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)

0
votes
0answers
18 views

Complex Analysis Dirichlet Problem

I have managed to answer (a) and (b). But so not know how to do the questions thereafter. For (c) could I tried to solve, with $argz=\frac{\pi}{2}$ and $\phi=\pi$, but that did not satisfy the ...
2
votes
1answer
42 views

Is it possible to write the function $f(x) = i \textrm{erf} (ix)$ (with $i$ imaginary unit) in a way that doesn't involve complex numbers?

Studying a physical problem I crashed into this differential equation (condition: $\lim_{x \to 0} = 0$) \begin{equation*} y' + A x y + B x^4 = 0 \end{equation*} where $x,A,B \in \mathbb{R}^+$. With ...
0
votes
0answers
25 views

why the numbers of poles and zeros of meromorphic function on the riemann sphere is finite?

why the numbers of poles and zeros of Meromorphic function on the Riemann sphere is finite? Can I use two statement below to conclude above question? if $f$ be a meromorphic function on ...
0
votes
2answers
39 views

Complex Analysis - Uniform Convergence

Question State The Weierstrass M-test, and use it to prove that if $\rho$ is a positive real number then the series $$\sum_{n=1}^\infty \frac n{e^{nz}}$$ is uniformely convergent on $\{x + iy ...
1
vote
1answer
30 views

Analysis of a Holomorphic function $f$ given $1 \geq |f '(z)|$.

Since $f$ is holomorphic we can use Cauchy's inequality. Thus for $n = 1$ we have $ |f'(z)|\leq \frac{M}{R} $ where is $M$ is the max value of $|f(z)|$ and $R$ is the radius of a random region. We ...
4
votes
0answers
36 views

Finding an analytic function such that real part is the given function.

I am reading the book Complex Analysis by Lars V Ahlfors. In the book he uses a nice method without involving integration to evaluate $f$ given that the real part of the function is $U$. The method ...
0
votes
0answers
29 views

Complex analysis, cutoff integration

The diff-invariant distance between $z'$ and $z$ is (for short distances) $e^{w(z)}|z'-z|$, so a diff-invaraint cutoff would be at $|z'-z|=\epsilon e^{-w(z)}$. Then $ \int ...
3
votes
1answer
115 views

The number of solutions of $z^5+2z^3-z^2+z=a$ for $a\in \mathbb{R}$

How we can calculate the number of solutions of $$z^5+2z^3-z^2+z=a\;\;,\;\;a\in \mathbb{R}$$ in the half-plane $\mathfrak {Re}(z)\ge 0$. Any hint would be appreciated.
0
votes
0answers
50 views

Zero moment of arc length measure

Suppose $\gamma$ is a simple smooth closed curve and is not a circle. Does there exist a monomial $z^n$ so that $\int_{\gamma}z^n ds(z)=0$ for some positive integer $n$? (In here, $ds$ is the arc ...
4
votes
1answer
125 views

Can the winding number be infinite?

Let $z$ be a point in the complex plane, and $\gamma$ be a closed curve. Is it possible that $$n(\gamma,z) = \frac{1}{2\pi i}\int_\gamma \frac{dw}{w-z}$$ becomes unbounded? In other words, is it ...
0
votes
0answers
24 views

Holomorphic and meromorphic functions on Riemann surfaces

On any domain $\Omega\subset \mathbb{C}$, the set of all holomorphic functions form an integral domain. Its field of quotient is the set of all meromorphic functions on $\Omega$. However this is not ...
2
votes
2answers
63 views

Poles of $\large e^{f(z)}$

$\fbox{1}$ If $z_0$ is a pole of $$f:U \subset \mathbb{C}\longrightarrow \mathbb{C}$$how to prove that $z_0$ can not be a pole of $\large e^{f(z)}$. $\fbox{2}$ If $z_0$ is an essential singularity of ...
2
votes
1answer
23 views

Showing that $\tan(\pi z) = z$ has exactly three solutions in the strip $|\Re(z)| < 1$

We can't use Rouche's theorem here directly, so we have to apply the argument principle. If $f(z) = \tan(\pi z) - z$ , then $f'(z) = \pi \sec^2(\pi z) - 1$. Choose the rectangle $\Gamma$ with ...
3
votes
0answers
45 views

Check my answer for find a formula for $\sum_{n=0}^{\infty} \frac{z^{n}}{4^{n+2}}$

The next question in John D'Angelo's text is exercise 4.9. I got an answer but wanted to check it because there's no solution manual: Find a formula $$ \sum_{n=0}^ {\infty} \frac{z^{n}}{4^{n+2}}. $$ ...
1
vote
1answer
36 views

Finding an explicit mapping

Here is a question from an old prelim exam in complex analysis that I am stuck on: Let $f: \mathbb{D} \rightarrow \mathbb{D}$ be analytic and satisfy $f(\frac{1}{2})= \frac{1}{2}$ and ...
0
votes
0answers
31 views

What advanced methods in contour integration are there?

It is well known how to evaluate a definite integral like $$ \int_{0}^\infty dx\, R(x), $$ where $R$ is a rational function, using contour integration around a semicircle or a keyhole. Most complex ...
2
votes
1answer
28 views

Complex Green's Theorem

I want to integrate $\int_{\partial R} |e^{zt}|dz$ where $R\subseteq \mathbb{C}$ is a rectangle whose sides are parallel to the coordinate axes. I want to use a complex version of green's theorem, but ...
5
votes
1answer
32 views

Is Cauchy's formula apt for evaluating this integral

I'm trying to evaluate the following. $$\frac{1}{2i}\int_{-\infty}^\infty \frac{s \sin{(sr)}}{(s-k)(s+k)}\mathrm{d}s,$$ with $k$ and $r$ being real constants. The integral could be written as ...
0
votes
1answer
22 views

maximum modulus principle question

Suppose that f is analytic on a domain D which contains a simple closed curve $\gamma$ and the inside of $\gamma$. If $|f|$ is constant on $\gamma$ then either f is constant or f has a zero inside ...
2
votes
1answer
29 views

$f(z) = \sum_{n=0}^\infty a_nz^n$ converges in the unit disk and $|f(z)| < 1$. Show that $|a_0|^2 + |a_1| \leq 1$.

The series $\sum_{n=0}^\infty a_nz^n$ converges in the unit disk $|z| < 1$ and defines a function mapping the unit disk into itself. Show that $|a_0|^2 + |a_1| \leq 1$. Only thing I've thought ...
2
votes
1answer
47 views

Factoring a complex polynomial

Factorize the polynomial : $$ p(x) = x^{5} - x^{4}+ 4x - 4 $$ In real factors in the lowest degree possible. So in previous questions I have been given at least one rot so that I can factorize it ...
0
votes
1answer
22 views

Weiestrass M-Test Complex Anal

Hi there I am struggling with the question above. I managed to prove that it converges $\mid z \mid \leq p$ using the Weierstrass M-test, with $M_{n}=\frac{z^{n}}{n(2-p)}$ followed by the ratio ...
1
vote
3answers
66 views

Complex Equations

The Equation: $$ z^{4} -2 z^{3} + 12z^{2} -14z + 35 = 0 $$ has a root with a real part 1, solve the equation. When it says a real part of 1, does this mean that we could use (z-1) and use ...
5
votes
3answers
89 views

Is entire function constant when $ |f(z)|\le \log|z|,\ |z|>1$.

Let $ f : \mathbb{C} \to \mathbb{C} ,$ entire and $|f(z)|\le \log|z|,\ |z|>1. $ Show that $f$ is constant. What first comes to mind is Louville's theorem, but log 's problems with analyticity ...
2
votes
0answers
32 views

Compute $[\Lambda,\ \bar{\Lambda}]$

I have a problem: We denote by $[X,\ Y]$ the commutator of $X$ and $Y$ defined by $$[X,\ Y]f(p)=X(Yf)(p)-Y(Xf)(p), \tag{1}$$ for any smooth function $f$ defined on a hypersurface $M$. ...
1
vote
1answer
19 views

Taylor series of an analytic function that maps the unit disk surjectively onto the upper half plane

Given only that $f(z)$ is analytic and maps the unit disk $|z| < 1$ surjectively to the upper half plane $\Im(z) > 0$, how much can we deduce about $f(z)$? In particular, can we find the radius ...
2
votes
1answer
28 views

$\int_{|z| = 2} \frac{1}{f(z)(1+f(z))^2} dz$ where $f(z) = z^{1/2}$ with branch such that $\Re f(z) \geq 0$

As the title states, the definite integral in question is $$\int_{|z| = 2} \frac{1}{f(z)(1+f(z))^2} dz,$$ where $f(z) = z^{1/2}$ with branch cut such that $\Re f(z) \geq 0$, i.e., the cut is the ...
2
votes
1answer
49 views

Image of a entire function.

Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be a non-constant entire function. by Liouville's Theorem, $f(\mathbb{C})$ is dence in $\mathbb{C}$. by the Open Mapping Theorem $f(\mathbb{C})$ is open ...
0
votes
2answers
57 views

If the imaginary part of an entire function is never zero, the function is constant

Let $f : \mathbb{C} \to \{z\in\mathbb{C}:\Im(z)\neq0\} $ entire . Show that $f$ is constant. I took $g(z)=\frac{1}{f(z)}$ and I think that g is bounded, therefore it is constant (due to Louville's ...
0
votes
1answer
42 views

The argument of complex numbers

Let w be a given real number and determine the argument: $$\frac{1}{(1+2iw)^{2}}$$ This is how far I came: $$\frac{(1-2iw)^{2}}{(1+2iw)^2(1-2iw))^2} = \frac{(1-2w^{2}) - 4iw}{(1+4w^{2})^{2}} = ...
3
votes
1answer
36 views

$f(z)$ and $g(z)$ are Meromorphic functions such $|f(z)|\le|g(z)|$ for all $z\in\mathbb{C} $ then $ f=ag$

We know that if $f(z)$ and $g(z)$ are entire functions such that $g(z)\ne0$ and $|f(z)|\le|g(z)|$ for all $z\in\mathbb{C} $ then by Liouville's theorem $$ f=ag$$ for some constant $a\in \mathbb{C} $ . ...
1
vote
1answer
36 views

Complex Analysis Liouville's Theorem [duplicate]

I am not sure how to solve the following problem: Use Liouville's theorem to prove that if f(z) is holomorhpic in the in entire complex plane and $f(z+1) = f(z)$, and $f(z+i)=f(z)$ for all $z$ in $C$ ...
1
vote
1answer
33 views

Entire function , guidance or advice

Let $f:\mathbb{C}\to\mathbb{C}$ entire and , $|f(z)|\le m\ e^{a\mathop{\rm Re} z}, z\in\mathbb{C},$ $a,m>0$ Show that $f(z)=Ae^{az}, A\in \mathbb{C}$ I think that most of these case are dealt ...
2
votes
2answers
75 views

Why is continuous differentiability required?

I have two questions. My book proves that if $f:\mathbb{C}\rightarrow \mathbb{C}$ is a holomorphic function, then it satisfies the Cauchy-Riemann equations, and if we look at the function as $F: ...
0
votes
0answers
30 views

Complex Analysis Estimation Theorem i

I am struggling with 65(ii) The part of the solution I dont understand is how $\mid e^{iz} \mid \leq1$ on gamma. Could someone please help me?
0
votes
1answer
27 views

Is saying that $Re(f(z))\to 0, z\to \infty$ “correspondent” to saying $Re(f(z))\le M, \forall z \in \mathbb{C}, M \in \mathbb{R}$ and $ M$ constant?

Let $f:\mathbb{C} \to \mathbb{C}$ entire . Is saying that $Re(f(z))\to 0, z\to \infty$ "correspondent" to saying $Re(f(z))\le M, \forall z \in \mathbb{C}, M \in \mathbb{R}$ and $ M$ constant?
0
votes
2answers
95 views

Does there exists an entire function with the following property: $f\left(\frac{1}{n}\right)= \frac{n^4}{1+n^4}, n =1,2,…$

Could anyone advise me on how to use the Identity theorem to determine whether there exists an entire function with the following property: $f\left(\dfrac{1}{n}\right)= \dfrac{n^4}{1+n^4}, n =1,2,...$ ...
6
votes
2answers
145 views

A strange answer for $\int _{-1}^1 \log x\; dx$

I typed $\int _{-1}^1 \log x\; dx$ on Wolfram Alpha. It is giving the answer to be $-2+i\pi$. Can someone please explain what is happening?
1
vote
1answer
47 views

Bounded meromorphic function on $\mathbb{C}$

I just want to make a clarification with regard to bounded meromorphic functions on the complex plane $\mathbb{C}$. Would they be constant? Here's what I do know: $(1)$ Liouville's Theorem states ...
5
votes
1answer
77 views

If $e^{i\theta}=e^{i\varphi}$, then $\theta-\varphi=2k\pi$

This is pretty easy I think but I am having a tough time trying to prove this in a satisfying way to me. I am trying to show that $$e^{i\theta}=e^{i\varphi} \Rightarrow \theta-\varphi=2k\pi,\, \text{ ...
4
votes
2answers
76 views

where does $\frac{1}{1-z}$ about the point $5i$ converge.

Hi: Th next question in John D'Angelo's text is exercise 4.8: where does the series for $\frac{1}{1-z}$ about the point $5i$ converge ? I understand that the expansion is : $\sum_{n=0}^{\infty} (z - ...
3
votes
2answers
65 views

Suppose $f: \mathbb{D} \rightarrow \mathbb{D}$ is analytic and $f(0)=a \neq 0$. Show that $f$ has no zeroes in the disk $\{z: |z|< |a|\}$.

I'm not sure how I could use Schwarz's Lemma to solve the following problem from an old complex analysis prelim: Let $\mathbb{D}$ be the unit disk and suppose we have $f: \mathbb{D} \rightarrow ...
1
vote
1answer
42 views

Complex Analysis Rouches Theorem

I am struggling to understand the solution below. I understand how to apply Rouches theorem when showing that there are a certain number of zeroes in a circle / annulus. In this example (where they ...
0
votes
1answer
23 views

Generalization of Montel's theorem?

I'm stuck with the following question: Let $\Delta$ be the unit disk and let $H$ be the upper half-plane. Show that any sequence of holomorphic functions $f_n:\Delta \rightarrow H$ either has a ...
1
vote
1answer
32 views

How to calculate complex residues

How would one best calculate the residue of $$f(z)=\frac{z^2}{z^6+1}$$ At its various poles? My method is to use L'hopital to calculate $\lim_{z\to root}(z-a)f(z)$ but this is rather slow and ...
1
vote
0answers
21 views

Divergent sequence for a non-constant meromorphic function - hint requested

I'm stuck with the following exercise and I'd appreciate a hint. Let $f$ be a non-constant meromorphic function. Show that either there exists $z_0\in \mathbb{C}$ such that $f(z_0)=0$ or there is a ...
4
votes
1answer
65 views

What does $|\mbox{d}z|$ mean?

Given the complex contour integral $\int_\alpha |z|\,|\mbox{d}z|$, with $\alpha(t)=\mbox{e}^{it}$, $0\leq t\leq 2\pi$. What does $|\mbox{d}z|$ mean? My guess is: $$\frac{|\mbox{d}z|}{|\mbox{d}t|}= ...
1
vote
1answer
36 views

If a real polynomial of degree $n\gt 1$ has a root of modulus exceeding all others, is that one a real root?

Suppose $a_nx^n+\ldots+a_1x+a_0=0\; (a_n\in \mathbb{R})$ has $n$ distinct roots $r_1,r_2,\ldots, r_n$ (no multiple roots), and if $\exists r_k$ s.t. $\forall r_i\in\{r_1,r_2\cdots r_n\}-\{r_k\}$, ...
0
votes
2answers
55 views

How should I calculate $\displaystyle\int_{-\infty}^\infty\exp\left\{-\frac{1}{2}(x-it)^2\right\}dx$?

I've read that the residue theorem would help to calculate $$I:=\displaystyle\int_{-\infty}^\infty\underbrace{\exp\left\{-\frac{1}{2}(x-it)^2\right\}}_{=:f(x)}dx$$ Since $f$ is an entire function ...
2
votes
2answers
36 views

Holomorphic function $|f| \geq 1$ is constant

Given $f:\mathbb{C} \mapsto \mathbb{C}$ is holomorphic on $\mathbb{C}$ and that $|f(z)| \geq 1$ for all $z \in \mathbb{C}$. Show $f$ is constant. The "equal" part of the problem is quite common but i ...