j \to x$ as $j \to \infty$, then $x \in F$ Okay so I really don't know where to start with ...
-1
votes
1answer
166 views

Explain that the complex sine function is not bounded.

That is, for any positive constant $M$, there exists a $z$ such that $|\sin z|>M$. Given $|\sin z|^2=(\sin x)^2+(\sinh y)^2$.
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votes
3answers
4k views

How to prove Lagrange trigonometric identity [duplicate]

I would to prove that $$1+\cos \theta+\cos 2\theta+\ldots+\cos n\theta =\displaystyle\frac{1}{2}+ \frac{\sin\left[(2n+1)\frac{\theta}{2}\right]}{2\sin\left(\frac{\theta}{2}\right)}$$ given that ...
5
votes
3answers
2k views

Proving $\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$

I am being asked to prove that $$\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$$ I have some progress made, but I am stuck and could use some help. What I did: ...
0
votes
2answers
97 views

Can we deduce that $h=f+g≠0$

Let us consider three complex functions $f,g,h$. Let $A$ be a set such that $f≠0$ and $g≠0$ in $A$. Can we deduce that $h=f+g≠0$ in $A$. If not can we add some conditions $f,g$ such that the property ...
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vote
3answers
296 views

Are the Cauchy Riemann conditions sufficient for analyticity

In reading a book All the mathematics you missed I came across this line: (The Cauchy Riemann equations coupled with the condition that the partial derivatives are continuous are sufficient to prove ...
1
vote
0answers
90 views

if f is analytic and it's mapping a region onto a circle

If $f$ is analytic on a domain $D$ and it's mapping the region onto a circle, then what can be said about $f$? I tried to write $f(z)$ as $(r\cos\theta+a)+(r\sin\theta+b)i$, and by using the Cauchy ...
1
vote
1answer
65 views

Essential singularity question

I'm asked to classify the singularity at the indicated poing and to find the residue at that point for $$f(z)=z^ne^{1/z}$$ for $$z_0=0$$ Here's what I have: ...
1
vote
1answer
71 views

Bromwich integral of $1/s^k$ with k real (non integer) and $1<k$

Is there a simple way to compute the inverse laplace transform of $1/s^k$ with k non integer using Bromwich integral (basically without using the known laplace transform of $t^n$)?
0
votes
0answers
58 views

Determining the disc of convergence in two series and determining at which points on the boundary of the disc the series converges.

The two series are as follows: $f(z) = \sum\limits_{n = 1}^\infty n(z+1-i)^{2n}$ and $f(z) = \sum\limits_{n = 1}^\infty n^{-1}z^{n}$ I have worked out that the discs of convergence are, ...
2
votes
1answer
70 views

Show, a holomorphic function is constant

I am given that $\left| \frac{g'(z)}{g(z)}\right|\leq \frac{1}{\left|z\right|^2} \hspace{0.3cm}(*)$. I want to show that if $g$ is holomorphic in $\mathbb{C}$, it is constant. I am not sure if ...
5
votes
5answers
92 views

Identity $\int_{-\infty}^{\infty}\frac{e^{uz}}{1+e^u} \mathrm{d}u=\frac{\pi}{\sin(\pi z)}$

I want to prove the identity $$F(z)=\int_{-\infty}^{\infty}\frac{e^{uz}}{1+e^u} \mathrm{d}u=\frac{\pi}{\sin(\pi z)}$$ First of all $F(z)$ defines an analytic function for $0<z<1$. I am little ...
0
votes
1answer
155 views

Perron Solution on Domain with Disk Removed

I have a homework question that asks Let $D$ be a bounded domain and let $h$ be a continuous function on $\partial D$. For $z_0 \in D$ and $\varepsilon > 0$, define $D_{\varepsilon} = D ...
0
votes
0answers
84 views

Prove $f(x,y,z)=e^{iy+z}$ is continuous on $\mathbb R^3$.

Prove $f(x,y,z)=e^{iy+z}$ is continuous on $\mathbb R^3$. I have already proved that other functions are continuous by using that $f, g$ are continuous implies $f+g$ and $fg$ are continuous. ...
1
vote
1answer
40 views

holomorphic function with integral coefficients

I'm trying to prove that an holomorphic function on $\{Z, |Z|<1\}$ and continuous on $\{Z, |Z|\leq 1\}$ with coefficients in $\mathbb Z$ is polynomial. I have tried to establish some partial ...
0
votes
1answer
43 views

Values of coefficients of polynomial

Suppose we have the polynomial on $\mathbb{C}$: $$p(z)=a_nz^n+a_{n-1}z^{n-1} + \dots + a_1z+a_0$$ and the factored form: $$p(z)=a_n(z-z_1)^{d_1}(z-z_2)^{d_2} \dots (z-z_r)^{d_r}, \sum_{i=1}^r{d_i} ...
3
votes
1answer
137 views

Counting roots of polynomial inside $S^1$

I would like to ask for a hint to this problem: Let $p$ a polynomial function on $C$ with no root on $S^1$. Show that the number of roots of $p$ with $|z|<1$ is the degree of the map $q: S^1 \to ...
0
votes
2answers
59 views

Continuity of $f(z)=u(x,y)+iv(x,y)$

If $u(x, y)$ and $v(x, y)$ are continuous (respectively differentiable) does it follow that $f(z) =u(x, y) + iv(x, y)$ is continuous (resp. differentiable)? If not, provide a counterexample. This ...
2
votes
1answer
80 views

Proving that $\exp(z_1+z_2) = \exp(z_1)\exp(z_2)$ with power series [duplicate]

Probably a simple question, but I wonder about the following: To prove that $\exp(z_1+z_2) = \exp(z_1)\exp(z_2)$, I use : $$\exp(z_1+z_2) = ...
1
vote
1answer
283 views

Complex mean value theorem

Complex mean value theorem: Let $g$ be a holomorphic function defined on an open convex subset $D_{g}$ of $ℂ$. Let $v$ and $u$ be two distinct points in $D_{g}$. Then there exist $z₁,z₂∈(u,v)$ such ...
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votes
2answers
105 views

Expressing numbers in cartesian form

I'm stuck at these 3 questions: Express the following in cartesian form: (a) $sin(3+i)$ (b) $sinh(1+ \frac{\pi}{2}i)$ (c) $cosh( \frac{\pi}{4}i)$ (d) tan(i) Please help me check my ans: (a) ...
1
vote
1answer
57 views

What is the value of the line integral $\int_{|z|=2}\frac{\overline{z}}{1-z}\mathbb dz$

I have done this two ways I know, and I keep getting zero. I tried changing to polar co-ordinates and integrating from 0 to $2\pi$ which came out as zero. Any suggestions / confirmations?
2
votes
1answer
118 views

Question about injective holomorphic functions on $\mathbb{D}$ and Koebe's quarter theorem

Let $f$ be an injective holomorphic function on $\mathbb{D}$ such that $f(0) = 0$, $f'(0) = 1$. The open mapping theorem implies that $f(|z| < 1)$ contains an open neighborhood of the origin. Then ...
5
votes
1answer
109 views

The “$\mathbb{Z}_n$-theta function” - what is it? Is it being studied somewhere?

The Jacobi theta function is well known: $$\theta(z, \tau) = \sum_{n=-\infty}^\infty \mathrm{e}^{\pi i n^2 \tau + 2\pi i nz}$$ In Shahn Majid's "Foundations of Quantum Group Theory", you'll find a ...
3
votes
0answers
152 views

on the convergence of a certain integral

If I have an entire function $\phi$ such that it is of exponential order zero. I.e for all $\rho > 0$ we get $|\phi(s)|\le C_\rho e^{|s|^{\rho}}$. Furthermore, I have an extreme decay in the Taylor ...
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votes
0answers
36 views

Real and Imaginary Parts of $z^z$

Apologies if I don't write this question clearly; this is my first time using the site and I'm not sure if I'm following all the proper protocols here. I'm working through Ahlfor's Complex ...
2
votes
1answer
47 views

Continuous functions. Second norm

Let $f:[a,b]\rightarrow \mathbb{R}^{d}$ be continuous. I need to prove that $\left \| \int_{a}^{b}f(x)dx \right \|_{2}\leq \int_{a}^{b}\left \| f(x)) \right \|_{2}dx$.
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votes
1answer
558 views

Calculus of residue of function around poles of fractional order (complex analysis)

The complex function $f(z)=\frac{1}{\sqrt{z^2+r_0z}}$ with $r_0>0$ has two poles (at $z=0$ and $z=-r_0$). But they are not simple poles. They are poles of fractional order. Am I right? How I can ...
0
votes
1answer
67 views

Uniqueness of singular measure for inner function

A singular inner function $M$ (an analytic function on the open unit disk without zeros which takes on unimodular boundary values almost everywhere) can be written as $$M(z)=c \exp\left(\int_0^{2\pi} ...
3
votes
1answer
60 views

$ds=\frac{2|dz|}{1-|z|^2}$ conformal invariant of the disc.

Suppose $w$, $z$, $w_0$ and $z_0$ are in the unit disc $D=\{z\in\mathbb{C} \ \big| \ |z| =1\}$ and satisfy $$\left| \frac{w-w_0}{1-\bar{w_0}w} \right| = \left| \frac{z-z_0}{1-\bar{z_0}z} \right|. $$ ...
3
votes
1answer
52 views

number of zeros in a disk of a holomorphic function

Let $f$ be a holomorphic function defined in a beighborhood of $\overline{D(0,R)}$ which has no zeros on $\partial D(0,R)$. Let $N$ be the number of zeros of $f$ inside $D(0,R)$. Prove that ...
1
vote
1answer
65 views

Old prelim exam problem: Suppose that $f$ is holomorphic on the unit disk. If $\exists$ $r \in (0,1)$ such that $|f(1/n)|\leq r^n$. Then $f=c$

Suppose that $f$ is holomorphic on the unit disk $|z|<1$. If $\,\exists$ $r \in (0,1)$ such that $|f(1/n)|\leq r^n$ for $n \in \mathbb{N}$. Then $f=c$ (constant). I think this problem could be ...
0
votes
1answer
90 views

Poisson integral on circle with jump discontinuity

This is related to a problem in Ahlfors' Complex Analysis book. Let $U(z)$ be piecewise continuous on $|z| = 1$ with a jump discontinuity at $\alpha = e^{i\psi}$ where $|\alpha| = 1$, such that $$ ...
2
votes
1answer
109 views

Perron Solution is Precisely Poisson Integral on Punctured Disk

I have a homework question that asks Define $D = \{ z \in \mathbb{C} : 0 < |z| < 1 \}$, and let $h(\zeta)$ be a continuous function on $\partial D$. Show that the Perron solution ...