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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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 ...
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1answer
72 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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22 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 ...
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1answer
107 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 ...
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14 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 ...
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2answers
52 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 ...
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1answer
20 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 ...
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43 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 $$ \sum_{n=0}^ {\infty} \frac{z^{n}}{4^{n+2}}. $$ ...
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1answer
30 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 ...
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0answers
24 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 ...
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1answer
27 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 ...
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1answer
29 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 ...
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1answer
21 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 ...
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1answer
27 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 ...
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1answer
42 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 ...
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1answer
18 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 ...
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3answers
64 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 ...
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3answers
74 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 ...
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0answers
30 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 $M$. ...
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1answer
17 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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14 views

Using an induction on $q$.

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 $M$. ...
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1answer
26 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 ...
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1answer
45 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 ...
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2answers
49 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 ...
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1answer
39 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}} = ...
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1answer
34 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} $ . ...
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1answer
31 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 $C$ ...
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1answer
30 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 ...
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0answers
12 views

Show that $\exists \beta(z,\bar{z},s)$ such that $\dfrac{1}{2i} [\Lambda, \bar{\Lambda} ]=\beta(z,\bar{z},s)\dfrac{\partial }{\partial s}$

I have a problem: We denote by $[X,\ Y]$ the commutator of $X$ and $Y$ defined by $$[X,\ Y]f(p)=X(Yf)(p)-Y(Xf)(p), \tag{*}$$ for any smooth function $f$ defined on a hypersurface $M$. ...
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2answers
73 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: ...
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29 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?
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1answer
20 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?
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2answers
90 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: $f\left(\dfrac{1}{n}\right)= \dfrac{n^4}{1+n^4}, n =1,2,...$ ...
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1answer
107 views

A strange answer for $\int _{-1}^1 logx dx$

I typed the integral of $\int _{-1}^1 logx dx$ on wolfram alpha. It is giving the answer to be $-2+i\pi$. Can someone please explain what is happening. Thanks.
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1answer
42 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
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1answer
74 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{ ...
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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 - ...
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2answers
63 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 ...
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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 ...
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1answer
22 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 ...
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37 views

Characteristic functions

Here $E(Y)$ means the expected value of $Y$. 1) Could any one explain for me how to get from (2.7) to (2.8) ? 2) Why does the author know to define $\phi_1(u)$ and $\phi_2(u)$ in such a way? ...
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1answer
31 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 ...
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20 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 ...
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1answer
63 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|}= ...
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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. $\forall r_i\in\{r_1,r_2\cdots r_n\}-\{r_k\}$, ...
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2answers
54 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
34 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 ...
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3answers
76 views

Guidance or advice with $I=\int_0^{2\pi}\frac{1}{4+\cos t}dt$

Let $$ \begin{align} I=\int_0^{2\pi}\frac{1}{4+\cos t}dt \end{align} $$ I would like to evaluate this integral using cauchhy's Integral formula, I understand that I have to convert this into a form ...
4
votes
2answers
49 views

Conformal map between $\mathbb{C}\setminus((-\infty, -1]\cup[1,\infty))$ and $\{z \in \mathbb{C} \mid 0 < \operatorname{Im}(z) < 7\}$

As it says in the title, I am looking for a conformal map from $\mathbb{C}\setminus((-\infty, -1]\cup[1,\infty))$ to $\{z \in \mathbb{C} \mid 0 < \operatorname{Im}(z) < 7\}$, but with the ...
1
vote
1answer
31 views

Entire functions such that $\limsup_{z \rightarrow \infty}\frac{|\log |f(z)||}{|z|} < \infty$

The problem I am working on is to find all entire functions satisfying $|f(z)| > 0$ for $|z|$ large and $$\limsup_{z \rightarrow \infty}\frac{|\log |f(z)||}{|z|} < \infty.$$ My guess is that ...