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, ...

learn more… | top users | synonyms (2)

2
votes
0answers
37 views

Analytic $N$th roots: can my statements here be generalized?

I'm generally having trouble seeing when and why analytic functions have analytic $N$th roots. I know the following statements to be true, but I know that they can also be generalized in various ...
8
votes
0answers
165 views

Help with the integral $\int_{0}^{\infty}\frac{\log(1\pm ix)^{2}}{\left(\frac{t}{2}\log(1 \pm ix) \right )^{2}-\pi ^{2}n^{2}}e^{-2\pi mx}dx$

Referring to a previous question, i want help with the integral : $$\int_{0}^{\infty}\frac{\log(1\pm ix)^{2}}{\left(\frac{t}{2}\log(1 \pm ix) \right )^{2}-\pi ^{2}n^{2}}e^{-2\pi mx}dx$$ Where $n,m$ ...
3
votes
1answer
86 views

Absolute Value of Complex Integral

Let $[a,b]$ be a closed real interval. Let $f:[a,b] \to \mathbb{C}$ be a continuous complex-valued function. Then $$\bigg|\int_{a}^{b} f(t)dt \ \bigg| \leq \int_{a}^{b} \bigg|f(t)\bigg| dt,$$ where ...
0
votes
0answers
75 views

How do I solve first order non-linear system of PDE: $\partial f^i(x,y)/\partial z = F^i(f^1,f^2,…,f^n)$?

Suppose that I have a system of PDEs of the following form: \begin{eqnarray} \frac{\partial f^i(x,y)}{\partial z} = F^i(f^1,f^2,...,f^n), \qquad i = 1,..,n \end{eqnarray} Where $z = x + iy$, ...
0
votes
1answer
23 views

When does $-\frac{\pi z}{2}\cot(\pi z)+\frac{1}{2}=0$ where $z$ is a complex variable?

Let $z$ be a complex variable. Is there someone who can show me when does :$$-\frac{\pi z}{2}\cot(\pi z)+\frac{1}{2}=0$$ Note: I have tried using trigonometric formulas but it didn't work. Maybe I ...
0
votes
1answer
56 views

Prove that $Z_1^2+Z_2^2+Z_3^2=Z_1Z_2+Z_1Z_3+Z_2Z_3$ [closed]

$Z_1,Z_2$ and $Z_3$ are affixes of points of equilateral triangle $ M_1 ,M_2$ and $M_3$. Prove that $Z_1^2+Z_2^2+Z_3^2=Z_1Z_2+Z_1Z_3+Z_2Z_3$.
2
votes
1answer
53 views

Do analytic functions on open subsets of $\mathbb{C}$ with an analytic square root form a sheaf? [duplicate]

I'm trying to learn algebraic geometry and am trying to think about what kinds of things are presheafs but not sheafs. One exercise I had was to show that bounded holomorphic functions on open ...
6
votes
0answers
166 views

Does $\displaystyle \lim_{m \to +\infty}f_{2,m}(x)$ converge?

This is related to a previous question where, as stated there, $f_{2}(n)$ gives the greatest power of $2$ that divides $n$. Specifically the sequence $\lbrace ...
3
votes
0answers
52 views

Construct holomorphic function from harmonic function

Let $h$ be a real valued harmonic function on the twice punctured plane $Ω=C$\ {0, 1}. Show that there exist unique real numbers $a_0, a_1$ such that $$u(z)=h(z)−a_0log|z|−a_1log|z−1|$$ is the real ...
3
votes
2answers
54 views

Iteration of analytic function

Suppose $f$ is analytic on the unit disc $D$ with $f(0)=0$ and $f(D)\subset D$. Define $f_n=f\circ f\circ\dots\circ f$. If $f$ is not a rotation, can we say $f_n\to 0$ uniformly on compact subsets of ...
1
vote
3answers
42 views

Is the boundedness necessary to extend harmonically?

"If $u$ is harmonic and bounded in the punctured disk $0<|z|< \rho$, then $u$ can be extended harmonically to the disk $|z|<\rho$ harmonically." This fact has been shown here. My Question ...
0
votes
2answers
35 views

Complex Analysis ( Open/Closed Set).

let $z = re^{i\theta}$ , How do we prove that , $0\leq \operatorname{arg}(z)\leq\dfrac{\pi}{4}$ ($z \neq 0$), is neither a open set nor a closed set. $\operatorname{arg}(z)$ is nothing but $\theta$ ...
1
vote
1answer
35 views

Uniqueness of harmonic function with Mixed Dirichlet Neumann condition

Let $u \colon \{\mbox{Im } z>0\}\subset\mathbb{C}\to \mathbb{R}$ be a positive harmonic function in the upper half plane, i.e $$ \Delta u=0,\,\, \mbox{for}\,\mbox{ Im } z>0. $$ Consider now the ...
0
votes
0answers
38 views

Complex dot product

I know that the complex dot product is defined as $\boldsymbol{a}\cdot\boldsymbol{b}=\sum_{i}a_ib_i^*$. Is there a standard name for the operator ...
0
votes
1answer
117 views

Show that $f(z)\not=0, \forall z\in \mathbb C$.

Suppose , an entire function maps the real line onto the circle $C=\{z:|z|=R\}, R>0$. Show that $f(z)\not=0, \forall z\in \mathbb C$. I thought through contradictory way but I could not think ...
0
votes
1answer
68 views

How to differntiate $\int_{0}^{2\pi} u(re^{i\theta}) d\theta$?

Suppose $u$ is a twice continuously differentiable function on $a< |z|<b, \ z\in \mathbb C,$ which is harmonic that is, it satisfies $u_{rr}+\frac{1}{r}u_r + u_{\theta \theta}=0.$ (If we put ...
2
votes
1answer
51 views

integrals of exponential functions over the real axis

How to evaluate the integral $$ \int_{-\infty}^\infty \exp(-\sqrt{1+x^2})dx? $$ I intend to change the variable $x$ to $\tan t$ but failed... How to solve it?
2
votes
1answer
45 views

Riemann mapping under which uncountably many boundary points correspond to a single point

I am interested in the following question, which is 10.4 from this list: Give an example of a domain conformally equivalent to the disc where uncountably many points on the unit circle ...
5
votes
4answers
98 views

integration of $\int_0^{2\pi} cos^{2n}(t)dt$

Show that for any $n \in \mathbb{N}$, $$\frac{1}{2\pi}\int_0^{2\pi}\cos^{2n}(t)dt = \frac{1 \cdot 3 \cdot 5 \cdots(2n-1)}{2 \cdot 4 \cdot 6 \cdots 2n}$$ To solve this problem, I was thinking that I ...
1
vote
1answer
27 views

If $\phi$ is entire and satisfies $|\phi(z)| \leq e^{|z|}$, then $|\phi'(z)| \leq c e^{|z|}$ for some $c > 0$.

If $\phi$ is entire and satisfies $|\phi(z)| \leq e^{|z|}$, then $|\phi'(z)| \leq c e^{|z|}$ for some $c > 0$. I saw this problem on a practice qual but I had no idea what to do. It looks ...
0
votes
1answer
49 views

Functions with real domain but complex range, do they have any use?

For example if we define the square root function like this: $$\text{Sqrt}({x})= \begin{cases} \sqrt{x} & x\geq 0 \\ i\sqrt{-x} & x<0 \end{cases}$$ Or we could have an exponential ...
1
vote
2answers
53 views

Show that $\lim\limits_{z \to a}\frac{\log|f(z) - f(a)|}{\log |z - a|}$ is an integer.

Let $f$ be analytic in a neighborhood of $a$. Show that $\lim\limits_{z \to a}\frac{\log|f(z) - f(a)|}{\log |z - a|}$ exists and is an integer. We have $$\frac{\log|f(z) - f(a)|}{\log |z - a|} = ...
0
votes
0answers
26 views

how to solve complex differential equation [closed]

how to solve this differential equation $$ a_1\cdot \phi'(x)^+ + a_2 \cdot \phi'(x)^-=c_2\cdot g(x) $$ where $\phi(x)$ is a complex analytic function thanks
1
vote
0answers
35 views

Holomorphic proper function on $\mathbb{C}$ is a polynomial

I want to show that every holomorphic proper map $f:\mathbb{C}\to \mathbb{C}$ is a polynomial. Since $f$ is continous and proper, it can be extended to a continous map $f:S^2\to S^2$, where $S^2$ is ...
1
vote
1answer
40 views

Requirement for a given function to be smooth

I have quite a basic question about the derivatives. My uncertainty comes mainly from the fact that I don't know how the complex logarithm behaves. Here is the description (this task is not ...
1
vote
1answer
35 views

Prove that a harmonic function is an open map.

I'm trying to solve the following exercise of the book Functions of one complex variable, John B. Conway on page 255: 4. Prove that a harmonic function is an open map. (Hint: Use the fact that the ...
0
votes
1answer
47 views

Variant of Riemann mapping theorem

Put $D=\{z\in \mathbb C: |z|<1\}$ (open disk) and let $\Omega$ be non empty open simply connected in $\mathbb C$ and $\Omega \neq \mathbb C.$ Then Riemann mapping theorem tells us that there exists ...
0
votes
2answers
34 views

A doubt regarding a property of a complex function

I just wanted to confirm that when we define a complex function $f (z) $, is it just a function in terms of $z $, or also that in terms of $\overline {z} $? This is because if the latter is true, ...
0
votes
1answer
36 views

Complex Analysis Dense Set Problem

The Problem: Suppose $f(z) = e^{i\theta}z$. Show that if $\theta$ is not a rational multiple of $\pi$, then the orbit of $ z \in \mathbb{C}$ is dense in the circle with radius $|z|$ and at the center ...
1
vote
0answers
49 views

integral from gradshteyn and ryzhik

I'm interested in evaluating the integral $$ \int_{a}^\infty e^{-x\cosh\alpha}\,K_{\nu}(x\sinh\alpha)\,\frac{dx}{x}, $$ where $a>0$ and $\nu$ is purely imaginary. Here $K$ denotes the MacDonald ...
0
votes
1answer
20 views

$ \exists f:\mathbb C \setminus D \to \mathbb C$ is bounded one-one holomorphic, how?

We note that there cannot exist bounded one-one holomorphic map $f:\mathbb C \setminus \{0\} \to \mathbb C.$ Put $D=\{z\in \mathbb C: |z|\leq1\}$ (closed disk). My Question: How to show there ...
1
vote
0answers
18 views

Analyticity of the outer function of an analytic composition

Let $\mathscr{U}$ be an open neighborhood of the origin of $\mathbb{C}$ and let $F(t,x)$ be a function that is continuous on $\mathbb{C} \times \mathscr{U}$ and that is holomorphic in $\mathscr{U}$ ...
1
vote
1answer
30 views

If $\lim_{|z|\to 1^-}u(z)=0$ then $u\equiv 0$

Let , $u(z)$ be a complex valued harmonic function in $|z|<1$ and $\lim_{|z|\to 1^-}u(z)=0$. Then show that $u(z)$ is identically zero in $|z|<1$. I am unable to understand that from where ...
0
votes
1answer
21 views

real coordinates of a complex manifold

I have a naive question about real coordinates of a complex manifold. Let's consider 1-dimensional case for simplicity. Let $X$ be a Riemann surface and $z$ be a local complex coordinate. Then one ...
5
votes
1answer
137 views

Prove that $f$ is a polynomial

If $f(z)$ is an entire function and $|f(z)|\ge1$ for all $z$ with $|z|\ge \pi$ then show that $f$ is a polynomial. I tried to apply Lioville's theorem on $f$. For $|z|\le \pi$ , $|f(z)|\le k$ for ...
0
votes
0answers
7 views

Characterization of Multivariate Polynomials with Unique Critical Point

I would like information about $\{f\in \mathbb{C}[x_1,\ldots,x_n]:Z(\nabla(f))=\{0\}\}$. Above, the $Z$ denotes the vanishing locus of a function, i.e. the set of points where it vanishes, and ...
0
votes
0answers
38 views

Show that a conformal map of an open disk onto any open disk is necessarily bilinear.

Show that a conformal map of an open disk onto an open disk is necessarily bilinear. Please help me with this proof. Any help will be appreciated. To clarify this, it's not just a map from a open ...
9
votes
1answer
150 views

For which complex $a,\,b,\,c$ does $(a^b)^c=a^{bc}$ hold?

Wolfram Mathematica simplifies $(a^b)^c$ to $a^{bc}$ only for positive real $a, b$ and $c$. See W|A output. I've previously been struggling to understand why does $\dfrac{\log(a^b)}{\log(a)}=b$ and ...
0
votes
0answers
25 views

Proof of Schwarz-Pick Theorem

This question has probably already been asked before, but since I can't find it here (it's probably labeled "Application of Schwarz lemma" among 30 others) I'll repeat it. Let $f: D \rightarrow D$ ...
0
votes
0answers
28 views

Proof of Sokhotski-Plemelj theorem

Sokhotski-Plemelj theorem states $$ \phi_i(z)=\frac{1}{2\pi i}\mathcal{P}\int_C\frac{\varphi(\zeta) \,d\zeta}{\zeta-z}+\frac{1}{2}\varphi(z), \, \\ \phi_e(z)=\frac{1}{2\pi ...
0
votes
1answer
27 views

Why does a conformal mapping create a full tiling of semi-infinite strips in the w-plane?

I know that, specifically for linear fractional transformations, symmetric points get mapped to symmetric points. So, if the real line gets mapped to a circle, then under a LFT, points symmetric ...
0
votes
3answers
23 views

Primitive of a meromorphic function

I found this statement that I cannot justify due to my lack of knowledge in complex analysis (this is not my field of study). Let $D\subset \mathbb{C}$ be the open unitary disc centered at $0$, let ...
1
vote
2answers
21 views

Quotient of space and a group of maps, Riemann surfaces

I've been attempting to study Riemann surfaces, and I have continuously run into this notion which eludes me. I see people write things like $ \mathbb H / <z\mapsto z+1>$ or $\mathbb D / PSL$. I ...
1
vote
4answers
135 views

Is $|z-i| = |z+i|$?

I computed a Mobius transformation $-\frac{z-i}{z+i}$ that maps the upper half plane to a disk, with i mapping to the center of the disk, $w = 0$. How do I know that the disk is a unit disk and not ...
-2
votes
1answer
64 views

Does there exists bounded one-one holomorphic map $f:\mathbb C \setminus \{ 0 \} \to \mathbb C$? [closed]

(1)Does there exists bounded one-one holomorphic map $f:\mathbb C \setminus \{ 0 \} \to \mathbb C$? (2)Let $X$ be a closed connected subset of $\mathbb C$ and which has more than one element. Does ...
0
votes
3answers
48 views

Why if a function is holomorphic and injective in neighbourhood of $x_0$ then $f'(x)\ne 0$ in neighbourhood of $x_0$?

Why if a function is complex differentiable and injective in some neighborhood of $x_0$ then its derivative is non zero in that neighborhood? I just don't see how why it is like that. Obviously in ...
14
votes
1answer
202 views

There exist $x_{1},x_{2},\cdots,x_{k}$ such two inequality $|x^b_{1}+x^b_{2}+\cdots+x^b_{k}|\ge 1$

This problem is a 2014 Sydney mathematics competition problem (11 grade). It seems difficult to solve. (I previously posted the n=2 case for which André Nicolas and Dan Robertson proposed solutions) ...
0
votes
1answer
46 views

Does there exists biholomorphic map(with suitable condition) from domain to open disk?

Put $D=\{z\in \mathbb C: |z|<1\}$ (open disk). We define $f:D\to D$ as $f(z)=z,$ for all $z\in D,$ which is clearly biholomorphic. My Question is: (1) For any $z_0\in D,$ can we choose a ...
1
vote
2answers
73 views

Evaluate $\frac{1}{2 \pi} \int_0^{2 \pi} \frac{1 - r^2}{1 - 2r \cos(\theta) +r^2}d\theta$ [duplicate]

Let $0 < r < 1$. Compute $$\frac{1}{2 \pi} \int_0^{2 \pi} \frac{1 - r^2}{1 - 2r \cos(\theta) +r^2}d\theta$$ The hint is rewrite this integral as a complex line, but I still don't know how to to ...
0
votes
1answer
37 views

Value of $\int_C\frac{e^z}{z}dz$ with $C$ unit circle

Compute the integral $$\int_C\frac{e^z}{z}dz$$ where $C$ denotes the unit circle with positive orientation. I was thinking that let $z = e^{it}$, $dz = ie^{it}$, then the integral will become ...