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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2
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2answers
39 views

Value of $z$ so that the series converges

$$ \begin{align} \sum_{n=0}^\infty \frac{1}{n^2}\left(z^n+\frac{1}{z^n}\right) \end{align} $$ Detrmine the value of $z$ so that the series converges I believe that the series converges when ...
1
vote
1answer
28 views

The range of $\arccos$

My question is whether or not the function $\arccos$ takes complex numbers to complex numbers? Specifically, if we identify $\mathbb{R}$ with the subset of the complex numbers which have zero ...
1
vote
1answer
44 views

'Identity theorem' for Meromorphic functions

If $f_1,f_2$ are meromorphic functions in $D$ and there exists a sequence of pairwise distinct points $z_n \in D$ such that $z_n \to z_o \in D$ and $f_1(z_n)=f_2(z_n),$ then $f_{1} \equiv f_2$ on $D.$ ...
2
votes
1answer
29 views

Radius of convergence of sum of complex power series

Could anyone advise me on how to find radius of convergence of $\sum^{\infty}_{n=1} [\frac{1}{n^2}+(-2)^n]z^n \ ?$ Thank you. My attempt: radius of convergence of $\sum^{\infty}_{n=1} ...
2
votes
1answer
35 views

Calculating Euler's Numbers

I've derived the finite series with binomial coefficients for Euler's numbers, as requested in John Conway's Functions of One Complex Variable, about p. 76, by deriving the expansion sec(z). But get ...
1
vote
1answer
22 views

Radius of convergence of powerseries containing $(\log n)^n$

$$ \begin{align} \sum_{n=2}^\infty (\log n)^n(z+1)^{n^2} \end{align} $$ What is the radius of convergence of this power-series? I tried applying the root test and the ratio test , but I couldn't ...
0
votes
2answers
38 views

Disk of convergence of the series $ \sum\limits_{n=1}^\infty n!\,(z-i)^{n!} $

$$ \sum_{n=1}^\infty n!(z-i)^{n!} $$ Find the disk of convergence of this powerseries. Can I set $n!=k$ and then deal with $\sum_{n=1}^\infty k z^k$ . On another note $\frac{z^{(n+1)!}}{z^{n!}}$ ...
0
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1answer
19 views

truncate power series to approximate holomorphic function by polynomial

Fix (open) polydisks $B' \subset B \subset \mathbb{C}^n$ and $\epsilon >0$. If $f$ is holomorphic on $B$, then there exists a polynomial $P$ such that $$\sup_{z \in\ B'}|f(z)-P(z)|<\epsilon.$$ ...
0
votes
1answer
40 views

Entire Function Which Tends to Zero At Infinity In All Directions

Say we have an entire function in the complex plane which tends to zero in all directions, i.e. $$f(z)\to 0 $$ as $$|z|\to \infty $$ Intuitively, this seems highly unlikely to me. There are many ...
2
votes
4answers
52 views

Writing the complex number $z = 1 - \sin{\alpha} + i\cos{\alpha}$ in trigonometric form

Now I can't finish this problem: Express the complex number $z = 1 - \sin{\alpha} + i\cos{\alpha}$ in trigonometric form, where $0 < \alpha < \frac{\pi}{2}$. So the goal is to determine both ...
6
votes
2answers
53 views

Geometric interpretation of complex path integral

Let's say that we want to make sense of integrating a function $f: \mathbb{C}\rightarrow\mathbb{C}$ over some path $\gamma$. I can imagine two reasonable ways of doing it. First, there's the way ...
0
votes
2answers
36 views

Radius of convergence of powerseries $\sum_{n=1}^\infty \frac{(-1^n)}{n!}z^n$

$$ \begin{align} \sum_{n=1}^\infty \frac{(-1)^n}{n!}z^n \end{align} $$ Find the radius of convergence of this powerseries. To determine the radius of convergence should I split it into two separate ...
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vote
2answers
59 views

When can an infinite sum and complex integral be interchanged?

Are there some conditions under which the following two are equal? $$\displaystyle \oint_C \sum f_n(z)= \sum \oint_C f_n(z)$$ In the case of real valued functions, the condition $f_n(z) \geq 0$ ...
5
votes
1answer
100 views

Prove $\int_0^1 \frac{\ln(1+t^{4+\sqrt{15}})}{1+t}\mathrm dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3})$

Prove that: \begin{equation} \int_0^1 \frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}\mathrm dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3}) ...
3
votes
3answers
46 views

radius of convergence for $\sum_{n=1}^{\infty} \frac{z^{n} n^{n}}{n!}$ and $\sum_{n=1}^{\infty} z^{n!}$

Exercise 4:10 in John D'Angelo's text is to find the radius of convergence for : A) $\sum_{n=1}^\infty \frac{z^n n^n}{n!}$ and B) $\sum_{n=1}^\infty z^{n!}$ I got half of an answer for A) which I ...
0
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0answers
15 views

Convergence of Series of Complex Numbers with Decreasing Modulo (non-zero imaginary part)

Let $(a_n)_{n \in \mathbb{N}}$ be a decreasing sequence of positive real numbers tending to zero. Show that for $\theta \in \mathbb{R}$, $\theta$ not a multiple of $2\pi$, the series $\sum_{n\geq1} ...
1
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2answers
33 views

rationalize the complex number multiplication rule

For a middle school student without previous knowledge of complex number, how do one introduce the multiplication rules of complex number? i.e., if we have two real number pairs of $(a,b)$ and ...
0
votes
1answer
25 views

Complex Analysis D shaped contour

Hi there. I am stuck on c. I proved (b) using Rouches theorem. To calculate the integral in c, I was not sure what to do. I am guessing you use the result in (b) somehow, but I thought that ...
0
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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
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1answer
35 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 ...
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0answers
24 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
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2answers
38 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 ...
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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
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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
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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.
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0answers
48 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
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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
29 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
31 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
46 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
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1answer
18 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 ...
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0answers
41 views

Using an induction on $q$.

I have a problem (It's a Proposition in Baouendi'book) 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 $f$ ...
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
56 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
35 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} $ . ...