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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127 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
40 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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2answers
180 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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1answer
34 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
152 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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2answers
190 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?
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1answer
150 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
105 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
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2answers
79 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
127 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
118 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
69 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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1answer
63 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 ...
4
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1answer
74 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
60 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
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2answers
68 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
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2answers
70 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
92 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 ...
5
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2answers
114 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 ...
2
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1answer
62 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 ...
2
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1answer
39 views

Is this problem worded correctly?

I'm working on a problem from an old complex analysis prelim, but while doing so, I'm not sure about whether it is worded correctly. The problem: Let $f(z)$ be a holomorphic function in the unit ...
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1answer
76 views

hexic polynomial question

I am faced with a polynomial of the form $$ ax^6+bx^3+cx+d=0, $$ where the coefficients are complex. I want to be able to say something about the roots of this polynomial (including finding them!). Is ...
2
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2answers
67 views

Is it possible to use complex logarithm to integrate $1/(z+i)$ along a path?

Evaluate the following on the path $\gamma_1$ with endpoints $[-1,1+i]$ $$ \begin{align} I_1=\frac{i}{2}\int_{\gamma_1} \frac{1}{z+i}dz -\frac{i}{2}\int_{\gamma_1}\frac{1}{z-i}dz \end{align} $$ Am I ...
2
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2answers
50 views

power expansion of $\frac{z}{z^{4} +9}$

The next question in John D'Angelo's text that I'm stuck on is Exercise 4.7: Find the power series expansion for $\frac{z}{z^{4} +9}$ about 0. Where does it converge ? I understand this section in ...
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0answers
90 views

Problem with calculating winding number in sum of curves

Let $$ \begin{align} \gamma= \gamma_1 +\gamma_2+\gamma_3,\\ \gamma_1(t)=e^{it}, t\in[0,2\pi] \\ \gamma_2(t)=-1+2e^{-2it}, t\in [0,2\pi]\\ \gamma_3(t)=1-i+e^{it},t\in [\frac{\pi}{2},\frac{9\pi}{2}] \...
0
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1answer
51 views

winding number of $\gamma$ and point exterior to $\gamma$

$$ \begin{align} n(\gamma,z_0)=\frac{1}{2\pi i}\int_\gamma\frac{1}{z-z_0}dz . \end{align} $$ Is it safe to say that $n(\gamma,z)=0,\forall z \in \mathbb{C}\backslash \gamma^*$
2
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1answer
53 views

A Submanifold $M$ of $\Bbb C^N$

I have a Proposition in my book, and I write here: For every $p \in M$, with $M$ be a hypersurface in $\Bbb C^N$ the following hold. \begin{align*} \mathcal V_p &= \left \{ X \in \Bbb C ...
3
votes
1answer
51 views

Contour Integrals evaluation verification

$$ \begin{align} \gamma(t)=2cost + isint, t\in[0,2\pi] \end{align} $$ Could someone verify my thinking and my results for the following Integrals and if I have properly justified my thoughts: $$ \...
3
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1answer
86 views

Finding all the possible values of an Integral in the Complex Plane

I am studying Complex Analysis by Lars V Ahlfors. I am unable to solve one of his exercises. It is: Find all possible values of $$\int \frac{dz}{\sqrt{1-z^2}}$$ over a closed curve. I do not have ...
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1answer
43 views

Stereographic projections - equation of a plane question

The proof I'm trying to understand. I don't get why $k$ is unique. When trying to find the equation of a plane, suppose we're given a normal vector $n=(x_o,y_o,z_o)$ and a point on the plane $p^*=(...
10
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1answer
158 views

Show that the set of one-to-one holomorphic maps $\Bbb{C}\setminus\{a,b,c\} \to \Bbb{C}\setminus\{a,b,c\}$ forms a finite group.

Let $\Omega = \mathbb{C}\setminus\{a, b, c\}$ be the complement of three distinct points in the complex plane. Show that the set of one-to-one holomorphic maps $f : \Omega \to \Omega$ forms a finite ...
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1answer
75 views

Mobius transformations are bijections proof

I don't understand the last line of this proof. To show a function is bijective we need to show it is one-to-one and onto. The proof shows that $f$ is one-to-one only. For some reason $f^{-1}$ ...
4
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3answers
74 views

Can $\frac1{z^2}$ be integrated on $|z+i|=\frac32$ using Cauchy's theorem?

$$ \begin{align} \int_{|z+i|=\frac{3}{2}}\frac{1}{z^2}dz=0 \end{align} $$ Is it safe to say the Integral is $0$ due to cauchy's Theorem? Does this apply for any $z_0$ that lies inside the circle ...
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1answer
56 views

Is this Integral Calculation correct? $\int_{|z|=1}(z^2+2z)^{-1}dz=\pi i$

$$ \begin{align} \int_{|z|=1}(z^2+2z)^{-1}dz=\int_{|z|=1}\frac{1}{z(z+2)}dz= \\ =\int_{|z|=1}\frac{1}{2z}dz+\int_{|z|=1}\frac{1}{2(z+2)}dz= \\ =\int_{|z|=1}\frac{1}{2z}dz+ 0= \\ =\frac{1}{2}\int_{|z|=...
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2answers
138 views

Show that $\sum \frac{z^n}{n}$ diverges if $z = 1$ but otherwise converges if $|z|=1$.

Hi: I'm reading John D'Angelo's textbook "An Introduction To Complex Analysis and Geometry" and trying ( emphasis on trying ) to work on the exercises in Chapter 4. I'm already stuck on only the ...
4
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3answers
753 views

The limit of complex sequence

$$\lim\limits_{n \rightarrow \infty} \left(\frac{i}{1+i}\right)^n$$ I think the limit is $0$; is it true that $\forall a,b\in \Bbb C$, if $|a|<|b|$ then $\lim\limits_{n\rightarrow \infty}\left(\...
2
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1answer
54 views

Change in the value of $f(z)= \cos(\pi·z^{2/3})$ as $z$ embraces around origin twice

This is a problem I confronted and I am really struggling hard with it. Can anyone please help me out? Consider the multi-valued function $f(z) = \cos (\pi·z^{2/3})$. Beginning with $f(1) = -1$ the ...
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1answer
31 views

Problem in showing that contours $\gamma_2$ is equivalent to $ \gamma $

Let $\gamma_2(t)= e^{-it^2}, t\in[0,\sqrt{2\pi}]$ and $\gamma(t)=e^{2\pi it}, t\in[0,1]$ Show that $\gamma_2 \sim \gamma $. I think that for the latter to be true $\gamma_2$ should be $\gamma_2(t)=...
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0answers
37 views

Cauchy-Riemann Equations - why $f'(z_o) = \frac{\partial f}{\partial x}(z_o)$ implies that f is differentiable at $z_o$

I'm trying to understand part b of this proof. The only line I don't understand is the sentence starting with "To prove the statement in (b)..." If someone could clarify why that line is true I ...
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0answers
593 views

Complex exponential integral: Mathematica and MATLAB give unexpected results

I currently compare analytical vs. numerical evaluation of the complex exponential integral and find mismatches: The imaginary part differs by $\pm \pi$ and the real part has a large error when ...
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0answers
316 views

Is there a book only about epsilon delta proofs?

I want to know if there is such book, with beautiful epsilon delta proofs of all kind.
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2answers
574 views

Why are people more interested in the Riemann hypothesis than Goldbach's conjecture? [closed]

One of my friends, a math professor, told me almost every one of his colleagues (in the math department) had attempted to prove the Riemann hypothesis at some point in their life (maybe secretly). ...
5
votes
2answers
221 views

Showing that $z^3 e^z = 1$ has infinitely many solutions

On an old complex analysis prelim, I encountered the following problem. Show that the equation $z^3 e^z =1$ has infinitely many solutions. How many are real? Well many sources in complex analysis ...
4
votes
2answers
195 views

Show there exists a value such that each partial sum equals its limit in modulus

For each $n \in \mathbb{N}_0$, and for all $z \in \mathbb{C}$, define $$p_n(z) := \sum^{n}_{k=0} {z^k \over {k!}}.$$ Show that for all $r > 0$ and for all $n \in \mathbb{N}_0$, there exists $z ...
0
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1answer
44 views

Does $\gamma(t)=e^{it^2} , t \in [0,\sqrt{2\pi}]$ represent the unit circle?

$$ \begin{align} \gamma(t)=e^{it^2} , t \in [0,\sqrt{2\pi}] \end{align} $$ Is my thinking correct that $\gamma$ represents the unit circle correct?
0
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1answer
60 views

Tricky(?) Singularity classification question

Find and classify isolated singularities of: $$\frac{1}{\sin z-\sin2z}$$ So I have found the singularities to be: $z=k\pi,(6k \pm1)\pi/3$ Now i could factor these out of $f(z)=\sin z-\sin 2z$ to ...
3
votes
1answer
53 views

Calculation of a Contour Integral

$$ \begin{align} \int_{|z|=1}(4-z^2)^{-1/2}dz \end{align} $$ The exercise hints at the usage of $e^{2Logz}$ , although any solution or methodology for such integrals would be welcome.
1
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2answers
70 views

Complex function defined by contour integral along a smoothly varying path

Let $D$ be a domain in the complex plane. Consider the function $F: D\to \mathbb{C}$, defined by $$ F\left( z \right) = \int_{\mathscr{C}\left( z \right)} {f\left( {z,t} \right)dt} . $$ Suppose that ...
1
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1answer
29 views

Complex Analysis Limits

Hi there. I am struggling to understand how the this equation is obtained in the working: How do you approach along a line when working out a limit? Thanks
1
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1answer
45 views

Find sup$\{|f′(3)| : f$ maps $Ω$ analytically into the unit disk $\}.$

Let $Ω=\{z=x+iy∈C : |y|<x\}.$ Find sup$\{|f′(3)| : f$ maps $Ω$ analytically into the unit disk $\}.$ Okay. So I can find a conformal map from $Ω\rightarrow \mathbb{D}$. I used the map $f(z) = ...