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
46 views

find set of complex numbers where $Arg(\frac{1}{z}) \neq -Arg(z)$

find set of complex numbers where $Arg(\frac{1}{z}) \neq -Arg(z)$ Def $Arg(z)=\theta :z=re^{i \theta } \wedge -\pi < \theta \leq \pi$ for z in first, second quadrant it holds except for ...
2
votes
3answers
51 views

How to show that $f$ is constant by using Liouville's theorem?

If $f$ is entire and $\mbox{Arg}(f(z))=-\frac{\pi}{2}$,when $|z|=1$ then show that $f$ is constant. All I need to prove is that f is bounded but I can't figure out how.I'd like someone's help.
1
vote
0answers
12 views

Inequality for complex Fourier' series coefficient

Let $f$ be $2\pi$ periodic, regular in $D_\beta=\{z\in \mathbb{C}\colon |\text{Im } z|<\beta\}$, $\beta>0$ and continuous until the boundary. I want to prove that function $f$ complex Fourier' ...
0
votes
1answer
27 views

Simply Connected sets

In my textbook it states, that the Union of two open docs is simply connected but not connected Why is this. I know simply connected means any closed path or loop can be shrunk to a point ...
0
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0answers
18 views

Help to understand the generalization of the Argument Principle

I'm reading Conway's complex analysis book and I'm trying to prove this theorem left to the reader on page 124: I tried to use integration by parts without success. I need some hint how prove this ...
0
votes
1answer
494 views

Prove the complex function is entire

The function $$ f(z) = e^{x^2 - y^2} (\cos 2xy + i \sin 2xy )=e^{z^2} $$ Then, how to prove it's analytic everywhere of complex plane of the exp function...?
17
votes
1answer
593 views

$-4\zeta(2)-2\zeta(3)+4\zeta(2)\zeta(3)+2\zeta(5)=S$

EDIT: Due to the solution below, I edited the answer of the post. Thanks!!!! Hi I am trying to calculate the infinite double sum $$ S:=\sum_{j,k=1}^\infty ...
3
votes
2answers
42 views

Using the Weierstrass M-test, show that the series converges uniformly on the given domain

$\sum_{k \geq 0} \frac{z^k}{z^k+1}$ on the domain $\overline{D}[0, r]$, where $0 \leq r < 1$ I'm honestly not sure how to do this. My text mentions the Weierstrass M-test but the example they ...
0
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0answers
20 views

Proof that in any set A such that A contains a circumference centered at zero you can't find a continuous square root function.

The following is an exercise in my textbook on complex analysis: Proof that in any set A such that A contains a circumference centered at zero you can't find a continuous square root function. ...
1
vote
2answers
55 views

why can't we define a branch of $\log f(z)$ in the whole complex plane?

My question is really simple. The only problem to define a branch of $\log f(z)$ in the whole complex plane is because we can have $f(z)=0$ for some $z\in \mathbb C$? In fact I think I don't ...
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0answers
31 views

Does this function belong to $H^1(\mathbb D)$?

$\mathbb D$ is the unitary disk centered at $0$. Does the following function belong to $H^1(\mathbb D)$? .$$f_\epsilon(z) = \frac{1}{(1-z)\left(\frac{1}{z}\log\frac{1}{1-z}\right)^{1+\epsilon}}, ...
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0answers
14 views

Exercise for proving maximum modulus principle

I have this exercise: let $D$ be a domain and $a \in D$ such that $D'(a,r)$ is a subset of $D$ (here $'$ for closure). suppose that $f$ is holomorphic on $D$ and let $A = \max|f(z)|$ with $|z - a| = ...
0
votes
1answer
26 views

Möbius transformation mapping problem [on hold]

Let $a,b,c,d\in \mathbb{R}$ be such that $ad-bc>0$. Consider the Möbius transformation $$T_{a,b,c,d}(z)=\frac{az+b}{cz+d}$$ and define ...
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0answers
40 views

A curious trigonometric equality

Let's consider the following expression: $(1)\cos(15\sqrt{2}^\circ) = \frac{1}{2}\sqrt[\sqrt{2}]{\frac{\sqrt3}{2}+\frac{i}{2}} +  \frac{1}{2}\sqrt[\sqrt{2}]{\frac{\sqrt3}{2}-\frac{i}{2}}$ The left ...
2
votes
0answers
44 views
+50

radius of convergence of Taylor series, function with branch cuts

Let $f(z) $ being the analytic continuation of some holomorphic function, having many branch points and isolated singularities at $\beta_1,\beta_2,\ldots,\beta_n,\ldots$ is the radius of convergence ...
0
votes
1answer
46 views

Help in this proof of the argument principle

I'm reading Conway's complex analysis book and on page 123 he made the following comment: Suppose that $f$ is analytic and has a zero of order $m$ at $z=a$. So $f(z)=(z-a)^mg(z)$ where $g(a)\neq ...
0
votes
1answer
16 views

Find the Laurent Series for $\frac{z-2}{z+1}$ around $z = -1$

Find the Laurent Series for $\frac{z-2}{z+1}$ around $z = -1$ I'm not sure how to do this because it is not something with a simple numerator. If it was something like $\frac{1}{(z-2)(z+1)}$ I ...
2
votes
0answers
21 views

$\frac{1}{2\pi}\int_0^{2\pi}\log|re^{it}-\zeta|dt$?

Prove that if $\zeta \in \mathbb{C}$ and $r>0$ then $\frac{1}{2\pi}\int_0^{2\pi}\log|re^{it}-\zeta|dt=\log |\zeta|$ if $r\leq |\zeta|$, and it is $\log r$ if $r> |\zeta|$. My Try: First I ...
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0answers
16 views

Complex analysis: given the radius of convergence of one series, how can I find radii of convergence for other series?

Suppose I know that: $\sum_{k = 0}^{\infty} c_k z^k$ is $R$. How can I use this to find the radius of convergence for something like: $\sum_{k = 0}^{\infty} 3^k c_k z^k$ $\sum_{k = 0}^{\infty} ...
0
votes
3answers
81 views

Evaluating an integral using the gamma function

My question regards an integral $$\int_0^\infty \frac{\sin(x^p)}{x^p}\mathrm{d}x$$ The answer should be $$\frac{1}{p-1}\cos(\frac{\pi}{2p})\Gamma(\frac{1}{p})$$ and I roughly know that I should apply ...
0
votes
0answers
11 views

Differentiation between the unit spheres and the hypersurfaces in $\mathbb C^n$

Let $\Sigma ^{n-1}$ be the complex unit sphere in $\mathbb C^n$, $$\Sigma^{n-1}=\{(z_1,...,z_n)\in \mathbb C^n; z_1\bar {z_1}+...+z_n\bar {z_n}=1\}$$ and let $S^{n-1}_{\mathbb C}$ be the hypersurface, ...
2
votes
2answers
85 views

closed path, winding number, Jordan contour

If $ D$ is a domain in $\Bbb C$, $z_0\in \Bbb C\setminus D$, and $\gamma$ is a closed, piecewise smooth path in $ D$ for which the winding number $n(\gamma, z_0)\ne0$, show that there is a Jordan ...
1
vote
1answer
62 views

What happens when $\beta_1 + \beta_2=1$ and when $0<\beta_1 + \beta_2<1$?

I have the following example of the Scwarz-Christoffel integral formula: $$S(z)=\int_0^z w^{-\beta_1}(1-w)^{-\beta_2}dw$$ with $0<β_1 <1, 0<β_2 <1$, and $1<β_1 +β_2 <2$ and I know ...
0
votes
0answers
16 views

Finding where a complex series converges absolutely, uniformly.

I need to figure out where the series converges absolutely and uniformly. I know that once I have absolute convergence on a region, then I know I also have uniform convergence on that region, so I ...
2
votes
4answers
58 views

Evaluation of the principal value of $\int\limits_{-\infty}^\infty \frac{\sin 2x}{x^3} \, dx$

I'm trying to evaluate an integral $\int\limits_{-\infty}^\infty \frac{\sin 2x}{x^3}\,dx$ using Cauchy's theorem. Considering an integral from $-R$ to $-\epsilon$, then a semicircular indentation ...
2
votes
2answers
62 views

Is $\sin z/z$ analytic at the origin?

For $z\in\Bbb C$ let $$ f(z) = \frac{\sin z}{z} $$ Along the real line this is well behaved, and approaches $1$ as $z\to 0$. But is $f(z)$ analytic at the origin ($z=0$)? I tried explicitly checking ...
0
votes
1answer
34 views

Evaluating $\int R(X)sin(x) dx$ with residue theorem.

The integral I am trying to evaluate is: $$I = \int_{-\infty}^\infty \frac{x}{1+x^2}\sin x\ dx = \int_{-\infty}^\infty f(x)\ dx$$ The standard approach to this is to realise $\sin x$ as the complex ...
1
vote
0answers
24 views

Integrate by parts $\int_0^\infty w' \bar w$; any nice expression for $w$ complex-valued?

Let $w$ be a complex-valued function of $t \in [0,\infty)$. At $t \to \infty$, it decays to zero. And $w_t(0)$ is prescribed. Is there any nice expression for the integral $$\int_0^\infty w' \bar w$$ ...
2
votes
2answers
61 views

Proving that two branch cuts can cancel out

Define the following functions $\mathbb{C}\to\mathbb{C}:$ $$u(z)=\frac{\log \left(z+\frac{1}{2}\right)}{z}\quad \left[-\pi\leqslant\arg \left(z+\tfrac12\right)<\pi\right];\quad v(z)=\frac{\log ...
1
vote
1answer
26 views

Satisfies CR-equations, but is not complex differentiable in 0

Consider the function $f$ on $\mathbb{C}$ given by: $$f(z) = \begin{cases} e^{-1/z^4} &\text{if } z \neq 0 \\ 0 &\text{if } z = 0. \end{cases}$$ Show that it satisfies the Cauchy-Riemann ...
3
votes
0answers
40 views

Any bounded region $G \subseteq \mathbb{C}$ with transitive automorphism group and sufficiently “smooth” edges is biholomorphic to the unit ball

Let $G$ be a bounded region in $\mathbb{C}$ (i.e. we have $G ≠ \emptyset$, and $G$ is open and connected), and let $G$ have a transitive automorphism group (that is, for each two points $z_1, z_2 \in ...
0
votes
2answers
48 views

Why the power series $\sum_{n=0}^{\infty}nz^n$ does not converge on the unit circle $\{z:|z|=1\}$?

For the power series $\sum_{n=0}^{\infty}nz^n$, I know its radius of convergence is 1 and it diverse on the boundrary of the disc of convergence. But I fail to prove the latter fact, i.e., it diverse ...
0
votes
1answer
38 views

What function does the power series $\sum_{n=1}^{\infty}\frac{z^n}{n^2}$ converge to in its disc of convergence?

For the power series $\sum_{n=1}^{\infty}\frac{z^n}{n^2}$, its radius of convergence is 1 which implies that this series is absolutely convergent in the the unit ball $\{z:|z|<1\}$. Since it is ...
1
vote
3answers
75 views

Continuity of $f(z) = \sin(\theta)$ - how to prove?

If $f: \mathbb{C}\to\mathbb{C}$ is defined by $f[r(\cos(\theta)+i\sin(\theta)]=\sin\theta$ if $r>0$, and $f(0) = 0$, then how does one prove that $f$ is discontinuous at $0$ and continuous ...
1
vote
1answer
49 views

$2^z$ behavior when changing real and imaginary components of $z$

I'm reading The Music of the Primes by du Sautoy and I've come across a section that I'm having difficulty understanding: Euler fed imaginary numbers into the function $2^x$. To his surprise, out ...
0
votes
1answer
36 views

Does there exists an analytic function from $D$ to $D$?

Let $D=\{z\in\mathbb{C}:|z|<1\}$. Which of the following are correct? There exists holomorphic function such that $f:D\rightarrow D$ with $f(0)=0$ and $f'(0)=2$. There exists holomorphic function ...
0
votes
2answers
23 views

Where does the sequence converge pointwise? Does it converge uniformly on this domain? (example)

I'm trying to learn about sequences of functions, which is a new concept to me, and I would like to have a simple example to go with what I already know from definitions. Unfortunately the notes that ...
0
votes
1answer
39 views

What is the order of the pole of $\frac{\mathrm{Log}(z)}{(z-1)^3}$ at $z=1$?

I read somewhere that the series for the principal branch of $$\mathrm{Log}(z) = \sum_{n=1}^{\infty}\frac{(-1)^{(n+1)}(z-1)^n}{n}$$ If this is true does it means that the order of the pole is equal to ...
1
vote
1answer
24 views

Stuck with epsilon-delta proof of existence of a limit

Given $f(z):=f(x+iy) := u(x,y) + iv(x,y)$, I'm trying to prove that if $\lim\limits_{z\to z_0} f(z) = L$, where $L\in \mathbb{C}$, $\lim\limits_{(x,y)\to(x_0,y_0)} u(x,y) = a$ and ...
0
votes
1answer
56 views

how to evaluate this definite integral $\int_0^\infty\frac{\sin^2(x)}{x^2}dx$? [duplicate]

For $\int_{0}^{\infty}\frac{\sin^2(x)}{x^2}dx$. I considered using residue theorem. But since the function inside is holomorphic except for a removable singularity at the origin. So whatever contour I ...
5
votes
1answer
54 views

What shapes, with boundary collapsed to a point, are homeomorphic to $S^n$?

Consider the following construction: Given a set $A\subseteq\Bbb R^n$, form the quotient space $A/\sim$ which identifies all the points on the boundary $\partial A$ (w.r.t $\Bbb R^n$). For which ...
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0answers
20 views

When can we take the integral limit from and to infinity?

I'm reading Conway's complex analysis book and on page 115 he writes this: I didn't understand why $\frac{x^2}{1+x^4}\ge 0$ implies this limit is true. What are the conditions to allow us to take ...
2
votes
0answers
30 views

Tori conformally equivalent

Let $L_1,L_2$ be the lattices generated by $\{1,\tau_1\},\{1,\tau_2\}$ respectively and $X_1=\mathbb C / L_1, X_2= \mathbb C / L_2 $ the corresponding complex tori. We look to prove that $X_1,X_2$ ...
1
vote
1answer
30 views

classfiying singularities

\begin{equation} h(z)=\frac{z^2e^{\frac{1}{z^2+1}}}{\sin(z^2)} \end{equation} It seems to me the function has essential singularity at $z=\mp i$ It is clear that $e^{\frac{1}{z}}$ has essential ...
2
votes
0answers
31 views

Compute $\int_{0}^{\infty} \frac{x^{1/3}}{(x^2 + 1)^2} dx$

So I want to compute $\int_{0}^{\infty} \frac{x^{1/3}}{(x^2 + 1)^2} dx$ using complex analysis, Cauchys theorem and the residue theorem. What I did was the following: define $g(z) = e^{1/3(\ln|z| + ...
1
vote
1answer
65 views

If $\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)}$ holds for $0<z<1$, then also for $0<\operatorname{Re}(z)<1$

In Special Functions p. 10, it has proven that $$\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)},$$ for $0<z<1$. Then it says that this equality implies for $0<\operatorname{Re}(z)<1$. I do ...
0
votes
0answers
23 views

ODE $Aw'' + iBw' + cw=0$ with complex coefficients, how to solve?

I have the ODE for $w:[0,\infty) \to \mathbb{R}$ : $$Aw'' + iBw' + cw=0$$ where $A, B \in \mathbb{R}$ and $c \in \mathbb{C}$ is a complex number. There is a boundary condition involving $w_y$. Also ...
1
vote
1answer
66 views

Proving that $\lim_{n \to \infty} (1+ \frac{i}{n})^n = e^i$

Prove that $\displaystyle \lim_{n \to \infty} (1+ \frac{i}{n})^n = e^i$ I was trying to proof in the same way of $\lim (1 + \frac{1}{n})^n = e$, but I couldn't proceed this way. Can someone give me a ...
0
votes
1answer
16 views

Composition with polynomial/ same type of singularity

Let $f\in O(D_1(0){}-\{0\})$ and $ p $ a non constant polynomial. Then $f$ and $p(f)$ have the same type of singularity at $z_o=0 $. I think its fairtly easy to Show that if $f$ has a ...
5
votes
2answers
49 views

Meromorphic, analytic, holomorphic and all that

I must have slept through something in my complex variables course, because all my life I have used the terms holomorphic, meromorphic, and analytic somewhat interchangeably. These are all also ...