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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Proof of Levi's problem in Hormander's book

In H$\ddot{\text{o}}$rmander's "An introduction to complex analysis in several variables," he gives a proof of Levi's problem for complex domain by $L^2$ methods. (Theorem 4.2.9) The proof is using ...
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23 views
+50

How to visualize bilinear transform of the form $S(z) = \frac {T}{2} \frac {z+1}{z-1}$

Note that this is the bilinear transform from a z-domain as appears in Z-transform to s-domain in Laplace transform Recall that bilinear transform has form $M(z) = \frac{az+b}{cz+d}$ with and has to ...
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1answer
48 views

Show $\left|{\frac{z_1-z_2}{1-z_1 \overline{z_2}}}\right| < 1$ if $|z_1| <1$ and $|z_2| < 1$

Show $$\left|{\frac{z_1-z_2}{1-z_1 \overline{z_2}}}\right| < 1$$ if $|z_1| <1$ and $|z_2| < 1$ Consider: $$\left|{\frac{z_1-z_2}{1-z_1 \overline{z_2}}}\right|^2$$ ...
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13 views

Roots of of the derivative of a polynomial are in the convex hull of the polynomial

$(def)$: The $\mathbf{Convex}$ $\mathbf{Hull}$ of a set $\{ z_1,\ldots,z_k \} \subset \mathbb{C}$ is the set $$ CH[z_1,\ldots,z_k] = \left\{ z \in \mathbb{C} : z = \sum_{j=1}^k \lambda_j z_j, \; \; 0 ...
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9 views

Let $B = \{(z_1, z_2) \in \mathbb{C}^2 : |z_1| \leq |z_2| \}$. Show that B is balanced, but that its interior is not.

I have the following definitions. The interior $E^o$ of $E$ is the union of all open sets that are subsets of $E$. A set $B \subset X$ is said to be balanced if $\alpha B \subset B$ for every ...
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2answers
85 views
+200

Integrating around simple pole and semicircle

Let $f$ be a holomorphic function on $\mathbb{C}$ with simple pole at $z_0$. Then if $\Gamma$ is a circle around $z_0$ oriented counter-clockwise with radius $r$ and $r\rightarrow 0$, then ...
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1answer
11 views

What is the range of this complex function: $f(z) = 2x^2+(1-x^2)(1+i)$ defined on $|z|\leq1$?

These range problems, I just don't get it. I tried to get this into a form where I could use the fact that $0\leq\theta\leq2\pi$, but I'm just not sure how to get it to that point. Any ${hints}$ ...
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3answers
79 views

Evaluation of integral $\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$

I'm trying to evaluate the following integral: $$\mathcal{J}=\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$$ Well there are $3$ poles , one lying on the real line the other on ...
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17 views

Does $\frac{1}{z^n}$ have a primitive on $\mathbb{D} - 0$ for $n < 1$?

I'm studying complex analysis from Stein and Shakarachi. There is a question that asks you to evaluate: $\int_{\gamma} z^{n}\ \mathrm{d}z$ for all integer n, where $\gamma$ is any circle centered ...
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3answers
18 views

evaluating a sequence of complex numbers

Let $\{z_n\}$ be sequence of complex numbers such that $$ |z_n - z_m| < \frac{1}{1+ |n-m|} $$ for all $n,m$ Given this information, can we compute $\lim_{n \to \infty} z_n $? Attempt: For sure ...
6
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2answers
56 views

Cauchy-Riemann Equations Written as Complex Conjugate

Apparently, it can be shown that the Cauchy-Riemann equations can be written simply as, $df/dz^*=0$. I do not understand how it does not immediately follow from this that $df/dz=0$. When we proved ...
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1answer
47 views

The Laplace transform of the Heaviside function

I am studying complex analysis but, because I'm an engineer, I have a lot of doubts. I'm going to present my doubts and it would be nice if someone helps me to see things clearly. Let's start with ...
2
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1answer
43 views

Prove for some $z_0 \in C$ the function $f(z)=|z-z_0|$ is continuous on all of $\mathbb{C}$

Let $z_0\in\mathbb{C}$ and $f(z)=|z-z_0|$. Show that $f$ is continuous on $\mathbb{C}$. I expect to see a proof using the triangle inequality. Note a function $f$ is continuous on $\mathbb{C}$ if ...
2
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1answer
22 views

Sketch $ z\in \mathbb{C}:0 < arg(z-(1+i)) < \frac\pi3 $

Sketch the following $$ z\in \mathbb{C}:0 < arg(z-(1+i)) < \frac\pi3 $$ I have considered this geometrically and ended up thinking that the complex numbers $z$ must satisfy $$0 < ...
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9 views

Determine the set of point $S$ in $C_∞$ such that the corresponding set $S'$ on the sphere is a circle that is equidistant from $z_1'$ and $z_2'$

Let $z_1$ and $z_2$ be 2 elements of $C _∞$ . Determine the set of point $S$ in $C_∞$ such that the corresponding set $S'$ on the sphere is a circle that is equidistant from $z_1'$ and $z_2'$ (the ...
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1answer
80 views

How to define Square Root

I'm trying to understand how to define the square root of a complex function "globally". Let's say we have some function from some set $X$ onto $\mathbb{C} - \{0\}$: $$ f:X\to\mathbb{C}-\{0\} $$ and ...
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1answer
34 views

Determine the set of points that satisfy the condition $Im \frac{z-z_1}{z-z_2 }=0$ where $z_1$ and $z_2$ are fixed complex number.

Determine the set of points that satisfy the condition $Im \frac{z-z_1}{z-z_2 }=0$ where $z_1$ and $z_2$ are fixed complex number. Since $Im\frac{z-z_1}{z-z_2 }=0$, then there is no imagining part in ...
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2answers
68 views

Principal value of Fourier Integral

I have tried to find the principal value of $$\int_{-\infty}^\infty {\sin(2x)\over x^3}\,dx.$$ As $ {\sin(2x)\over x^3}$ is an even function, its integral may not be zero in the given limits. I ...
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3answers
30 views

Finding $f(i)$ for an Entire Function $f$ that Maps a Line to a Subset of Itself and Sends $1$ to $0$.

Consider the line $L=\lbrace x+iy: x=y\rbrace\subset \mathbb{C}$, suppose that $f:\mathbb{C}\to\mathbb{C}$ is entire and satisfies $f(L)\subset L$. Given $f(1)=0$, find $f(i).$ (Ans: $f(i)=0)$ I am ...
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1answer
79 views

Integration in complex analysis using Residue Theorem

Prove that for $a>0$, $$ \int_{-\infty}^{\infty} \frac{1}{x^4+a^4}dx=\frac{\pi}{a^3\sqrt{2}} $$ I think I'm supposed to use Cauchy's Residue Theorem somehow, but I don't know what closed path to ...
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1answer
132 views

Residue Theorem Question

I am having trouble with this integral. Any help would be greatly appreciated! $$\int_0^{\infty} \frac{\sqrt{x}}{x^2+2x+5}dx $$ using the ``keyhole'' contour. I know that we want to obtain our ...
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1answer
108 views

inequality and complex integration

$\gamma_r$ is the part of $|z-1|=r$ inside $|z|=1$, $\gamma_1$ is the part of |z|=1 outside $|z-1|<r$. prove that 1) If $|z|<1, |z-1|=r(0<r<1),$ then $$|\frac{\log (1-z)}{z}|\leq ...
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1answer
72 views

Find all holomorphic functions $f:\mathbb{C}\setminus\{0\}\rightarrow \mathbb{C}$

Find all holomorphic functions $f:\mathbb{C}\setminus\{0\}\rightarrow \mathbb{C}$ such that $$\limsup_{z\rightarrow 0}\left|\frac{f(z)}{\sin z}\right|<\infty$$ and $$\limsup_{z\rightarrow ...
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1answer
17 views

Show that a complex map is onto

I consider $\mathbb{C}$ as a real vector space. For $(a,b) \in \mathbb{C}^{2}$, consider the map : $F_{a,b} \, ; \, \mathbb{C} \, \rightarrow \, \mathbb{C}^{\ast}$ such that : $$ \forall z \in ...
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3answers
30 views

Find taylor of $\psi (z)$ where $(e^z-1)^2=z^2 \psi (z)$ - first 3 terms

I was asked to find the first three terms in the taylor series of $\psi (z)$ around $z=0$ where $(e^z-1)^2=z^2 \psi(z)$ and I'm having a few difficulties. My original idea was to say $\psi ...
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0answers
20 views

Continuous complex function from Rudin's Real and complex analysis

Lemma 10.29 from Rudin's Real and Complex Analysis, p. 314 of the third edition states that "if $f \in H(\Omega)$, then $g:\Omega \times \Omega \to \mathbb{C}$ defined by \begin{equation} g(z, w) = ...
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23 views

Understanding Green's Theorem

When looking at Goursat's theorem in complex analysis, I came across the Wiki proof which involves beautiful application of Green's theorem. I saw Greens theorem simply as "connection between line ...
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3answers
92 views

Video Lessons in Complex Analysis

Does anybody have some link for good video lessons of a complete course in Complex Analysis? Grateful.
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1answer
41 views

Are complex numbers a trivial lie group of itself? [on hold]

Let $z$ be a complex number, then let's define a map $e^{T(*)}$. Let $w = e^{T(z)}$, where $T$ is some real number. Then is $z$ a lie group of $w$?
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64 views

Determine the set of points $z$ that satisfy the condition $|2z|>|1+z^2|$

Determine the set of points $z$ that satisfy the condition $|2z|>|1+z^2|$ I tried with polar coordinate. So let $z=r(\cos \theta +i \sin \theta)$. the LHS $=|2z|=2|z|=2r$ The RHS, I do some ...
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3answers
905 views

A sine integral $\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$

The following question comes from Some integral with sine post $$\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$$ but now I'd be curious to know how to deal with it by methods of ...
2
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2answers
47 views

Inverting complex cosine

I have been working out problem 3a in chapter 1 section 3 in Basic Complex Analysis by Marsden. He asks to solve $$ \cos z=\frac{3}{4}+\frac{i}{4} $$ After putting cosine in its exponential form and ...
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1answer
23 views

Redundancy in the Laplace transform and Mellin's inverse formula

As I understand it, Mellin's inverse formula relates a sufficiently 'nice' function $f$ and its Laplace transform $F$ as follows: $$f(t)=\frac1{2\pi i}\lim_{T\to\infty}\int_{-T}^{T}e^{i\omega ...
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1answer
40 views

How to indentify the power series I need to use?

Let $$ f(z) = \frac{1}{(z - 4)(z + 8i)} $$ a) Find the domains where f(z) is valid b) Find its power series at such domains Considering three singularities, I believe the domains are: $$ D_{1} = ...
2
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3answers
42 views

Prove the intersection of a compact set and a set with no accumulation points is finite

Let $S\subset\mathbb{C}$. We say that $z_0$ is an accumulation point of $S$ if for every $r>0$, the intersection $D(z_0,r)\cap S$ is an infinite set. Let $U\subset\mathbb{C}$ be an open set such ...
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2answers
366 views

integrating $\oint_C \frac{3z^3 + 2}{(z-1)(z^2 + 9)}dz$ on $|z|=4$

I am doing $\oint_C \dfrac{3z^3 + 2}{(z-1)(z^2 + 9)}dz$ on $|z|=4$ and I find that there are poles within the contour at $z = 1$ and at $z = 3i$, both simple poles. I find that the integral $I = 2\pi ...
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2answers
239 views

finding residue with $\oint_C \frac{3z^3 + 2}{(z-1)(z^2 + 9)} dz$

I am doing the integral $\oint_C \dfrac{3z^3 + 2}{(z-1)(z^2 + 9)} dz$, and I am trying to find the residue at the pole $3i$;I am unsure how to do this. Could I factor $z^2 + 9$ further?
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35 views

Show that $e^z-az^n=0$ has exactly $n$ zeros in $B(0,1)$

Let $a\in\mathbb{C}, |a|>e, n\in\mathbb{N_1}$ I have to show that $e^z-az^n=0$ has exactly $n$ zeros in $B(0,1)$ First, $f(z)=e^z$ and $g(z)=-az^n$ are entire. On $\partial B(0,1)$ we have ...
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1answer
39 views

Integrating $\frac{1}{z\sin(z)}$ on a path

What is the path integral of $\dfrac{1}{z\sin(z)}$ around $D_1(0)$? I've been trying to exploit residuals or use parametrization, but I can't come out of it. What I actually need to do is to find the ...
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3answers
95 views

Show that $\frac{x}{e^x-1}$ is non-singular near zero

Show that $\dfrac{x}{e^x-1}$ is non-singular near zero. Does this show boundedness near zero?
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1answer
22 views

Evaluation of an integral using nonrigorous methods

I was trying to solve the following integral $$ G(\alpha,m,n)=\int_0^{\infty}\cos(2nx)e^{-\alpha x}x^{m-1}dx;n\in N,\alpha>0,m\ge1. $$ By doing a change of variable I brought it to the integral $$ ...
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1answer
20 views

Mapping of region into a line

Define a function $f:\mathbb{C}\rightarrow\mathbb{C}$, and let $z=x+iy$, then $$f(z)=u(x,y)+iv(x,y)$$ Given that $f$ maps a region into a line, what is the relation between $u(x,y)$ and $v(x,y)$? I'm ...
4
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1answer
51 views

Find all holomorphic functions $f$ satisfying $f(1-f(z))=f(z)$?

Find all holomorphic functions $f$ (on $\mathbb{C}$) satisfying $f(1-f(z))=f(z)$? First, every constant function $f(z)=w$ is holomorphic and satsfies our condition. Now assume $f$ is not constant. ...
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1answer
57 views

Analyze convergence of the series $\sum_{n=1}^\infty \frac{1}{r^n - \frac{1}{r^n}}$ for $r \in (0, \infty) \setminus \{1\}$

I would like to have a little help determining for which $r \in (0, \infty) \setminus \{1\}$ the following series converges: $$\displaystyle \sum_{n=1}^\infty \frac{1}{r^n - \frac{1}{r^n}}.$$ I'm ...
0
votes
1answer
91 views

Sketching complex image

Let $f = e^{2-z}$. Find and sketch the image $f(S)$ of the strip $$S=\{1 < \mathrm{Re}(z) \leq 2, -\pi/4 < \mathrm{Im}(z) \leq 0\}.$$ I got radius of $f$ is bound by $e^3 \leq r \leq e^4, 0 ...
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1answer
85 views

Finding the largest set where a complex function is analytic

I am considering the function $$ f(z) = \frac{e^z}{\sin z - \cos z}. $$ So I solved for $\sin z - \cos z = 0$ and got $\pi/4$. But why is it $\pi/4 + k\pi$ and not $\pi/4 + k2\pi$ for the part of the ...
3
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2answers
29 views

Complex Analysis holomorphic function question

I have a Complex Analysis assessment question about holomorphic functions: Let f be a function on a plane and satisfies $f'(z) = f(z)$ and $f(0) = 1$ i) Give an example of a function with this ...
2
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0answers
278 views

why $\pi \cot \pi z = \sum _{n=-\infty}^{\infty} \frac{1}{z+n}$

In prove of the above claim I need prove: $\cot (\pi\cdot z)$ is bounded for $\{y \geq 1 , 0.5 \geq x \geq -0.5\}$ In stein book a function has defined: $$\Delta ( z ) = \pi \cot \pi z - \sum ...
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0answers
15 views

Coupling complex functions

After several calculations I end up with two complex functions: $$g(z)=zA(z)+\overline{z}A(\overline{z})+z^{-1}B(z)+\overline{z^{-1}}B(\bar{z})$$ and ...
2
votes
2answers
129 views

calculate the integral

Compute $$I=\int_C\frac{e^{zt}}{1+z^2}dz$$ where $t>0$, a real number, and $C$ is the line $\{z\mid \mathrm{Re}(z)=1\}$ with direction of increasing imaginary part. I tried to integral along the ...