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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23 views

Residue as coefficient of Laurent Series

Suppose we are given: Find the residue at $z=n$ of $$(\psi(-z) + \gamma)^2$$ Where $\psi(-z)$ means the digamma function. $$(\psi(-z) +\gamma)^2 = \left( 1 + \frac{1}{2} + \frac{1}{z} + \frac{1}{3} ...
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1answer
64 views

How to perform this contour integration with $\log$ in the denominator?

Let $k > 0$ and $ a>1$ be constants. As far as I can tell, the integral $$ J = \int_{-\infty}^\infty dx\frac{e^{i k x}}{1+x^2}\frac{1}{\log(a - ix)} $$ converges, since the argument of the ...
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2answers
44 views

Residue of $\frac{(\psi(-z) + \gamma)^2}{(z+1)(z+2)^3}$ at $z=n$

Find the residue of: $$f(z) = \frac{(\psi(-z) + \gamma)^2}{(z+1)(z+2)^3} \space \text{at} \space z=n \space \text{for a general} \space n$$ How can I start? Using advice from Jack D Aurizio, I get: ...
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1answer
38 views

Fubini-Study measure on a product

I read that if $\Omega_2$ is the Fubini-Study form on $\mathbb{P}_1\times\mathbb{P}_1$, and $\Omega$ the Fubini-Study form on $\mathbb{P}_1$, then for all $(x,y)\in\mathbb{P}_1\times\mathbb{P}_1$, one ...
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0answers
17 views

$\lim_{t \to \infty} \left(\ln(f(t))e^{-ic t}-\ln(f(-t))e^{ic t}\right)$

What is the limit of \begin{align*} \lim_{t \to \infty} \left(\ln(f(t))e^{-ic t}-\ln(f(-t))e^{ic t}\right) \end{align*} where $c>0$ is constant and $t \in \mathbb{R}$ and $\lim_{t \to ...
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1answer
25 views

Finding the Residue at $z=n$

Find the residue of: $$f(z) = \frac{(\psi(-z) + \gamma)}{(z+1)(z+2)^3} \space \text{at} \space z=n$$ Where $n$ is every positive integer because those $n$ are the poles of $f(z)$ This is a simple ...
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19 views

Coupled Complex Dynamics

Consider two dynamical systems $$Z_{n+1}=f(W_{n},Z_{n-1}) $$ $$ W_{n+1}=g(Z_{n},W_{n-1})$$ where $w_0, w_{-1}, z_{0}$ and $z_{-1}$ are given complex numbers. If $f$ and $g$ are two functions on ...
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1answer
112 views

Integration of $\int_{-\infty}^{\infty}\frac{\cos x}{a^2+x^2}dx$ [duplicate]

I'm trying to find the integral $$\int_{-\infty}^{\infty}\frac{\cos x}{a^2+x^2}dx$$ Wolfram alpha says this is $$\frac{\pi e^{-a}}{a}$$ But how do you get this result? I tried using partial ...
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2answers
53 views

Singular solution to ODE and singularities

I'm a bit confused about those two, i'll try to explain. I get an ODE like $$ \frac {y \cdot dy }{\sqrt{y^2+1} } + \frac {x \cdot dx}{ \sqrt {x^2 +1}} = 0 $$ What I'm not sure about : 1) at y=0 the ...
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1answer
56 views

Some inequalities for an entire function $f$

Let, $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ be an entire function and let $r$ be a positive real number. Then, which is(/are) correct? (a) $\sum_{n=0}^{\infty}|a_{n}|^{2}r^{2n}\le sup_{|z|=r} ...
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1answer
30 views

Assume $\arg(z_1)-\arg(z_2)=2n\pi$. Show that this implies $|z_1+z_2|=|z_1|+|z_2|$

I am really lost here. Can anyone please give me a hint or two?
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3answers
35 views

Assume $|z_1+z_2|=|z_1|+|z_2|$. Show that this implies $\arg(z_1)-\arg(z_2)=2n\pi$

The hint I am given is that the relationship of $|z1||z2|$ implies $\arg(z1)-\arg(z2)=2n\pi$ is to be used somewhere. I think the only way this can be done is to square it but after that I'm not ...
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1answer
26 views

Assume $\theta_1-\theta_2=2n\pi$. Prove that $\text{Re}(z_1 \bar z_2)=|z_1||z_2|$

I proved the other way around already but I can't seem to prove this way. Both $\theta_1$ and $\theta_2$ are the arguments of $z_1$ and $z_2$ respectively. Can anyone help me out here? Edit: The ...
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1answer
36 views

Epsilon delta proof in Complex variables

I am having big difficulty with this problem ; although the book I am working with gives several similar examples :Problem :Using epsilon delta prove that $$ \lim_{z \rightarrow 1+i} ...
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0answers
9 views

Sequence in Complex

Let the next sequence $f (z) = \dfrac{(z + | z |)}{2} $, $f_2(z)=f(f(z))$,...,$f_n(z)=f(f_{n-1}(z))$. Find $\lim_{n \to \infty} f_{n}(i)$ I think this sequence converges to zero
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1answer
48 views

Finding roots of a complex polynomial in $\{\operatorname{Re}(z) < 0 \} $

How many roots does $P(z)=2z^4+z^3-5z^2+z+2$ have in $\{ \operatorname{Re}(z)<0 \}$? I was told that I should compute $P(it)$ for $t \in \Bbb R$ which is: $P(it)=2t^4 +5t^2 +2 +it(1-t^2) $. ...
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1answer
30 views

Existence of a function

Let $D=\{z\in\mathbb{C}:|z|<1\}$ How can one show that there exist function $f:[0,1]\times D \rightarrow \mathbb{C}$ satisfying the following properties: (i) $f(\cdot,z)$ is continous on $[0,1]$ ...
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0answers
27 views

Possible values of contour integral

Let $f$ be an entire function, $\gamma$ be a closed curve such that $f(z) \neq 0, \forall z \in [\gamma].$ What are the possible values of $\int_{\gamma}\frac{f'}{f}dz$? I'm kinda confused. ...
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1answer
48 views

Prove that a certain entire function is constant.

How can one show that if $f$ is entire and $f(\mathbb{C})\subset \mathbb{C} \setminus p$ where $p$ is any ray in $\mathbb{C}$ then $f$ is constant. It's obvious result from the Little Picard ...
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1answer
42 views

Cauchy Residue Theorem and Cauchy integral formula

Is it true that you can use the Cauchy Residue Theorem and the Cauchy integral formula interchangeably? I believe that the functions that satisfy the conditions of one, will indeed satisfy the ...
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1answer
40 views

Function holomorphic except for real line and continuous everywhere is entire

1) I've already shown that if $f:\mathbb{C}\rightarrow\mathbb{C}$ is holomorphic everywhere except for a single point and if it continuous on whole $\mathbb{C}$ then it is entire. It was quite easy. ...
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1answer
41 views

Finding the coefficients of $h(z)$ laurent series

Consider: $$h(z) = \frac{\psi(-z)}{(z+1)(z+2)^3}$$ Find the coefficients $a_n$ of the Laurent Series of $h(z)$ centered at $z=-2$ I got this from the approach here: Infinite sum complex analysis ...
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1answer
25 views

How to Derive this Digamma identity?

I dont see the transition from $(-z)^k$ in the fist sum to the transition to $(z+2)^k$ in the second sum? How is that derived?
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1answer
17 views

Residue Formula in complex analysis

I understand the residue formula but I just can't understand the cancelling down of $$ \operatorname{res}_{z=z_1} (f)= \lim \limits_{z \to z_1}(z-z_1) \frac {z^2}{z^4+1} = \frac {z_1^2}{4z_1^3}.$$ ...
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0answers
31 views

Is the integral $\int_1^{\infty} {A(t)}{t^{-s-1}}dt$ a holomorphic function of $s$?

My question is whether, for $Re(s)=\sigma > 3/4$, $$s\int_1^{\infty} \dfrac{A(t)}{t^{s+1}}dt$$ is holomorphic, where $A(x)=O(x^{3/4})$. Under absolute value, it is easy to see that the integral ...
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1answer
46 views

Find the residue at $z=-2$ for $g(z) = \frac{\psi(-z)}{(z+1)(z+2)^3}$

Find the residue at $z=-2$ for $$g(z) = \frac{\psi(-z)}{(z+1)(z+2)^3}$$ I know that: $$\psi(z+1) = -\gamma - \sum_{k=1}^{\infty} (-1)^k\zeta(k+1)z^k$$ Let $z \to -1 - z$ to get: $$\psi(-z) = ...
2
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0answers
22 views

Residue of $f(z)$ using Laurent Series at $z=-2$ [duplicate]

Calculate the residue of: $$f(z) = \frac{\psi(-z)}{(z+1)(z+2)^3} \space \text{at} \space z=-2$$ Where $\psi(z)$ is the digamma function, and $\zeta(z)$ is the Riemann-zeta function (below). The ...
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1answer
10 views

Classifying singularities and finding their residues

How would one find the residues of: $f(z)=z/cos(z)$ I believe that the singularities are $z=\pi/2 + 2k\pi$ where k is an integer, but I'm not sure how to go about classifying them and then finding ...
2
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0answers
34 views

Prove that $\mathrm{Re}(z_1\overline{z_2})=|z_1||z_2|\;\mathrm{iff}\;\arg(z_1)-\arg(z_2)=2n\pi$, where $n\in\mathbb{Z}$.

I've been sitting here for an hour thinking about how to approach this problem. I can't seem to connect $z_1$ $z_2$ with its arguments, I mean can't $z_1$ and $z_2$ be anything? Can someone please ...
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1answer
39 views

A complex integration arround the boundary of a rectangular region

Let, $u(x,y)$ be thereal part of an entire function $f(z)=u(x,y)+iv(x,y)$ for $z=x+iy\in \mathbb C$. If $C$ is the positively oriented boundary of a rectangular region $R$ in $\mathbb R^{2}$ then the ...
3
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1answer
39 views

Computing Fourier Transform of $\frac{1}{t^2+a^2}$

I know this should be relatively simple, but I'm not getting the complete answer correct when I check with Wolframalpha. Here is my attempt. Going straight from the definition, with $x,t,a \in ...
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1answer
33 views

Conformal/Biholomorphism equivalence classes in $\mathbb{C}^n$

Recently I have got interested in the topic of conformal equivalence classes of complex domains, mostly one-dimensional ones. Here by conformal map $f: U \rightarrow V$ I mean a complex holomorphic ...
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1answer
40 views

Is this logarithmic inequality true?

Assume we have two complex variables $h_i$ and $h_d$ which satisfy the following relationship $$ 2\ |h_i|^2\leq \ |h_d|^2$$ can we say that $$\log\left( 1+ \frac{\big||h_d| - ...
3
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3answers
53 views

Closed form of product of complex numbers [duplicate]

I'm stuck in a proof where I want to get a closed form of something. This is the last thing I need to complete my proof: Apparently for small $n\geq2$, the product $\prod\limits_{k=1}^{n-1} ...
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3answers
61 views

Adding complex exponentials

Can somebody please explain $$e^{-\frac{3}{4}\pi i}+e^{-\frac{9}{4}\pi i}+e^{-\frac{15}{4}\pi i}+e^{-\frac{21}{4}\pi i}=0$$ WolframAlpha Computation.
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1answer
20 views

A simple question about a cauchy integral

Let $L$ be a closed simple contour in the complex plane, denote the interior and exterior domains respectively by $D_+$ and $D_-$. I need to prove that if $f(z)$ be analytic in $D_-$ and $z \in D_+$ , ...
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27 views

An application of the residue theorem

How would one find the residue of the function around an arbitrary point $z_0$ and using the branch cut $z\in(0,2\pi]$ $$R(z) = \frac{1}{(\sqrt{z+a}+\ln(z+b))^n}$$ Where $a,b\in\mathbb{R}$ and $n$ ...
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2answers
62 views

Imaginary-Order Derivative

I would like to find the imaginary-order derivative of a function (let's just focus on a simple function for now). There is the Riemann-Liouville fractional-derivative: $$ _{a}D^{i}_{t} f(t) = ...
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1answer
56 views

Solve $z^3 + 5z^2 + (9 - 5i)z + 10 - 10i = 0$ [duplicate]

Solve $$z^3 + 5z^2 + (9 - 5i)z + 10 - 10i = 0$$ I have never dealt with equations with complex numbers in them so this is interesting; first Ill expand. $$ \implies z^3 + 5z^2- 5iz + 9z + 10 - 10i = ...
3
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1answer
65 views

Solution to the Basel Problem in complex analysis pole issue.

Solve: $$\sum_{n=1}^{\infty} \frac{1}{z^2}$$ Before you mark as duplicate, I have a problem with only the consideration of the pole, please read carefully! Consider: $$f(z) = \frac{\pi \cot(\pi ...
4
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3answers
165 views

$\sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)^3}$ using complex analysis

Evaluate: $$S = \sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)^3} \space \text{using complex analysis}$$ This my question: we need to consider a $f(z)$ such that, $$\frac{1}{2\pi i} \cdot\oint_{C_N} f(z) ...
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1answer
38 views

Guidelines for choosing integrand to get a sum.

The idea was to find: $$\sum_{n=1}^{\infty} \frac{\coth(n\pi)}{n^3}$$ As you see in the solution, they conveniently choose a $f(z)$ they chose: $$f(z) = \frac{\pi \cot(\pi z)\coth(\pi z)}{z^3}$$ ...
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0answers
29 views

Why is taking the sum of residues required?

As you see in the solution, I am confused as to why the sum of residues is required. My question is the sum: $$(4)\cdot\sum_{n=1}^{N} \frac{\coth(\pi n)}{n^3}$$ Question #1: -Why is the ...
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1answer
27 views

Radius of Convergence of $\sum_n \frac{z^{2n}}{n}$

The problem (exercise) is to find radius of Convergence of $\sum_n \frac{z^{2n}}{n}$. The way I proceed was: put $z^{2}=w$. Then the series is $\sum_n \frac{w^n}{n}$, which (I know) converges if ...
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2answers
24 views

regarding the integral form of the inverse Laplace transform

The integral form of the inverse Laplace transform is given as $$f(t)=\frac{1}{2\pi i}\int_{s'-i\infty}^{s'+i\infty}e^{st}F(s)ds$$ where $s'$ is larger than the real parts of all the possible ...
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votes
1answer
33 views

Find all possible values of $\dfrac{1}{2\pi i}\int_\gamma \dfrac{2i}{z^2+1} \, dz$

let $\gamma$ be a closed and continuously differentiable path in the upper half plane not passing through $i$. Describe the set of all possible values of: $$\dfrac{1}{2\pi i}\int_\gamma ...
16
votes
5answers
232 views

What is $0^{i}$?

$$\lim_{n\to 0} n^{i} = \lim_{n\to 0} e^{i\log(n)} $$ I know that $0^{0}$ is generally undefined, but can equal one in the context of the empty set mapping to itself only one time. I realize that in ...
0
votes
1answer
26 views

Integrate $\int_C{\tan{z}\ dz}; C: y=x^2$ (complex numbers)

Integrate $$\int_C{\tan{z}\ dz}$$ $C$ is the parabola arc $y=x^2$ that connects the points $z=0$ and $z=1+i$. This is what I've done so far: I know that $\tan{z}=\dfrac{\sin{z}}{\cos{z}}$ And ...
0
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0answers
23 views

Calculating $I = \iint_{\mathbb{R}^2}dxdp \ xW_0(x-b(t)x_0\sqrt{2},p) $ by shifting the contour.

I want to calculate the following integral: $$I = \iint_{\mathbb{R}^2}dxdp \ xW_0(x-b(t)x_0\sqrt{2},p) $$ Where $W_0$ is a real valued function, and $b(t)=b_0e^{-i\omega t}$, with $b_0$ real. Also, ...
0
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0answers
31 views

Singularities of a function, complex calculus

I need to find all singularities and their type on $C \cup \{ \infty \}$ of function : $$f(z)={1 \over \exp(1/z)-1}.$$ So far I was only able to tell that function $$f(z)= {1 \over ...