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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39 views

Showing that for continuous logarithms $g_1, g_2$ of a function on a connected set, the difference $g_1 −g_2$ is a constant

If $S$ is connected, $\ f$ is continuous and has continuous logarithms $g_1$ and $g_2$ on $S$, and continuous arguments $\theta_1$ and $\theta_2$, then $g_1 −g_2$ and $\theta_1-\theta_2$ are constant ...
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1answer
46 views

Value of the path integral $\int_{\partial K(0,1)} \frac {dz} {(z-a)(z-b)}$ along the unit circle $\partial K(0,1)$, where $|a|,|b| < 1$.

I want to determine the value of the path integral along the unit circle $\partial K(0,1)$, where $|a|,|b| < 1$: $$\int_{\partial K(0,1)} \frac {dz} {(z-a)(z-b)}$$ Assuming $a \neq b$, I compute $...
2
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2answers
118 views

Simplify $Im \left(\frac{az+b}{cz+d}\right)$

Let $z \in \mathbb{H}$, where $\mathbb{H}$ denotes the half plane $\mathbb{H}=\{z \in \mathbb{C}:Im(z)>0\}$. Let \begin{equation*} f(z)=\frac{az+b}{cz+d} \end{equation*} which is called a Mobius ...
3
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2answers
233 views

Holomorphic functions: is it true that $f(\bar{z})=\overline {f(z)}$?

Is it true that $ f(\bar{z})=\overline {f(z)}$, Where z is complex? I think it holds when $f(z)$ is holomorphic since we have $f(z)=p(x,y)+iq(x,y)=p(z,0)+iq(z,0)$ Any help...
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2answers
119 views

Analytic function in open unit disc.

How to prove that there exist an analytic function $f$ of $|z|<1$ onto itself such that $f(0)=1/2$, $f(1/2)=1/3$ and $f(1/3)=1/4$.
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1answer
41 views

Estimating a power series for the order of an entire function

Let $0<s<1$ and consider the power series $$\sum_{n=0}^{\infty}\frac{r^n}{(n!)^{1/s}}.$$ I need to show that for any given $\epsilon>0$, there exists $R>0$ such that for all $r>R$, $$\...
4
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2answers
285 views

Evaluating contour integral without using Residue Theorem

Find the value of the integration without using Cauchy integral formula/Residue theorem: $\int_{C}\cfrac{dz}{z^2+1}$ where C is a simple closed contour oriented in counter clockwise direction ...
2
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1answer
41 views

The total number of poles of an elliptic function in $P_0$ is always $\geq$ 2

I'm trying to follow this proof from Stein & Shakarchi "Complex Analysis". The statement of the theorem is in the title of the question. The Proof is as follows: Suppose that $f$ (an elliptic ...
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1answer
471 views

Give an example of a continuous function on an open interval which is not integrable

Here is the full question: Prove that every continuous function on a closed interval is Riemann integrable. Give an example of a continuous function on an open interval which is not integrable. I ...
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1answer
55 views

If $\lvert f(z) \rvert$ is bounded, does that imply $f(z)$ is bounded?

In the proof that if $f(z)$ is entire and the real part of $f$ is bounded ($Re(f) \leq M$), then $f$ is a constant, the first line of the proof is that we define $g = e^f$, so $\lvert g \rvert = \...
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2answers
30 views

metric on the set of complex sequences

Let X be the set of complex sequences $(a_n)_{n\in\mathbb{N}}\in \mathbb{C}$. Show that the transformation: $$ d((a_n), (b_n)) := \sum_{n=0}^\infty \frac{1}{2^{n+1}} \frac{|a_n - b_n|}{1 + |a_n - b_n|...
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1answer
32 views

Simple calculus series question; convergence of $\sum_{j = -\infty}^{\infty} \frac{1}{z - j}$

So I am having a brain fart and I cannot rigorous write (in my head) why the series in the question does not converge. I know it has to do with the harmonic series. Is it because if $|z| \to \infty$, ...
1
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2answers
56 views

Taylor/Laurent series question for $\cot(\pi z)$;where did $1/n$ come from?

I am having a panic attack right now so I can't see this immediately. But where did $\frac{1}{z - n} + \frac{1}{n}$ come from (yes I know its from the sum, but where did the summand come from?). Do ...
2
votes
2answers
219 views

Can every continuous function on complex domain be approximated by polynomials pointwise?

Do you know any theorem that will help me with this question: Let $f$ be any continuous function on complex plane. Show that there is a sequence $(P_n)$ of polynomials such that $P_n$ converges ...
2
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3answers
336 views

Holomorphic function

Let $f(z)$ be a holomorphic function that maps the unit disk to the unit disk. Prove that $$|f^{(5)}(0)| \leq 120.$$ I use some concrete example it seems that this statement work out but i ...
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2answers
35 views

Find the radius of convergence of $\sum_{n=0}^{\infty} \frac{\ln(n+1)} {n!} (z+2)^n$

I am having a little difficulty with this. I need to find the radius of convergence of this problem: $$\sum_{n=0}^{\infty} \frac{\ln(n+1)} {n!} (z+2)^n$$ Using the root test I have $$\lim_{x\to\...
3
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1answer
97 views

Understanding proof of fundamental theorem of algebra

So this is the proof I have: If $p(z)$ is a non-constant polynomial, then there exists a $z \in \Bbb Z$ such that $p(z) = 0$. Let $p(z) = z^n + a_{n-1}z^{n-1} +a_{n-2} z^{n-2} + ... + a_0$ ...
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1answer
46 views

Find the radius of convergence of this power serie

$\sum_{n=1}^{\infty} \frac{n^n}{n!} \cdot (z-1)^n$. Here is how i start. Since i know that $$lim|\frac{a_{n+1}}{a_n}| = |a_n|^{\frac{1}{n}}$$ so i did $$\frac{\frac{(n+1)^{n+1}(z-1)^{n+1}}{(n+1)!}}{\...
3
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2answers
134 views

Index in complex analysis is an integer : Intuition?

In complex analysis course, we prove that given a closed path $\gamma$ and $a\notin \gamma^*$ the following number: $$ \frac{1}{2i\pi}\int_{\gamma}\frac{dz}{z-a} $$ is an integer. The integral can ...
4
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1answer
177 views

Image under an entire function.

Let $f$ be an entire function and $B$ be a bounded open set in $\mathbb {C} $. Prove that boundary of image of $B$ under $f$ is contained in image of boundary of $B$. Does the same result is true for ...
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1answer
51 views

Surjectivity of meromorphic functions with a pole of order 1

Let $f(z)$ be a meromorphic function having pole of order $1$. For every $\tau \in \mathbb{C}$ does there exist a $z_o$ such that $f(z_o)= \tau$? If not in $\mathbb{C}$ then does it hold on a Riemann ...
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1answer
51 views

Mittag-Leffler Proof. Rudin notation question and some basic real analysis/topology

Proof is below What is meant by $\sum_{\alpha \in A_n}$? I thought $P_\alpha$ is a sum already. What is this open set he is talking about? Because $A_n$ isn't open. Oh he means the ...
2
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1answer
58 views

No Generalization of Mean Value Property for harmonic functions?

The Mean Value Property for harmonic functions tells us that the value of a harmonic function evaluated at the center of $D(P,r)$ equals its weighted integral over $\partial D(P,r)$. I am wondering if ...
2
votes
3answers
68 views

Finding the sum of the trigonometric serie:

There are two series: $$1) 1+\dfrac{\cos{x}}{p}+\dfrac{\cos{2x}}{p^2}+...+\dfrac{\cos{nx}}{p^n}=\sum_{k=0}^{n}{\dfrac{\cos{kx}}{p^k}}$$ $$2) \dfrac{\sin{x}}{p}+\dfrac{\sin{2x}}{p^2}+...+\dfrac{\sin{...
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2answers
21 views

What is $P$ and $X$ is supposed to be in this analysis question?

Source page 626. Can someone explain what is $\| P\| $ mean? Is that partition or what? Also why bother with $\epsilon/2$ if the giant expression in the middle proves the lemma. Finally, what is $X$...
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3answers
82 views

Why does the complex equation $z=Ae^{it}+Be^{-it}$ represent an ellipse?

Why does the complex equation $z=Ae^{it}+Be^{-it}$ represent an ellipse?, with $A,B \in \mathbb C$ How can it be described?
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2answers
104 views

Question related to Liouville's Theorem

Prove or disprove: Let $D$ be an unbounded domain in $\mathbb{C}$. If $f$ is a bounded, analytic function on $D$, then must $f$ be constant on $D$? This is clearly related to Liouville's Theorem, yet ...
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0answers
29 views

Existence of nice exhaustion - Rudin.

This is taken from Rudin's Complex Analysis/Real Analysis Can someone tell me why $K_n \subset \Omega$? I agree it is compact, but why does it follow that it is a subset of $\Omega$? WLOG, I ...
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1answer
62 views

Calculating the residue of a function

Let $f(z) = \frac{1+z}{1-\cos(z)}$ I wish to calculate the residue of $f$ at $0$, $2\pi$ and $-2\pi$. I believe this can be done by the following since $f$ has simple poles at these points $Res(f, ...
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1answer
44 views

Why Riemann Mapping Theorem is not valid for $U=\mathbb{C}$

If we take simple connected domain $U=\mathbb{C}$, in the statement of Riemann mapping theorem, then why is it not valid. What is the proper justification?
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1answer
85 views

Evaluate $\int_\gamma z^ne^{1/z}dz$, where $\gamma$ is the unit circle.

I need to evaluate $\int_\gamma z^ne^{1/z}dz$, where $\gamma$ is the unit circle traveled in the counterclockwise direction. I'm thinking about writing the function as a Laurent series and then ...
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0answers
45 views

Line integrals of $\frac{1}{z^2}$ and $\frac{e^z}{z}$ without calculations.

I have learned that : 1) if $f$, a holomorphic function on $U\subset \mathbb{C}$, open and simply connected, then $f$ has a holomorphic antiderivative on $U$ 2) if $f$, a holomorphic function on $U\...
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1answer
96 views

sub mean value property of plurisubharmonic function

It is well known that a plurisubharmonic function $\varphi$ defined in a domain $\Omega\subset \mathbb C^n$ satisfies the sub mean value property. Now if $\varphi$ is defined on a complex manifold $X$,...
3
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1answer
53 views

Riesz projection as a Cauchy-type integral

Let \begin{equation*} f(\zeta)=\sum_{k\in\mathbb{Z}}\widehat{f}(k)\zeta^k \end{equation*} be a complex-valued function on unit circle $\mathbb{T}=\{ \zeta\in\mathbb{C}:|\zeta|=1\},$ where $\{\...
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0answers
55 views

continuity of the complex square root function

I want to show that there is no continuous square root function in the complex plane, i.e. a function $f:\mathbb{C}\rightarrow\mathbb{C}$ with $f(w)^2=w$ for all $w \in \mathbb{C}$. I already ...
2
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1answer
42 views

A quadratic polynomial bounded by another

Suppose $p(x)$ and $q(x)$ are two quadratic polynomials in real coefficients such that: $$\lvert p(x) \rvert \leq \lvert q(x) \rvert ~ ~ ~ \text{for all} ~ x \in \mathbb{R} \tag{1}$$ Is the above a ...
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2answers
93 views

How to show there exists $E$ such that $E \cap K_n$ is dense for every $n$?

Let $\Omega$ be a region (nonempty connected open subset of the complex plane). Let $K_n$ be a sequence of compact sets whose union is $\Omega$, such that $K_n \subset \mathring{K_{n+1}}$ (the ...
3
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2answers
73 views

Laurent series of $e^{e^{\frac{1}{z}}}$ around $z=0$

Actually I need only the $res(f;0)$ where $f = e^{e^{\frac{1}{z}}}$ I thought of finding the Laurent series of $e^{e^{\frac{1}{z}}}$ around $z=0$ Any other Ideas if you have ?
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2answers
52 views

Suppose $a \in \mathbb{C}$, $|a| < 1$, and $f(z) = \dfrac{z - a}{1 - \overline{a}z}$. How to prove dependence of $|f(z)|$ on $|z|$? [duplicate]

Let $a \in \mathbb{C}$, $|a| < 1$. Also let $f(z) = \dfrac{z - a}{1 - \overline{a}z}$. I am asked to prove that $|f(z)| < 1$ if $|z| < 1$ and that $|f(z)| = 1$ if $|z| = 1$. What is a good ...
2
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1answer
57 views

Heine borel theorem on the complex plane

I'm trying to understand this proof of the Heine-Borel theorem on the complex plane. I'm reading Lang's Complex Analysis (page 22): I didn't understand the converse. Why there is a convergent ...
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1answer
31 views

Please could someone check my results for principal values of the complex logarithm?

I solved an exercise in my book and would greatly appreicate it if someone would check my result and tell me if it is correct:. The exercise: Find the principal values of the logarithm for the ...
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1answer
31 views

Showing this is analytic and finding its derivative $f(z)= \frac{4z+1}{z^3 - z}$

How to show the following is analytic and find it's derivative? $$f(z)= \frac{4z+1}{z^3 - z}$$ I am having trouble solving the above, since I am not sure how to break this into terms of $u,v$ for my ...
2
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2answers
157 views

Determining whether $f(z)=\ln r + i\theta$ (with domain $\{z:r\gt , 0\lt \theta \lt 2\pi\}$) is analytic [duplicate]

Define $$f(z)=\ln r + i\theta$$ on the domain $\{z:r\gt , 0\lt \theta \lt 2\pi\}$. This domain is just a punctured disk of radius $\ln r$, correct? How does one determine whether this is analytic, ...
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2answers
41 views

Proving $\prod_k \sin \pi k / n = n / 2^{n-1}$

I am stuck trying to prove $$\prod_{k=1}^{n-1} \sin {\pi k \over n} = {n \over 2^{n-1}}$$ and I'd appreciate help. What I have done so far: $z^n - 1 = \prod_{k=1}^n (z - \xi^k)$ where $\xi = e^{...
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1answer
36 views

IS $f(z) = x^3 + i(1-y)^3$ analytic and where is it differentiable?

Where is $f(z) = x^3 + i(1-y)^3$ analytic and where is it differentiable? I have taken Cauchy-Riemann equations as follows: $$u(x,y) = x^3$$ $$v(x,y) =(1-y)^3$$ $$\frac{\partial u}{\partial x}=3x^2$...
1
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1answer
132 views

Using de l'Hopital for complex functions?

I was wondering if de l'Hopital's rule also applies to complex functions. Some background information: This question came up as I was trying to calculate $\displaystyle \lim_{z \to 1} {z^n - 1 \over ...
1
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1answer
15 views

Prove that there is a postive integer $n_0$ such that all the $z_n$ are nonzero for $n \le n_0$

Assume that a sequence $(z_n)$ of complex numbers converges to a nonzero limit. Then Prove that there is a postive integer $n_0$ such that all the $z_n$ are nonzero for $n \le n_0$ I know I should ...
0
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1answer
43 views

Prove that $|\int_C f(z)dz| \le M |z_2 - z_1|$ where $M \gt 0$ such that $|f(z)|\le M; \ \forall \ z \in \Omega$

Let $z_1$ and $z_2$ be any two points in $\Omega$ and let $C$ be any oriented contour in $\Omega$ from $z_1$ to $z_2$. Also, assume that $f:\Omega \to \Bbb{C}$ is analytic on an open convex set $\...
1
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0answers
24 views

Use of Cauchy's intergral theorem (and consequences).

Let $f$ be an analytical function, with $|f(z)|\leq\displaystyle\frac{1}{1-|z|}$ for $|z|<1$. I have to prove that : $\left|\displaystyle\frac{f^{(n)}(0)}{n!}\right|\leq(n+1)\left(1+\frac{1}{n}\...
1
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1answer
36 views

$f$ holomorphic, calculate $f(1+i)$ with two informations about $f$

Let $f(x+iy)=u(x,y)+iv(x,y)$ be a holomorphic function, knowing that : 1) $Im(f'(x+iy))=6x(2y-1)$ 2) $f(0)=3-2i$ Find $f(1+i)$. There is nothing in my notes, but I have read online that $f'(x+iy)=\...