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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48 views

The series $\sum\limits_{n\ge1}n^{-z}$ converges locally normally

Show that the series $\sum\limits_{n\ge1}n^{-z}$ converges locally normally on the half plane $\{z:\text{Re}(z)>1\}$ $\displaystyle ...
2
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0answers
43 views

Why is the bold text true??

Ok I have stared at this for nearly 30 minutes now, and can't figure out why the bold text is true. Problem: If $z \in \mathbb{C}$ and $\mathrm{Re}(z^n) \ge 0$ for $n \in \mathbb{N}$, show that $z ...
4
votes
1answer
120 views

Find the derivative of a polylogarithm function

I was trying to find to which function the next series converges. $$ \sum_{n=1}^{\infty} \ln(n)z^n $$ If we take the polylogarithm function $Li_s(z)$ defined as $$ Li_s(s)=\sum_{n=1}^{\infty} ...
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votes
0answers
34 views

image of $|z|= \pi\ $ under $ z \mapsto\ e^z\ $

I have been asked to sketch the image of $|z|= \pi\ $ under the mapping $ z \mapsto\ e^z\ $ ; I think I should use the identity that $ z = |z| e^{i\theta}$ , and replace z in $ e^z $ with this ...
7
votes
1answer
133 views

Showing a complex analytic function is unbounded

This was one of the problems on a previous year's Complex Analysis final exam. Assume $f\in \mathcal O (\mathbb H )$, non-constant, and $f(\frac {i}{\sqrt n})=0, \forall n\in \mathbb N$. Prove that ...
3
votes
1answer
88 views

Schwarz 's lemma and sharp upper bound

Let $f$ be a holomorphic function on $|z|<1$ with $|f(z)|<1$ for all $|z|<1$. (1) Find necessary and sufficient conditions for equality of $$\frac{|f'(z)|}{1-|f(z)|^2} \leq ...
0
votes
1answer
144 views

Question about graduate textbook and class.

I am a senior in mathematics, and I have had Advanced Cal I, but currently go to a no name school (there were only three people in the class). I have also taken Advanced Cal II as an independent ...
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3answers
50 views

Number of solutions for equations with complex variables

A question about the number of solutions for the following equation: $$z^2+(1-i)z-3i=0$$ So the solutions are: $$z_{1,2}=\frac{-1+i \pm \sqrt{10i} }{2}$$ But $\sqrt{10i}$ has two options with ...
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0answers
62 views

Convincing proof of quotient rule using Landau's notation

I'm taking a Complex Analysis course this semester which follows D. Ulrich's Complex Made Simple. One of the earliest exercises in Chapter 0 is to re-derive the usual Calculus rules for ...
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vote
4answers
62 views

Why does $f = u+iv$ holomorphic $\implies$ $-if = -iu + v$ holomorphic?

If we multiply both sides of a holomorphic complex-valued function $f = u(x,y) + iv(x,y)$ by $-i$, why is it true that the resulting equation is also holomorphic?
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1answer
185 views

Determine $\lim _{z \to 0}\frac {e^{-1/z}}{z}$ =?

It appears to be $0$. But the classical L'Hôpital does not work because we end up with pretty much the same: $\lim _{z \to 0}\dfrac {e^{\frac {-1}{z}}}{z^2}$ ....
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0answers
16 views

Is the unit ball of $H^\infty(\mathbb{D})$ a metrizable topological semigroup under multiplication?

The space $H^\infty(\mathbb{D})$ of all bounded holomorphic functions on the open unit disc carries many different topologies. One such topology is given by uniform convergence on compact subsets; ...
0
votes
1answer
36 views

Find a sequence of complex polynomials with certain properties. (Hardy spaces over unit circle)

Let $\lambda\in \Bbb S^1$. Find a sequence of complex polynomials $p_n(z)$ such that for any $c>0$ the following inequality does not hold: $$|p_n(\lambda)|\le c\cdot \|p_n\|$$ where ...
1
vote
1answer
36 views

Prove that $\vert z_1-z_2\vert$ is more than or equal to $\vert z_1 \vert - \vert z_2 \vert$

Prove that $\vert z_1 - z_2 \vert \geq \vert z_1\vert - \vert z_2 \vert$ where $z_1,z_2$ are complex numbers. I know that you have to use the triangle inequality for say $\vert z_1+z_2 \vert \leq ...
1
vote
2answers
51 views

Existence of complex polynomial with modulus on $|z|=1$ less than 1

I wonder if there exists a complex polynomial $P(z),z\in \mathbb{C}$ s.t $$\forall |z|\leq 1, P(z)<1.$$ I know that using modulus maximum principle, we only need to find $$P(z)<1, \forall ...
0
votes
1answer
24 views

Modulus of $c^z$

For a positive real number $c$ and a complex number $z$, we define $c^z=e^{z\ln(c)}$. Express the modulus $|c^z|$ in terms of real and imaginary parts of $z$ I get ...
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1answer
98 views

Find all holomorphic functions with $\Re(f(x+iy))=2xy$

Find all holomorphic functions $f:\mathbb C\to\mathbb C$ such that for all $x,y\in\mathbb R$ $\Re(f(x+iy))=2xy$ So $f$ must be complex differentiable, which is equivalent to that ...
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0answers
91 views

Relation between being holomorphic in $\Delta\times\Delta$ and in every relatively compact polydisk.

I asked to some professors of my university and no one was able to help me (the one who held the course is abroad for a period, otherwise I'd ask him, obviously). My problem is that simply I don't ...
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0answers
45 views

Exercise: Refelxive Space and Hahn-Banach Theorem

Let $X$ be a Banach space, $(f_i)_{i\geq 1}\subset X'$. Prove that for any $x\in X$, $\sum_{i\geq 1}|f_i(x)|<\infty$ if and only if $\sum_{i\geq 1}|F(f_i)|<\infty$ and $F\in X''$. I just ...
2
votes
1answer
68 views

Exercise: Compact Operator on Banach Space

Let $X$ be a Banach space, $A\in\mathbb{B} (x)$ and let $B\in\mathbb{K} (x)$ be a compact operator on $X$. show that $$\sigma(A+B)\subset\sigma(A)\cup\sigma_p(A+B)$$ (where ...
0
votes
1answer
36 views

If $u,v$ are Harmonic Conjugates in $\mathbb{C}$ and $u^3-3uv^2\ge 0$, then $u,v$ are Constants.

Let $u,v$ be harmonic in $\mathbb{C}$ and assume that $v$ is the harmonic conjugate of $u$. Assume that $$u^3-3uv^2\ge 0\tag{$1$}$$ in $\mathbb{C}$. Prove that $u,v$ are constants. I am not sure ...
3
votes
2answers
183 views

Residue Theorem and Homologous to zero

This is a very basic question and I couldn't find it posted yet but here it goes; The Residue Theorem states that if $f:G\to \mathbb{C}$ is analytic on $G$- a region and $f$ has isolated singularities ...
2
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0answers
24 views

Is the following complex value differential equation always has a solution?

Let $a(z)$ be a fixed complex value complex variable function, not necessarily holomorphic. Consider the following differential equation $$ \frac{\overline{\partial}f}{\partial \overline{z}}+af=0. $$ ...
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1answer
73 views

sum of a complex power series

I have to find the sum of a complex power series inside radius of convergence, for simplicity let's say the series looks something like that: $f(z)=\sum_{n \geq 2} \frac{z^n}{n(n-1)}$ Then after ...
0
votes
1answer
147 views

Star-shaped proof complex

How do we formally prove that a convex set is star-shaped, and how to exhibit a non-convex set which is star-shaped? Also that an open star-shaped set is a region? I guess its easily proven ...
0
votes
1answer
105 views

Complex integration?

I’ve read that you can integrate some definite integrals using complex variables. I’m curious as to how this happens in practice - for example, an integral such as $\int_0^{2\pi} \frac{\cos 2 ...
3
votes
2answers
104 views

Computing Residue for a General, Multiple-Poled function?

I'm trying to compute the residue of the following function at $a$. I'm having a little trouble seeing which poles are relevant: Compute $\,Res_f(a)$ for the following function: $$f(z) = ...
2
votes
2answers
55 views

Laurent series of $f(z)=\frac{1}{1-z}-\frac{1}{2-z}$

I have to find the Laurent series of $f(z)=\frac{1}{1-z}-\frac{1}{2-z}$ on $D_1(0)=\{z\mid |z|<1\}$, on $C_{1,2}(0)=\{z\mid 1<|z|<2\}$ and on $C_{2,\infty }(0)=\{z\mid |z|>2\}$. For ...
4
votes
3answers
2k views

Poisson's integral formula

Evaluate $$\int_{0}^{2\pi}\frac{1}{\rho^2+r^2-2r\rho \cos(t-\theta)}dt.$$ I found this under some exercises about Poisson's integral formula, to my surprise the problem looks simple but I do not ...
2
votes
2answers
703 views

Fourier transform of a complex exponential with quadratic argument

I'm a PhD student who is starting to work right now in the well-established field of ultra-fast optics. The thing is that, in most of the papers I have been reading during the past few days, there is ...
3
votes
1answer
62 views

Maximum value of the absolute value of a holomorphic function

Consider the holomorphic function $f(z) := \frac{1}{z}(e^z - 1) = \sum_{k=0}^\infty \frac{z^k}{(k+1)!}$ with $\text{Re}(z) \leq 0$ and let $g(z) := |f(z)|$. Show that the maximum of $g$ is attained at ...
1
vote
1answer
28 views

Uniform bound for $cos(nz)$.

One of my friend's professor gave him a question that $\cos(nz)$ is uniformly bounded in the open unit disk. I do not believe it is since setting $z=1/2i$, $\cos(nz) = \frac{e^{-n/2}+e^{n/2}}{2}$ ...
3
votes
3answers
57 views

Residue of two functions

Let be $f,g$ functions analytic in $z_0$, with $z_0$ a zero of order one of $g$ and $f(z_{0})\neq 0$. Show that $$ \operatorname{Res}\Bigl(\frac{f}{g},z_0\Bigr)=\frac{f(z_{0})}{g'(z_{0})} $$ My ...
0
votes
1answer
41 views

zero of order one

Let be $f,g$ functions analytic in $z_{0}$, with $z_{0}$ a zero of order one of $g$. Show that for small r, we have $$z_{0}=\dfrac{1}{2\pi i}\int_{\gamma}{\dfrac{zg'(z)}{g(z)}}dz$$ Where $\gamma$ is ...
3
votes
3answers
60 views

Finding the roots of $(1 + i)^{\frac{1}{4}}$

The professor says that the $n = 4$ roots of this are in the form: $\cos(\frac{\theta + 2k\pi}{n}) + i\sin(\frac{\theta + 2k\pi}{n})$, where $k = 0, 1, 2, 3$. So to find $\theta$, we find the $r = ...
0
votes
1answer
27 views

Show that h is harmonic iff $\frac{\partial h}{\partial \overline z}$ is conjugate harmonic

I want to solve the following: Show that h is harmonic iff $\frac{\partial h}{\partial \overline z}$ is conjugate harmonic My attempt: $h$ is harmonic iff $\frac{\partial^{2} h}{\partial z ...
0
votes
1answer
37 views

Taylor Series to the Power 1/z

I am attempting to find the Taylor Series for $(\frac{\sin{z}}{z})^{\frac{1}{z^2}}$. While I can plug this into Wolfram and use the output, I want to understand how to calculate the Taylor Series ...
4
votes
0answers
43 views

Show $\frac{\partial^2}{\partial ^2 x}+ \frac{\partial^2}{\partial ^2 y}= 4 \frac{\partial^2}{\partial z \partial{ \overline{z}}} $

I want to solve the following exercise: Show that: $$\frac{\partial^{2}}{\partial ^{2}x}+ \frac{\partial^{2}}{\partial ^{2}y}= 4 \frac{\partial^{2}}{\partial z \partial{ \overline{z}}} $$ My ...
0
votes
1answer
14 views

Residue Calculation Using given Conditions

Assume $f(z)$ is holomorphic on a punctured domain (a is removed), and that $f(z)$ has a pole of order n greater than or equal to $1$. Need To Compute residue at $z=a$ of $f'(z)/f(z)$ How to look ...
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0answers
26 views
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23 views

Branch cut choice for $(z+\frac{1}{z})^{\frac{1}{3}}$

$$f(z)=(z+\frac{1}{z})^{\frac{1}{3}}$$ It has branch points at $z=0$ , $z=+i$ , $z=-i$. I am trying to select a branch cut in such a way that $g(z)$ is analytic in the annulus $|z|>2$. I have ...
0
votes
1answer
26 views

Cauchy integral description

If we have $z(t)=re^{it}, 0\leq t \leq 2\pi $ and $r >0$, How can we show an analytic description of the interior bounded by $\gamma$ which is a closed path, and show that it is star-shaped? ...
1
vote
1answer
44 views

Unique continuous complex log of a function nowhere zero

Consider a function $\phi : \mathbb{R}^d \rightarrow \mathbb{C}$ which is continuous, satisfies $\phi(0)=1$ and is nowhere zero. I am reading a book where the following claim is made: Fix $z \in ...
0
votes
2answers
28 views

$f$ holomorphic on $\Delta$ the open unitary disk $\Rightarrow\lim_{z\to\partial\Delta}$ exists finite in $\Bbb C$

Let $f:\Delta\to\Bbb C$ holomorphic, where $\Delta=\{z\in\Bbb C\;:\;|z|<1\}$ is the open unitary disk. How can I prove that $\lim_{z\to z_0}$ exists finite in $\Bbb C,\;\;\;\forall ...
1
vote
1answer
213 views

Are Möbius transformations holomorphic or meromorphic?

In my previous question it was pointed out to me that an "automorphism" of the projective line/Riemann sphere (=that is a Möbius transform) is a bijection that is meromorphic in the local coordinate ...
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votes
1answer
20 views

Complex conjugate of the following expression

suppose that we have f is a scalar. And we have the expression $H=Re(f)+Im(f)$. If I want to take the complex conjugate of $H$, does this become $\bar{H}=Re(f)-Im(f)$ or this doesn't make sense?
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vote
1answer
45 views

How to define complex powers of $0$?

I'm studying Complex Analysis, and I've seen the definition of the set-valued power function as follows Let $z,w \in \mathbb{C}$, then $z^{w} \equiv \exp(w\log z)$. If I recall correctly. Now it ...
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vote
0answers
160 views

Generalization of the argument principle

This exercise is from big Rudin: Let $f \in H(U)$ and $D(a,r)\subset U$ be a disk s.t. $f$ has no zero on the boundary of the disk. Let $\gamma$ be a curve parametrizing the boundary of ...
0
votes
1answer
40 views

Given the entire function $f$, prove that $f(z)=u(x)+\textrm{i}v(y)$ is a polynomial of degree one

So i have written out the Cauchy Riemann equations and have seen that $u_y=0=v_x$. I am trying to think of the relation of these partial derivatives but I'm not so sure how to word my thoughts. I ...
2
votes
1answer
28 views

Show that the function tends to zero

Let be $\gamma_{\epsilon}$ the circle $|z|=\epsilon$, positively oriented, $0<\alpha<1$ and $f$ a entire function. Show that $$\displaystyle\lim_{\epsilon \to ...