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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4
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2answers
140 views

How to find the radius of convergence?

The function is $\dfrac {z-z^3}{\sin {\pi z}} $. How to find the radius of convergence in $ z=0 $?
3
votes
1answer
117 views

Finding complex power series with interesting boundary behavior

I need to find one (or more) interesting complex power series to give to my students for their analysis exam. Ideally, this would be a power series that has interesting behavior at the boundary, i.e. ...
3
votes
1answer
80 views

Showing a holomorphic function is bounded by |tan z|

Suppose $f$ is holomorphic on $U=\{|\Re z| < \frac{\pi}{4} \}$ and that $f$ maps 0 to 0, and $|f(z)|<1$ on $U$. Show $|f(z)| \le |\tan z|$ on $U$.
2
votes
1answer
70 views

uniformly bounded sequence of non constant holomorphic functions

Let $\{f_n\}_{n=1}^{\infty}$ be a uniformly bounded sequence of nonconstant holomorphic functions in a connected open set $\Omega$. Let $f \not \equiv 0 $ be a holomorphic function in $\Omega$. ...
0
votes
1answer
424 views

Defining single-valued branches

These questions come from 2.2 of Ahlfor's famous text. I admit that defining branches of power functions and log functions in $\mathbb C$ has been conceptually difficult for me, and I think having a ...
5
votes
4answers
196 views

With hypotheses of Schwarz's lemma, estimate the radius around zero where $f$ must be one-to-one

Suppose $f(z)$ is analytic in the open unit disc and $|f(z)|<1$ there. Suppose further that $f(0) =0$ and $f'(0) = a \neq 0$. Show that there is a disc of positive radius $|z|<\rho$ such ...
-1
votes
2answers
93 views

What would be the best *domain* for defining *complex-differentiation*?

Before i start, please know that i don't know any concepts about "smooth manifold". I skimmed some texts and wikipedia, so i guess it would be the best concept to treat differentiation, but i don't ...
3
votes
2answers
152 views

Behavior of $f(z)=\int_0^1\mathrm{e}^{\alpha t^2}\sin(tz)\,dt$ when $\alpha <0$

Define $$f(z)=\int_0^1\mathrm{e}^{\alpha t^2}\sin(tz)\,dt,$$ where $\alpha \in \mathbb{R}$. If $\alpha >0$ then $f(z)$ has infinitely many real zeros and at most a finite number of complex ...
2
votes
1answer
56 views

Holomorphic functions on the product of open sets.

Is it true that $$ \mathcal H(\mathrm U \times \mathrm V) \simeq \mathcal H(\mathrm U) \widehat{\otimes} \mathcal H(\mathrm V) $$ for open two open affine sets $\mathrm U$ and $\mathrm V$? Edit: I ...
4
votes
1answer
151 views

Convergence of $\sum\frac{\tan(nz)}{n^2}$ to an analytic function…what if $z\in \mathbb{R}$?

For which values of $z$ does $$\sum_{n=1}^\infty \frac{\tan(nz)}{n^2}$$ converge? For which values of $z$ is the limiting function analytic? One can show, as in this answer, that ...
2
votes
1answer
113 views

Expand a complex function in a series

I want to expand complex function $f(z) = \frac{e^{\frac{1}{1-z}}}{e^z - 1}$ in series in the neighborhood of $z_0 = 1$ to find order of the pole. We can make replacement $\xi = z - 1$ and expand ...
3
votes
1answer
42 views

A question on a sequence in a Banach algebra [duplicate]

If $\{u_{k}\}_{k=1}^{\infty}$ is a sequence in an Banach algebra (and more specifically, in the set of all the bounded linear operators of a Banach space $X$). If ...
3
votes
1answer
139 views

Sheaf of a complex analytic function

Let $$ F(U) = \left\{ \mbox{ all complex analytic functions } f \mbox{ on } U \mid z \frac{df}{dz}=1 \right\}$$ for any domain $U$ in $\mathbb{C}$. I want to show that: $F$ is a sheaf. The stalk ...
1
vote
2answers
58 views

An easy question on complex

Let $\{u_{k}\}_{k=1}^{\infty}$ be a complex number sequence. If $\sum_{k=1}^{\infty}\lambda^{k}u_{k}=0$, for each $\lambda\in \mathbb{D}(0, 1/3)$(where the $\mathbb{D}(0, 1/3)~$denotes an open disc ...
4
votes
2answers
337 views

Why is Riemann integration used in complex analysis and not Lebesgue integration?

In the development of complex analysis you use Riemann integration and not Lebesgue integration to define line integrals. My questions are: Are the theories developed the same? (i.e. does it not ...
1
vote
2answers
81 views

Visualizing a complex valued function of one real parameter

I'm looking for a way to capture/graph or visualize it in my head, but I can't find how.. a 2-dimensional path won't do, because it doesn't reveal the rate-of-change.. 2 1-dimensional graphs on top ...
8
votes
1answer
282 views

Entire function $f(z)$ bounded for $\mathrm{Re}(z)^2 > 1$?

Let $z$ be a complex number and $\mathrm{Re}$ denote the real part. Does there exist a nonconstant entire function $f(z)$ such that $f(z)$ is bounded for $\mathrm{Re}(z)^2 > 1$ ?
2
votes
1answer
63 views

Least area of the image of a simply connected domain under holomorphic maps

I found the following statement in the book "The Kernel Function and Conformal mapping" at page 23-24 by Stefan Bergman: Let $\Omega$ be a bounded, simply connected domain in $\mathbb{C}.$ Let ...
1
vote
1answer
105 views

Prove $\sum_{m \geq 1} {\frac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \frac{t^m}{(1-x)^{2m-1}}} $is divergent

How do I show that the following power series is divergent? $$ u(t,x) = \sum_{m \geq 1} {\dfrac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \dfrac{t^m}{(1-x)^{2m-1}}} $$ where $t$ is complex 1-dimensional, $x$ ...
1
vote
4answers
173 views

A quick question about analytic functions on the unit disk with $Re(z) \leq 0$

Suppose f is analytic in the domain $$D(0; 1) = \{z ∈ \Bbb C : |z| < 1\}$$ and $\operatorname{Re} f(z) ≤ 0$ for all $z \in D(0; 1)$. If $\operatorname{Re} f(0) = 0$, show that $f$ is constant on ...
4
votes
0answers
356 views

Joukowski Aerofoil Plot

I've just had a go at plotting flow around aerofoils and I've come across a problem where I can't spot where I've gone wrong. I've previously worked out that the complex potential flow around a disk ...
1
vote
1answer
114 views

estimate of a holomorphic function in the unit disc with a zero of order 4 at the origin

Define $\mathcal{F}$ to be the family of holomorphic functions which map the open unit disc to itself and which together with their first three derivatives vanish at $0$. Find $\sup_{f \in ...
3
votes
1answer
182 views

Finite order function in the complex analysis.

Assume that an entire function $f$ be finite order with finitely many zeros. Please show that either $f(z)$ is a polynomial or $f(z) + z$ has infinitely many zeros. Thank you. And I know the ...
14
votes
2answers
197 views

Maximum of $|(z-a_1)\cdots(z-a_n)|$ on the unit circle

Let $a_1,\ldots,a_n$ be points on the unit circle. Let $P(z)=(z-a_1)\cdots(z-a_n)$. The maximum principle or Rouche's theorem can be used to show that there exists a point $b$ on the unit circle such ...
2
votes
3answers
170 views

Map $\{x+iy \mid x^2+y^2<1 \text{ and } x^2 + (y-1)^2<2\}$ conformally to UHP

From an old qualifying exam: Let $D$ be the domain $$D :=\{x+iy \mid x^2+y^2<1 \text{ and } x^2 + (y-1)^2<2\}.$$ Map the domain onto the upper half-plane. Obtain a function $f(z)$ ...
2
votes
1answer
143 views

Explicit contour integration gone wrong.

Consider the function $f(z):\mathbb{C}\to\mathbb{C}$: $$f(z)=\frac{4z}{1+z^2}$$ There are a few properties evident: The anti-derivative (with integration constant $c=0$) is given by: ...
0
votes
2answers
60 views

An equivalence concerning analytic continuation along arcs (Ahlfors)

In Ahlfors' Complex Analysis text, page 295, he discusses the Monodromy theorem. As he begins to prove the theorem he states: To begin with we note that continuation along an arc of the form ...
4
votes
0answers
93 views

Probably Riemann surface integral

Here is the integral: May you please suggest some beautiful idea on using Riemann surface, or some Gauss-Ostrogradsky at the beginning. Also, the initial integral looks really symmetric, so maybe ...
0
votes
1answer
61 views

Laurent series of $f(t)$

Prove that for any Laurent series $f(t)$ one has $\operatorname{Res}\{f'\} = 0$? I know for a Laurent series of a complex function f is a representation of that function as a power series which ...
4
votes
2answers
140 views

meromorphic function in the unit disc with only one pole of order n

Let $f$ be meromorphic in a neighborhood of $\{|z| \leq 1\}\setminus \{1/2\}$ and have a pole or order $n$ at $1/2$. Suppose that $|f| < 3$ on $\{|z|=1\}$. Show that for any $\phi \in \mathbb{R}$, ...
2
votes
2answers
165 views

Convergence of $\sum_{n=1}^\infty \frac{\cos(nz)}{e^n}$

Determine where $$\sum_{n=1}^\infty \frac{\cos(nz)}{e^n}$$ is convergent. I believe it should converge for $\text{Im}(z)<1$, diverge for $\text{Im}(z)>1$. My only question comes when ...
1
vote
1answer
138 views

Asymptotic behavior of complex function

So I have a function $f(x)$ which I know behaves like $\alpha \sqrt{x} + \frac{\beta}{\sqrt{x}} $ for large $x$. I want to extend $f(x)$ to $f(x + i y) = f(z)$, and I was hoping that the asymptotic ...
0
votes
2answers
38 views

Constructing an antiderivative of a function if the contour integral depends on initial and final point

I am working on the following problem: Let $D \subset \mathbb C$ be a domain, $f: D \to \mathbb C$ a continuous function and $\gamma : [\alpha, \beta] \to D$ a contour. Assume that $\int_\gamma f$ ...
4
votes
2answers
154 views

If $f$ is entire and $\exp(f(z))$ is a polynomial, then $f$ is constant.

In a recent question that was just deleted, @danielfischer gave at the end of his answer the following exercise: for entire $f$, $$e^{f(z)} \text{ is a polynomial} \iff f \text{ is constant}$$ I was ...
1
vote
1answer
238 views

$f(z)$ has infinitely many zeros and that each zero is simple.

Let $f(z)=e^z-z$ I want to check $f(z)$ is finite order. And how to show that $f(z)$ has infinitely many zeros and that each zero is simple. Dfn: an entire function f is finite order if ...
3
votes
1answer
71 views

Please help me with this complex variables integral

Compute the following :$$\int z^n (1-z)^m dz $$ for any integer n and m. (Integral is done on the circle $|z|=2$) I am just stuck from the start. Should I use Cauchy's Theorem of integral? Since n ...
2
votes
1answer
152 views

Residue Theorem for trigonometric integrals.

I am working on the following statement. Let $Q = Q(x,y): \mathbb R^2 \to \mathbb R$ be a rational function, which is continuous on the unit circle $S_1(0)$. Let furthermore $f: \mathbb C \to ...
1
vote
2answers
72 views

Limit of the general term of a power series in a pole

Let $\Omega = D(0,2)/\{\frac{1}{2}\}$ , where $D(0,2)$ is a disc, $f$ holomorphic in $\Omega$. $\frac{1}{2}$ is a simple pole for $f$ with residue $1$, calculate $$ lim_{n \to \infty} ...
1
vote
0answers
46 views

Find a function $f$ such that $f$ is harmonic on $D$ and $f|_{\partial D}$.

I understand its solutions in general. But my question is how to decide whether I sould take $Im z^4$ or $Re z^4$? I have two similar examples. And in one example, the real part is taken, but in ...
1
vote
1answer
111 views

positive harmonic function has a zero limit at a point on the boundary

Let $u$ be a positive harmonic function in $\{ \Re{z} > 0\}$ such that $\lim_{r \rightarrow 0^+} u(r) = 0 $. Prove that then $\lim_{r \rightarrow 0^+} u(re^{i\theta}) = 0 $ for any $\theta \in ...
0
votes
0answers
70 views

Laurent series expansion of sine

The problem is : find the expansion of $ \frac {z}{\sin z} $ in the ring $\pi < abs (z) < 2\pi $, if the expansion in $abs (z)<\pi $ is known. How to solve such problems?
1
vote
1answer
100 views

holomorhicity implies harmonic function in several variables

I had read somewhere that it follows by cauchy riemann equations that any holomorphic or anti-holomorphic function $f$ from an open subset of $C^n$ to $C$ is harmonic i.e $\sum_{i=1}^n ...
0
votes
1answer
44 views

Singularity structure of function in the complex plane.

Consider a piecewise differentiable function $f(x):\mathbb{R}\to\mathbb{R}$. Now, analytically continue this function ($x\to z$) to complex argument and values $f(z):\mathbb{C}\to\mathbb{C}$. For such ...
4
votes
3answers
606 views

Let $f(z)$ be a one-to-one entire function, Show that $f(z)=az+b$.

Let $f(z)$ be a one-to-one entire function, Show that $f(z)=az+b$. My try : Because $f$ is entire it has a taylor series around zero (in particular). $f(z)=\sum^{\infty}_{k=0} a_kz^k$ Proof by ...
1
vote
1answer
87 views

Find the root of the polynomial?

Consider the root of the polynomial $p(x) = x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots+a_1x -1$. Suppose that $p(x)$ has no roots in the open unit disc in a complex plane and $p(-1)=0$. Show that ...
10
votes
4answers
898 views

Suppose $f$ and $g$ are entire functions, and $|f(z)| \leq |g(z)|$ for all $z \in \mathbb{C}$, Prove that $f(z)=cg(z)$.

Suppose $f$ and $g$ are entire functions, and $|f(z)| \leq |g(z)|$ for all $z \in \mathbb{C}$, Prove that $f(z)=cg(z)$. My try : I consider $h(z)=\frac{f(z)}{g(z)}$. If I prove that $h(z)$ is ...
1
vote
1answer
58 views

How to prove that $|F(z)|\le A e^{B |z|^2}$

Let $$\large F:\mathbb{C}\longrightarrow\mathbb{C}$$ $$\large F(z)=\prod_{n=1}^{\infty}(1-e^{-2\pi n}\cdot e^{2\pi i z })$$ How to prove that $$\large|F(z)|\le A e^{B |z|^2},\forall z \in ...
2
votes
2answers
48 views

Hunt for a function.

I am looking for any nontrivial function $f(z): \mathbb{C}\to\mathbb{C}$ such that: $f(z)$ is an entire function. A $z_p\in\mathbb{C}$ exists for which $\Re(f(z))\geq\Re(f(z_p))~\forall ...
1
vote
1answer
55 views

Images of two arcs under $z+1/z$

Let $C$ be the circle $|z-ai|=\sqrt{1+a^2}$, where $a>0$. Let $f(z)=z+\frac{1}{z}$, where $z\in \mathbb{C}$. Let $C_1, C_2$ be the two arcs on $C$ determined by $-1$ and $1$. I am looking for an ...
1
vote
2answers
589 views

Laurent series for $(\sin 2z)/z^3$

I have to find the Laurent series for $(\sin2z)/z^3$ in $|z|>0$, but I really don't know how to start. And I thought that in this area it's a Taylor serie because the singularity isn't in the area, ...