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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29 views

limit of a complex expression

Suppose that $$R(r)=\left[(a_0-r^2a_2)e^{ir\tau}+(b_0-r^2b_2)\right]^{\frac{1}{r}}\;,$$ where $a_0,a_2,b_0,b_2\in\mathbb{R}$ and $\tau>0$. What is $\displaystyle\lim_{r\rightarrow\infty}R(r)$? I ...
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1answer
18 views

Find the residue of $e^{\frac{1}{z^2-1}}\sin(\pi z)$ at $z=1$

I'm dealing with the following problem (from an old qualifying exam): Let $\gamma$ be a closed curve in the right half-plane that has index $N$ with respect to the point 1. Find $$ ...
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1answer
24 views

Why are compact complex manifolds Liouville?

I know this is true but strangely can't find references. Also, consider the trivial $n$-bundle over any connected compact manifold, does Liouville imply that all holomorphic sections are constant? ...
-4
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2answers
72 views

Nontrivial entire $f(z)$ never equal to $0$

I'm looking for nonconstant entire functions $f(z)$ such that $f(z)\neq 0$ for any $z$. More specifically I'm looking for nontrivial cases. So $\exp(z),\exp(z^2),...$ is not what I am looking for. ...
6
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2answers
950 views

the zeros of $\sin(z)$, where $z$ is a complex number

How do I find the zeros of $\sin(z)$, where $z$ is a complex number? I know that along the real line we have zeros along $k\pi$, where $k$ is an integer. But what about the rest of the plane? The ...
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0answers
35 views

Using estimation theorem to show integrals are zero.

How would I use the Estimation theorem to show the last two integrals on the right equal zero? \begin{equation} \int_{-R}^{R} e^{-\pi x z^{2}}dz= \int_{-R+i \frac{m}{x}}^{R + i \frac{m}{x}} e^{-\pi x ...
6
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1answer
125 views

What proof uses both the Riemann Hypothesis and its negation?

Some time ago I happened to see a proof that was remarkable in that it used both the Riemann Hypothesis and its negation. That is, it considered the two cases: RH is true, and RH is false, obtaining, ...
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3answers
46 views

Question about the Fourier Inversion Formula

We have $$\hat{f}(\xi)=\mathcal{F}f(\xi):= \int_{-\infty}^{\infty}f(x)e^{-2\pi i\xi x}dx,$$ with $f\in L^{1}$, and the Fourier inversion formula says that ...
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2answers
34 views

Why aren't these two properties of complex powers the same?

Let $z\in\mathbb{C}$ s.t. $z=u+iv$. As an example, take the square in this trivial manner: $(u+iv)^2=u^2-v^2+2iuv$. On the other hand taking the square using the properties of complex powers, i.e. ...
3
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1answer
89 views

If $f$ has pole of order $m$, then $\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$

Statement: Let $$f(z):=\sum_{k=-\infty}^\infty a_kz^k$$ have a pole of order $m$ at $z_0$. Then $$\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$$ ...
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1answer
49 views

Showing a function map to itself

Let $ D = \{ z \in \mathbb{C} : |z| < 1\}$. Fix $ w \in D$ and define $f: \bar{D} \to \mathbb{C}$ by $$f(z) = \frac{w-z}{1-\bar{w}z}$$ Show the following: $f$ maps $D$ to $D$ and $\partial D$ to ...
2
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2answers
70 views
+50

Show that there is no analytic bijection from the unit disc to $\mathbb{C}$

Show that there is no analytic bijection from the unit disc to $\mathbb{C}$. I am quite unsure how to proceed here. I know for a fact that there is no analytic function from $\mathbb{C}$ to the open ...
1
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2answers
22 views

Closedness of a complex $1$-form defined by homogeneous functions

How can I prove this 1-form is a closed one within the specific subset $A\subset \mathbb C$ ? $$\omega=\frac{f(x,y)}{xf(x,y)+yg(x,y)}dx+\frac{g(x,y)}{xf(x,y)+yg(x,y)}dy$$ Where $f, g\in C^1$ are ...
2
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1answer
39 views

Roots of polynomial outside a vertical strip of $\mathbb C$

Let $P(z)$ be an arbitrary polynomial with real coefficients. I'd like to guarantee that all roots of $P$ have real parts outside the interval $(0, 1)$. Is there some simple condition on P that will ...
5
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1answer
659 views

Why is every conformal bijection between disks a linear fractional transformation?

Why is every conformal bijection between disks actually a linear fractional transformation? I thought I could justify this claim with the following idea. Suppose $f$ is a conformal bijection from a ...
7
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4answers
172 views

Calculate $\int _0^\infty \frac{\ln x}{(x^2+1)^2}dx$

Calculate $$\int _0^\infty \dfrac{\ln x}{(x^2+1)^2}dx.$$ I am having trouble using Jordan's lemma for this kind of integral. Moreover, can I multiply it by half and evaluate $\frac{1}{2}\int_0^\infty ...
8
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1answer
92 views

For which complex $a, b, c$ does $(a^b)^c=a^{bc}$ hold?

Wolfram Mathematica simplifies $(a^b)^c$ to $a^{bc}$ only for positive real $a, b$ and $c$. See W|A output. I've previously been struggling to understand why does $\dfrac{\log(a^b)}{\log(a)}=b$ and ...
2
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2answers
57 views

Weierstrass's M-test example for uniform convergence and switching Sum and Integral.

How would I go about finding $M_n$ in \begin{equation} \sum_{n=1}^{\infty} \int_{0}^\infty x^{\frac{s}{2}-1}e^{-\pi n^{2}x}dx \end{equation} to show that it is uniformly convergent?
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1answer
34 views

Separate real and imaginary part of $\arccos(z)$

Beginning with $$i \cos \left[ \frac{1}{n} \arccos \left( \frac{i}{\epsilon} \right) + \frac{m \pi}{n} \right]$$ where $m,n \in \mathbf{Z}$, $\epsilon >0$, $\epsilon \in \mathbf{R}$ and $i$ is ...
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0answers
18 views

Smooth interpolation for complex variable function. [on hold]

Is there any smooth interpolation function $T(z)$ that could smoothly connect two complex variable rational polynomial function $H_1(z)$ and $H_2(z)$, for example $$ H(z) = \begin{cases} ...
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1answer
24 views

How can I calculate the singularities and residues of…?

$$\frac{e^z}{z^3(z-1)}+\frac{1}{z^3}$$ I have problems specially for $z=0$ Can anyone show me how to do it?
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39 views

$\eta(1) = \ln(2)$ proof using Abel's Theorem

Hi I was just wondering how does one justify $\eta(1) = \ln(2)$. Looking at the power series for $\ln(1+x)$ we have \begin{equation} \ln(1+x)= \sum_{n=1}^{\infty} \frac{(-1)^{n+1}x^{n}}{n} ...
4
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0answers
71 views

Find the natural boundary of $\sum_{n=1}^\infty \frac{z^n}{1-z^n}$

I'm asked to prove that the natural boundary of $\sum_{n=1}^\infty \frac{z^n}{1-z^n}$ is the unit circle. My try: First, use root test to show that the series converges for $|z|<1$. Then I have ...
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1answer
34 views

Evaluating $\int_{\gamma} \frac{z}{\cosh (z) -1}dz$

Evaluate $\int_{\gamma} \frac{z}{\cosh (z) -1}dz$ where $\gamma$ is the positively oriented boundary of $\{x+iy \in \Bbb{C} : y^2 < (4\pi^2 -1)(1-x^2)\}$. I just learned the residue theorem, ...
3
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3answers
56 views

How do limits work in complex functions?

I don't quite understand one example in my notes it says. My query is this: I don't understand what the significance of $\theta$ is. Why does it matter that $\theta \in (-\pi,\pi]$? I see the ...
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1answer
38 views

Integrating $\sin^2(x)/x^2$, discrepancies with other solutions

I know this is an old problem and it has been answered some times, but I encountered some discrepancies with former solutions and I am not certain what the underlying mathematical reason is. Here ...
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16 views

The bilinear mapping that maps the given three points onto three given points

I need to solve this question: Find the bilinear mapping that maps the given three points onto the three given points in the respective order $$ 1,i,5\quad\text{into}\quad i,-2i,2. $$ The way I can ...
2
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1answer
79 views

Verifying the complex integral: $\int_0^{\infty}\frac{\cos{ax}}{1+x^4}dx$

Verifying the integral: $$\int_0^{\infty}\dfrac{\cos{ax}}{1+x^4}dx$$ I started considering: $$\cos{x}=\dfrac{e^{ix}+e^{-ix}}{2}\implies \cos{ax}=\dfrac{e^{iax}+e^{-iax}}{2}?$$ So: ...
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0answers
44 views

Can the triangle function approximate the Gaussian curve for complex numbers?

I was thinking about approximating the Gaussian curve with a triangular curve. The graphs look like this: their respective functions are: $$ y_1(x) = t(x) = max(0, 1 - |x|)$$ $$ y_2(x) = e^{ - ...
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2answers
37 views

Proof composition of analytic functions is analytic

Title says it all I looked for a proof on this site but couldn't find one. Prove if $f$ is analytic on $D$ and $g$ is analytic on $\Omega$ containing the range of $f$ show $g(f(z)$ is analytic. ...
2
votes
1answer
160 views

Why are conformal mappings necessarily 1 to 1?

Say, by the Riemann Mapping Theorem, there exists a biholomorphic, conformal mapping from the upper half plane to the (open) unit disk (since the UHP is simply connected and is not the entire complex ...
4
votes
1answer
416 views

What are applications of Lagrange's identity?

I recently proved for homework the following identity on $\mathbb{C}$: if $a_1, \ldots , a_n, b_1, \ldots, b_n\in\mathbb{C}$, then $$ \left|\sum_{i=1}^na_ib_i\right|^2 = ...
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2answers
73 views

Find the Laurent series of $f(z)=\frac{1}{z(1-z)}$

I am having difficulties finding Laurent series of the above function, around these two domains: $$0<|z-1|<1$$ and $$|z-1|>1$$ The function $f(z)$ takes the form ...
5
votes
1answer
118 views

Definite integral with logarithm and arctangent inside of arctangent

How to prove $$\int_0^1 \left[ \frac{2}{\pi }\arctan \left(\frac 2 \pi \arctan \frac{1}{x} + \frac{1}{\pi }\ln \frac{1 + x}{1 - x}\right) - \frac{1}{2} \right]\frac{\mathrm{d}x} x = \frac{1}{2} \ln ...
6
votes
4answers
208 views

Help with the contour for this integral using residues

$$ PV \int_0^\infty \frac{dx}{\sqrt{x}(x^2-1)} $$ A keyhole contour can't be used because we have a pole in the real positive axis, isn't it?
2
votes
1answer
39 views

Harmonic complex function

Can anyone help me with this question? Show that a $C^2$ function, $f:U\longrightarrow \mathbb C$, is harmonic iff $\frac{\partial }{\partial z}\frac{\partial f}{\partial \bar z}\equiv0$. Thank you. ...
1
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1answer
48 views

Mapping the upper half plane conformally onto a semi-infinite strip,

Map the upper half y>0 of the z-plane conformally onto the semi-infinite strip u>0, $-\pi<v<\pi$ in the w-plane. I would like some hints for now, please. I'm not sure how to even get started ...
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1answer
19 views

Determine a meromorphic function satisfying certain conditions

Suppose $f$ is meromorphic on the Riemann sphere, and suppose also that $f(0) = 0$, $f(-1) = 2$, $f(3) = 3$, $f$ has a simple pole at $1$ with residue $1$, and $f$ has a triple pole at $2$ with ...
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1answer
63 views

The “argument” of a quaternion

My question is pretty simple. I've been trying to read a pretty introductory text on Clifford algebras, and I encountered how they define the "argument" of a quaternion as an ordered quadruple ...
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1answer
31 views

Solving $\int_{0}^{+ \infty} \frac{x \cos(x)}{x^4 + 4 a^4} dx$ with residues

We also have the condition $a > 0$. My attempt was to, as usual, define $f(z) = \displaystyle\frac{z e^{iz}}{z^4 + 4 a^4}$. Then I tried to integrate $f$ over a curve $\gamma$ which goes from $0$ ...
3
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2answers
70 views

Complex analysis: Prove a meromorphic function to be rational.

I come across a problem about complex analysis: Show that a meromorphic function on the complex plane, which achieves any complex number no more than fixed given times, must be rational. The only ...
6
votes
1answer
90 views

Convergence of $\prod_{n=1}^{\infty}\left(1-\frac{z^2}{n^2}\right)$

I'm looking at some notes from my previous complex variables course and I need help verifying some things about the convergence of $$ \prod_{n=1}^{\infty}\left(1-\frac{z^2}{n^2}\right) $$ on compact ...
4
votes
1answer
58 views

Generating series - Finite groups of order $n$

I am wondering if something of interest can be said about one of the two series $$G_1(x)=\sum_{n=1}^{+\infty}{\mathcal{G}(n)z^n}$$ $$G_2(s)=\sum_{n=1}^{+\infty}{\frac{\mathcal{G}(n)}{n^s}}$$ where ...
3
votes
3answers
41 views

Where do these Mobius transformations map the coordinate half-planes?

They are $$\frac{z-1}{z+1}, \frac{z+1}{z-1},\frac{z-i}{z+i},\frac{z+i}{z-i}.$$ All four look virtually identical, so I would like to know how to best distinguish between them. For example, the ...
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1answer
21 views

Proper definition of concyclic?

Let $z_1,z_2...,z_n$ be points in the complex plane, then if there exists $Z$ such that $$\vert Z-z_k\vert=a\in\{\text{Real Numbers}\}$$for all $k\in \{1, 2, 3...,n\}$, then $z_1,z_2...,z_n$ are ...
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3answers
93 views

Solve integrals using residue theorem? [on hold]

$$\int_{0}^{\pi}\frac{d\theta }{2+\cos\theta}$$ $$\int_{0}^{\infty}\frac{x }{(1+x)^6} dx$$ My problem is that I don't know how to start solving these integrals, or how to convert them into usual ...
3
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1answer
38 views

How to show that $\int_{S}\frac{1}{z}\frac{1}{\cos(2\pi i a)-\cos(2\pi z)}dz\rightarrow 0$ as the sides of the square $S$ go to $\infty$.

I have a question where I am asked to show that the following sum is $$\sum_{k=-\infty}^\infty\frac{1}{a^2+k^2}=\frac{\pi}{a}\frac{e^{2\pi a}-e^{-2\pi a}}{e^{2\pi a}+e^{-2\pi a}-2}$$ by integrating ...
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0answers
46 views

Using Rouché prove that $a_0\geq a_1 \geq … \geq a_n > 0$ the polynomial $P(z)=a_n z^n + … + a_1 z + a_0$ has no roots $|z|<1$

As asked in the title I am trying to prove this using Rouché's theorem. But I am having a hard time trying to find a second polynomial $f(z)$ such that $$|P(z)-f(z)| < |f(z)|$$ on the unit ...
2
votes
1answer
647 views

inverse Laplace transform of $e^\sqrt{as}$

I am trying to find the inverse Laplace transform of $e^\sqrt{as}$ for $a>0$. So we need to solve $\oint_B dz \: e^\sqrt{az} e^{z t}$ (Bromwich contour), but not sure how to start. How do we even ...
0
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0answers
19 views

Total derivative of a complex function using Wirtinger derivatives

The mathworld.wolfram page on Cauchy-Riemann Equations states that given a complex function $f(z) = f(x,y) = u(x,y) + iy(x,y)$ has the derivative: $$\frac {df} {dz} = \frac {\partial f} {\partial x} ...