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)

1
vote
1answer
51 views

Has anyone ever explored $(\sin{x})^x$ , $(\cos{x})^x$, etc?

I've come across a problem that involves something very close to: $$\int(\cos{x})^xdx$$ and I have no clue as to how to proceed with any kind of analysis. It occurred to me that complex analysis ...
0
votes
1answer
16 views

Show a certain analytic function must exist

Suppose that $f$ is holomorphic on $D - \{0\}$, where $D$ is the open unit disk. Suppose that $f$ has a pole of order one at $0$, with a residue equal to $n$ for some positive integer $n$. Show there ...
1
vote
3answers
41 views

Prove that if $f$ and $g$ are analytic at $w$, then so is $fg$

Prove that if $f$ and $g$ are analytic at $w$, then so is $fg.$ My main attempt was using the Cauchy-Riemann equations on the product in this manner but this did not work out. My thinking: ...
0
votes
1answer
14 views

Determine all open sets on which $f$ is analytic.

Given $$f(z) = \frac{1}{x} + i(-\frac{1}{y})$$ Determine all open sets in which $f$ is analytic. My attempt: $$f(z) = u(x,y) + iv(x,y)$$ where $u(x,y) = \frac{1}{x}$ and $v(x,y) = ...
0
votes
1answer
25 views

Prove that a given function is analytic.

We are given the following function : $g(z)=\sqrt{r}e^{i\frac{\theta}{2}}$ , $(r>0,-\pi <\theta <\pi)$ Also , $g(z)$ is analytic in its domain with derivative : ...
0
votes
1answer
21 views

Proving two domains are not conformally equivalent

Let $D$ be the open unit disk. Show that $D - [-1/2, 1/2]$ and $D - \{0\}$ are not conformally equivalent. Thoughts so far: I'm not sure where to begin, but a hint would most helpful to get me ...
0
votes
1answer
16 views

Reference about conformal map

I am here because I want to know if someone knows of some good e fast books or references about conformal map . More precisely I need of the propeties of the conformal maps on manifolds with boundary. ...
0
votes
3answers
45 views

Is $\sin( | z^{2}| )$ ,where z is complex, analytic?

I know sin is analytic, but I got myself confused in regards to the $| z^{2}| $. I want to say it is since any input sin takes is fine but I feel there's something I missed. Thanks.
0
votes
0answers
13 views

Bilinear transformations

I'm having a problem going about these questions Let $T$ be the bilinear transformation that maps $\infty$ to $0$, $\iota$ to $1$ and $2\iota$ to $2$. Derive a formula for $T$. Obtain the image ...
1
vote
1answer
25 views

Recover the holomorphic function from one of its components using Cauchy-Riemann equations

I got two similar questions: Find the holomorphic function $f(x+iy)$ if $\Re(f(x+iy))=x(3-2y)\text{ and }f(i)=2i$ Find the holomorphic function $f(x+iy)$ if $\Im(f(x+iy))=3(x-1)^2y-y^3\text{ and ...
1
vote
2answers
24 views

Verifying Cauchy-Riemann equations for $f(z) = \bar{z}^2/z$ at the origin

We're given a two variable function as follows : $$ f(z) = \begin{cases} \dfrac{\bar{z}^{2}}{z} , & z\neq0 \\ 0\:\:\:, & z=0 \\ \end{cases} $$ We need to show that the ...
0
votes
1answer
19 views

Evaluating This Complex Line integral

I'm trying to evaluate the following: $$\int_{\mathcal{C}}z^3 e^{-z^4}\,dz $$ along the path $\mathcal{C}=\left\{\sin(t^2)-i\frac{2t^2}{\pi}:0\leq t\leq\sqrt{\frac{\pi}{2}}\right\}.$ I tried using ...
0
votes
1answer
16 views

Find a domain (open and connected set) in which $f(z) = (z-2)arg_0 (z)$ is continuous.

Find a domain (open and connected set) in which $f(z) = (z-2)arg_0 (z)$ is continuous. Note: $$arg_\phi(z) = arg(\phi),~~~~ \text{where }\phi < arg(z) \le \phi + 2\pi$$ \begin{align}f(z) ...
-1
votes
1answer
54 views

Are all derivatives of sinc function bounded on real axis?

It seems that all derivatives of $sinc$ function ($sinc(x)=sin(x)/x$) are bounded on real axis. Is it true or no?
-6
votes
0answers
34 views

How can I prove $e^{i\theta}=\cos\theta+i\sin\theta$? [duplicate]

How can I prove $$e^{i\theta}=\cos\theta+i\sin\theta$$
-1
votes
0answers
10 views

a,b and c are on complex plane are on the unit circle [on hold]

Suppose that a, b and c are on the unit circle in the complex plane and a + b + c = 0. Prove that a, b and c are the vertices of an equilateral triangle. Find the general expression ...
0
votes
1answer
53 views

Show that there exists holomorphic $f$ such that $f^2=\frac{\sin z}{z}$

I have to show that there exists a holomorphic function $f$ on a neighborhood of $0$ such that $f^2(z)=\frac{\sin z}{z}$ on this neighborhood. Furthermore, I have to find the radius of convergence of ...
3
votes
2answers
27 views

Find the image of $|z+1|=2$ under $f(z) = \frac{1}{z}$ where $z \in \mathbb C$

Find the image of $|z+1|=2$ under $f(z) = \frac{1}{z}$ where $z \in \mathbb C$ My attempt: Let $z = x + iy$ $\displaystyle |z+1|=2 \iff | (x + iy)+1|=2 \iff |(x+1) +iy|=2 \iff (x+1)^2 + y^2 = ...
2
votes
1answer
21 views

Limit of a sequence of holomorphic functions

Let $f_n$ be a sequence of holomorphic functions on a domain $D \subset \mathbb{C}$ converging to a function $f$, and also converging uniformly on compact subsets. Suppose each function has at most ...
0
votes
1answer
59 views

prove that if $ h = |f_1|^2 \cdots + |f_n|^2$ is constant then $f_i$ is constant

Let G be a domain, and let $f_1 \ldots f_n$ analytics in G such that $$ h = |f_1|^2 + \ldots + |f_n|^2$$ is constant prove that every $f_i$ is also constant in G the question has a hint to ...
2
votes
1answer
51 views

find all zeroes of $p(z) = \sum_{k=0}^n \frac{z^k}{k!}$

show that all the zeroes of $$p(z) = \sum_{k=0}^n \frac{z^k}{k!}$$ where z is a complex number $z\in \Bbb{C}$ are inside the ring $$ \{\frac{n}{2e} < |z| <2n\}$$
1
vote
1answer
19 views

Is there a holomorphic function on the unit disk that satisfies a certain condition?

Is there a holomorphic function on the unit disk that satisfies $f(1/n) = 1/\sqrt{n}$? Thougthts so far: I know that $f(z) = \sqrt{z}$ won't work, as it is not analytic at $0$. My intuition says that ...
3
votes
1answer
25 views

Primitive of $dz/z$ is a branch of log

Let $D$ a connected open set of $\mathbb{C}$. A continuous function $f:D\to \mathbb{C}$ is a branch of log if $e^{f(t)}=t$ on $D$. In my book (Cartan) it is written that if $F$ is a primitive of the ...
1
vote
1answer
33 views

Is it necessary for the Imaginary-axis to be perpendicular to the Real-axis?

The Real number line is in one dimension. If you want to map a complex number, you would have to add a second dimension to that number line- the Imaginary-axis. The Imaginary-axis is always ...
0
votes
0answers
34 views

About Defintion of an Open Mapping

Definition: A function $f$ is said to be an open mapping if the image of every open set in its domain is itself open. So: If we have $f:K \rightarrow \mathbb{C}$, where $K \subseteq D$ , $D$ is the ...
0
votes
0answers
16 views

How to choose grid for a numerical integral of complex function?

I need to numerically integrate a complex function $f(x)$ on R, i.e. to approximate $\int_{-\infty}^\infty{f(\xi)d\xi}$. Performance is crucial as the integration is repeated a high number of times ...
0
votes
2answers
28 views

Maximal Modulus Principle Theorem for a Compact Set

The theorem says: The maximum of the absolute value of an analytic function over some compact set in a domain of analyticity is attained at the boundary of that compact set. Could someone please ...
0
votes
1answer
10 views

Uniform convergence on punctured disc => analytic

Suppose $f(z)$ is analytic on the punctured disc $D'=\{z\in\mathbf{C}:0<|z|<1\}$. Suppose further that there are functions $\{f_n(z)\}_{n=1}^\infty$ that are analytic on all of ...
1
vote
1answer
35 views

An alternative method to find $\sum_{k=1}^{2n-1} | \beta ^k - 1|$

Let $\beta \in \mathbb{C}$ such that $\beta ^n = 1$ but $ \beta ^k \neq 1$, $\forall k=1,2,\cdots, n-1$. Find the value of $$ \sum_{k=1}^{2n-1} | \beta ^k - 1|.$$ I came across such a question ...
1
vote
1answer
39 views

Set on which a holomorphic function is bounded is simply connected

I came across this problem and couldn't figure it out: Let $U$ be a simply connected domain, and let $f:U\to\Bbb{C}$ be holomorphic on $U$. For $c>0$ define $V_c:=\{z:|f(z)|<c\}$. Show that ...
2
votes
1answer
42 views

How to determinate poles and residues of this function.

I have the following function $$f(z)=\frac{e^{iz}}{e^z+e^{-z}}$$ Using the notaion $z= \rho(cos\theta+isen\theta)$ I found that the poles are $z_1=\frac{\pi}2i$ and $z_2=-\frac{\pi}2i$. To determinate ...
0
votes
1answer
43 views

How Can I Proof $1+2\cos(\theta)+2\cos(2\theta)+2\cos(3\theta)+.. $ [duplicate]

Proof $1+2\cos(\theta)+2\cos(2\theta)+2\cos(3\theta)+...+2\cos((n-1)\theta)=\frac{2\sin((n-\frac{1}{2})\theta)}{2\sin(\frac{1}{2}\theta)} $
0
votes
2answers
24 views

calculating complex numbers - help needed [on hold]

Probably it is simple, but I am blind right now and I do not see how to solve this task: $e^{i \frac{2\pi}{3}}+e^{i\frac{4\pi}{3}}+1$
1
vote
1answer
16 views

Lipshitz constant of bounded function on disc

Let $f(z)$ be analytic in the unit disc $\{z\in\mathbf{C}:|z|<1\}$ and have bounded modulus in the sense that $|f(z)|\leq M$ for all $z$ in the disc. Let $0<r<1$. Find a constant $C$ ...
1
vote
1answer
33 views

Find the number of elements of a complex subset

How many elements does the set $\{z\in \mathbb C:z^{60}=-1,z^k\not=-1\text{ for } 0<k<60\}$ have ? $z^{60}=-1=\cos(2k\pi+\pi)+i\sin(2k\pi+\pi)$. Then , ...
0
votes
0answers
13 views

An application of Rouche. when will roots lie in the annulus

Suppose $0<\rho<1$. Show that for $n\geq N(\rho)$, $nz^{n+1}-(n+1)z^n+1$ has all the zeros in $\rho<|z|<1/\rho$. Provide an estimate of $N(\rho)$. We wish to show that on $|z|=1/\rho$, ...
0
votes
0answers
49 views

$S_n\to S$ implies $\sigma_n\to S$ [duplicate]

Let $(a_n)_{n\ge0}$ be a sequence of complex numbers and $S_n=\sum_{k=0}^na_k$, set $\sigma_n:=\frac{S_0+S_1+\dots+S_n}{n+1}$, if $\lim_{n\to\infty}S_n=S$ then show that ...
0
votes
1answer
20 views

Given sequence of harmonic functions converges uniformly on compact subsets

Suppose that $u_n$ is a sequence of harmonic function on an open, connected subset $D \subset \mathbb{C}$ such that $u_n(z) \in (0, \infty)$ for all $z \in D$ and with $u_n(z_0) \to 0$ for some $z_0 ...
0
votes
1answer
19 views

Estimate when $f$ is locally one to one

Suppose $f$ is analytic in $|z|\leq 1$ and $|f(z)|<1$, $f(0)=0$ and $f '(0)=a\neq 0$. Show that there exists a disc of radius $\rho$ s.t for any $z_1,z_2$ in the disc, $f(z_1)=f(z_2)$ implies that ...
-1
votes
0answers
24 views

entire and meromorphic functions

Let $f$ be an entire function with simple zeros at $z_1,...,z_n,...$ a sequence with no finite limit point. Let $g(z)$ be a meromorphic function with simple poles at the same points $z_n$. (a) What ...
-2
votes
0answers
19 views

Conformal mapping of complement of a line to upper half plane

Find a one to one analytic function that maps the complex plane with the segment $[-1,1]$ removed from the real axis, onto the upper half plane, and leaves the point $z=i$ unchanged. How do we get a ...
0
votes
0answers
9 views

Extending a holomorphic function to boundary

Recently I am reading stein's Complex Analysis and it talks about extending a holomorphic function to boundary. My question appears in Example 1. in p.231: Define $f:\mathbb H=\{z\in\mathbb ...
0
votes
0answers
32 views

Weird inequality bounding $|f(0)|$ and $|f(z) - f(0)|$, probably application of Stchwarz lemma

I recently encountered a weird inequality which says that $$|f(0)| + \frac{1 - |z|}{2|z|} |f(z) - f(0)| \leq 1$$ where $f$ is a holomorphic function on the unit disk to itself (and of course for $z ...
0
votes
0answers
31 views

If $|f(z)|<1$ for $|z|=1$ then $f(z)=z^n$ has $n$ solutions in $\mathbb E$

If $f$ is analytic in an open set $D$ containing $\mathbb E:=\{z\in\mathbb C:|z|\le1\}$ and $|f(z)|<1$ for $|z|=1$ then $f(z)=z^n$ has $n$ solutions in $\mathbb E$ and exactly one fixed point ...
0
votes
1answer
24 views

Degrees of freedom of a complex vector space $V$ and its conjugate $\bar V$?

As an easy example consider the complex vector space $\Bbb C^2$. We can consider $\Bbb C^2$ as vector space over $\Bbb R$ and thus have the four basis vectors $$ \hat e =\{(1,0), (i,0), (0,1), ...
0
votes
0answers
57 views

the complete set of half-period formulas in terms euler's formula

Given the elliptic modulus $k$ ,such that the complementary modulus is defined by $$k'\equiv \sqrt{1-k^2}$$,the jacobi amplitude $$\phi\equiv am(u|k)$$ and $K(k)$, is the complete elliptic integral of ...
1
vote
2answers
39 views

About complex power series

I have a really big doubt. I'm trying to find all the values of $z$ for which the next power series converges: $$\sum_{n=0}^{\infty} \frac{z^{3n}}{8^{n}(1-in)} $$ Using the root test I have that ...
3
votes
2answers
47 views

Holomorphic function with reals to reals

Suppose that $f$ is an entire function and that there is a bounded sequence of real numbers $a_1, a_2, ... $ such that $f(a_n)$ is real for all $n$. Show that $f(x)$ is real for all real $x$. ...
0
votes
1answer
39 views

In what sense do complex functions have norms?

Can someone please correct my misunderstanding about bound and norms? By Liouville Theorem, a complex function is bounded if and only if it is a constant function. By this logic wouldn't all ...
1
vote
2answers
33 views

$\sum_{k=1}^\infty f(z^{n_k})$ is analytic

Suppose $f(z)$ is analytic on $D=\{z:|z|<1\}$ with $f(0)=0$. Suppose $\{n_k\}_{k=1}^\infty$ are positive integers such that $n_k<n_{k+1}$ for all $k$. Show that on $D$ the function ...