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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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 ...
3
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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 ...
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4answers
62 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 ...
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2answers
64 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 ...
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1answer
68 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 ...
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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 ...
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0answers
26 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 $$\int_0^\infty w' \bar w$$ ...
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3answers
85 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 ...
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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 ...
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2answers
49 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 ...
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1answer
40 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 ...
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1answer
50 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 ...
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3answers
78 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 ...
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1answer
40 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 ...
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1answer
25 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 ...
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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 ...
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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 ...
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2answers
63 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 ...
5
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1answer
76 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
21 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 ...
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0answers
43 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 ...
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0answers
35 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| + ...
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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 ...
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1answer
17 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 ...
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2answers
51 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 ...
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1answer
67 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 ...
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0answers
14 views

Any convex Reinhardt domain is logarithmically convex

I have the following question in Shabat p.59: Prove that any convex Reinhardt domain is logarithmically convex. I think I have a good idea about how to show this, but need to be clear on the ...
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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 ...
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0answers
26 views

Continuity of a function of two complex variables

Let $\Omega$ be a domain and a continuous function $f: \overline{\Omega}\rightarrow\mathbb{C}$ which is holomorphic on $\Omega$ and $f'$ extends continuously on $\overline{\Omega}$. Is the function ...
3
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0answers
26 views

Spherical Harmonics and $L_+$ and $L_-$ operators

I have the spherical harmonics $Y_{m}^{l}\left(\theta,\varphi\right)$ and I want to show that the operators $L^{\pm}$ act as "creation" and "annihilation" operators such that ...
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1answer
21 views

Residue of a non-identically zero function

Assume f(z)is analytic in the complex plane and let f be a complex function which is not identically zero.Then,show that Res(1/f(z^3),0)=0. I know that the residue is calculating for only ...
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1answer
28 views

On $\zeta(s)=h^2(s)$ when we presume that the Riemann zeta function has no zeros for $\Re s>\frac{1}{2}$

By specialization of a theorem from complex analysis, one has that on assumption that the Riemann zeta function $\zeta(s)$ has no zeros with $\Re s>\frac{1}{2}$, then there exists an analytic ...
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0answers
29 views

Laurent Series, How it is done

Suppose that a series $$\sum_{n=-\infty}^{\infty}x[n]z^{-n}$$ converges to analytic function $X(z)$ in some annulus $R_1<|z|<R_2$. That sum $X(z)$ is called the z-transform of $x[n]$ $(n=0,\mp ...
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1answer
34 views

How can I prove that it is an Entire Function

Prove that if $$ f(z)=\left\{ \begin{array}{ll} \frac{\cos z}{z^2-(\pi /2)^2} & \hbox{when} \; z\neq \mp \pi/2\\ -\frac{1}{\pi}, & \hbox{when} \;z= \pi/2. \end{array} \right. $$ ...
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1answer
36 views

Challanging problems on [Grade-12]Complex Number [on hold]

recently we are introduced to interesting world of complex number but except for 3-5 problems in the my books,all the problems are just plug-and chug,expression manipulation,etc.. which bores me out ...
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0answers
29 views

Power series representation with Gamma function

This is taken from Stein and Shakarchi's Complex Analysis (Chapter 6, Exercise 4): Prove that if we take $$f(z) = \frac{1}{(1-z)^\alpha}$$ for $|z|<1$ (defined in terms of the principal branch ...
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0answers
31 views

Non-trivial inverse Laplace transform

I'm trying to compute the inverse Laplace transform of $f(s) = s^c/(N + s^{ir} )$ where $c,N \in \mathbb{C}$ and $r \in \mathbb{R}^+$ using the Bromwich integral $$ F(t) = \frac{1}{2 \pi i} \int_{- ...
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4answers
67 views

How do i write the analytic function $f(z)$ in terms of $z$?

I have an entire function, consider the function : $f(z)= (3x^2 + 2x - 3y^2 - 1) + i(6xy + 2y)$ I want to write $f(z)$ in terms of $z$.
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2answers
40 views

Contour integration with logarithms

I'm having trouble calculating the below integral to get the right answer: $$\frac{1}{2\pi i}\int_\gamma \frac{3}{z-2}\; dz$$ where $\gamma$ is parametrised by $\gamma(t)=3e^{it}, t\in [0,2\pi]$. So ...
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0answers
37 views

$|p(z)|=1$ contains no circles [duplicate]

Help with the following problem would be appreciated: Let $p(z)$ be a polynomial over $\mathbb{C}$ with at least two distinct roots. Prove that no connected component of the set $\{z \in \mathbb{C} : ...
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0answers
22 views

About the proof of a corollary of Arzela-Ascoli Theorem.

This is from Scheidemann, Complex Analysis. Theorem (Arzela-Ascoli): Let $K$ be a compact separable metric space, $E$ a finite-dimensional Banach space and $(f_j)_{j\in\mathbb{N}}\subseteq C(K,E)$ ...
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2answers
47 views

Is this a legitimate way to compute a contour integral?

I wish to calculate $$\int_{\Gamma}\cos(z)\sin(z)~\text{d}z$$ where $\Gamma$ is the line segment given by $\gamma(t)=\pi t+(1-t)i$ for $0\leq t \leq 1$. Here is what I did: We have that $$\int ...
2
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1answer
65 views

Laurent series of $\frac{e^z}{z^2+1}$

I cant figure out the laurent series of the following function. Let $f(z)= \frac{e^z}{z^2+1} $ and $|z|\gt 1$ $$\frac{1}{z^2+1}=\sum_{n=0}^{\infty}(-1)^nz^{-2n-2}$$ and $$e^z = ...
5
votes
1answer
67 views

Entire function $f$ such that $\lim\limits_{z\rightarrow \infty}f(z)=0$ and $f(0)=1$?

The question is this: Does there exist an entire function $f$ such that $\lim_{z\rightarrow \infty}f(z)=0$ and $f(0)=1$. I immediately would point to $f(z)=e^{-z}$. It is entire and satisfies the ...
0
votes
1answer
24 views

How to show $f(z)=x^2+y^2+i2xy$ is differentiable at $z_0=x_0+i0$?

How to show $f(z)=x^2+y^2+i2xy$ is differentiable at $z=x_0+i0$? Here is what I have done we know by the Cauchy Riemann (its it very easy to verify) that these can only hold for $z_0=x_0+i0$ that is ...
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0answers
35 views

Radius of convergence of the series $\sum\limits_{-\infty}^{\infty}(2^{-n}+4^{-n}) z^n$

I'm trying to find for what values of $z\in\mathbb{C}$ the series $$\sum_{n=-\infty}^{\infty}(2^{-n}+4^{-n})z^n$$ converges. My main methods are the nth root test and the ratio test. I believe it can ...
0
votes
0answers
28 views

When finding Laurent series when to use partial fractions?

When finding the Laurent series of $$f(z):=\frac{1}{z(z-1)(z-2)}$$ valid in the region $1<|z-2|<2$ for example do we just use partial fractions to break $f(z)$ up and the just find the Laurent ...
0
votes
1answer
26 views

Evaluating an integral using Cauchy Integral Formula and a further application

Question: $i)$ Evaluate $$\int_{\gamma}\frac{e^{2z}}{z}dz$$ Where $\gamma=${$z\in \Bbb{C}: \lvert z\rvert$=1} $ii)$ Hence find $$\int_{0}^{2\pi}{e^{2\cos(t)}}.\cos(2\sin(t) dt$$ My attempt: $i)$ ...
0
votes
2answers
49 views

Holomorphic function with $f(z)^2=z$

Is there an holomorphic function $f:B_1(0)\setminus\{0\}\rightarrow\mathbb{C}$ with $f(z)^2=z$?
0
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1answer
31 views

Help in this inequality in Conway's complex analysis book

I'm reading Conway's complex analysis book and on page 118 he write the following inequality: Why is this inequality true?