Questions tagged [residue-calculus]

Questions on the evaluation of residues, on the evaluation of integrals using the method of residues or in the method's theory.

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Residue of $ze^{\frac{1}{z-1}}$

I want to calculate the residue of $ze^{\frac{1}{z-1}}$ at $z=1$. My idea is the following: $$ ze^{\frac{1}{z-1}}=((z-1)+1)e^{\frac{1}{z-1}}=(z-1)e^{\frac{1}{z-1}}+e^{\frac{1}{z-1}} $$ Around $z=1$: $$...
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Need help on solving the limit of a contour integral of a semicircle around $z = 0$ as the radius $\to \infty$

$$f(z)=\frac{e^{-iz}}{z}$$ What is the value of: $$\frac{1}{\pi i }\int_{C} f(z) dz$$ if ${C}$ is the arc of the semicircle with radius $R \to \infty$ ,going counterclockwise from point $(R,0)$ to $(-...
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Visualisation of the residue theorem - contour integration

I was trying to solve this integral: $$\int_{-\infty}^{+\infty}\frac{\xi_{0}^{2}e^{-iEt}dE}{\left(E-E_{0}\right)^{2}+\left|\xi_{0}\right|^{4}\pi^{2}}$$ For that i used partial fractions to find the ...
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2 votes
2 answers
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Calculating complex integral with Residue - where is my fault?

\begin{align}\int_0^{2\pi} \frac{\cos(x)}{13+12\cos(x)} dx & = \displaystyle\int_0^{2\pi} \frac{(z+1/z)\frac{1}{2}}{13+12(z+1/z)\frac{1}{2}}\frac{1}{iz} dz \\ & = \cdots \\ &= -i\...
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1 answer
19 views

Simplification of a Residuum

I had to find the Residuum $Res(f|z_k)$ for all Singularities $z_k$ for $f(z)=tan(z)$. I got that. which is: $Res(f|\frac \pi2) = \lim_{z \to \frac \pi 2} (z-\frac \pi 2)\ tan(z) = \lim_{z \to \frac \...
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3 votes
4 answers
218 views

Complex Analysis to solve this integral? $\int_0^{\pi/2} \frac{\ln(\sin(x))}{\sqrt{1 + \cos^2(x)}}\text{d}x$

Complex Analysis time! I need some help in figuring out how to proceed to calculate this integral: $$\int_0^{\pi/2} \frac{\ln(\sin(x))}{\sqrt{1 + \cos^2(x)}}\text{d}x$$ I tried to apply what I have ...
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0 votes
1 answer
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Compute integral using Residue theorem, not by manipulating the denominator

I need to compute $$\int_{-\infty}^{\infty}\frac{1}{x^2+2x+2}dx$$ using the residue theorem. I know I can manipulate this to $$\int_{-\infty}^\infty \frac{1}{(x+1)^2+1}dx$$ Then make a u sub and BAM ...
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To prove $\lim_←n(R/π^mR((X)))/(X^nR/π^mR[[X]])=R/π^mR((X))$

Let $L$ be a finite extension of $ \Bbb{Q}_p$, let $R$ be integer ring of $L$.Let $π$ be uniformizer of $L$. For given integer $m$, I want to prove $\lim_←n(R/π^mR((X)))/(X^nR/π^mR[[X]])=R/π^mR((X))$. ...
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Integral $\int_{-\infty}^{\infty}\frac{x^2\cos(x)}{x^4-1}dx$

I want to compute $$\int_{-\infty}^{\infty}\frac{x^2\cos(x)}{x^4-1}dx$$ by using Residue Theorem. The poles are given by $A:=\{-i,i,1,-1\}$. Now, it seems to me that semicircle contour won't work, ...
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Why is the value of r taken as $1$ when we try to find the pole of $\frac{1}{1+z^4}$?

I am trying to find the poles of $ \frac{1}{1 + z^4}$. The pole is at $z^4 = -1$. We know that $z = re^{i\theta}$. $\therefore (re^{i\theta})^4 = -1$ $\implies \theta = \frac{1}{i}ln\left(\frac{(-1)^{\...
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2 votes
1 answer
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How to calculate this integral using the residue theorem?

$\displaystyle \frac{1}{2\pi i}\int \limits _\gamma \sin ^2\frac{1}{\xi}\,d\xi$ $\gamma (t)=Re^{it}$, $R>0$, $0\leq t\leq 2\pi$ Could you help me solve this equation? I did not understand this ...
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1 vote
1 answer
32 views

breaking integral into 2 sub integrals

is it possible to break $$\oint_{|z|=2}\tan(z)+\frac{e^z}{z-4}\,dz$$ into $$\oint_{|z|=2}\tan(z)\,dz+\oint_{|z|=2}\frac{e^z}{z-4}\,dz$$ or these kinds of integral doesn't support this feature?
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  • 27
2 votes
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Integral of a function with essential singularity

I have to determine the value of $$\int_{\partial D_r(0)}\frac{\sin(\pi/z)}{z^2-1}dz$$ after having specified for which values of $r$ the integral is well defined. I don't know how to find the values ...
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  • 342
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0 answers
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residue of $f(z)=\frac{e^{1/z}}{z-1}$ [duplicate]

find the residue of $$f(z)=\frac{e^{1/z}}{z-1}$$ in it's singular points my thoughts: I think we should calculate it's residue in $z=1$ and $z=0$ but In order to do so I have to write $f(z)$ with it'...
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3 votes
2 answers
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Joining $\infty$ and -$\infty$ in complex contour integral.

When solving the following integral $$\int_{-1}^1 \frac{1}{(x^4+1)\sqrt{1-x^2}} \, dx$$ using complex contour integration, I decided to take the keyhole contour, looping around the branch point at $-1$...
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2 answers
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What is the residue of $\frac{1}{e^z+1}$ at its singular point? [closed]

What is the residue of $\frac{1}{e^z+1}$ at its singular point? The singular point is $z=\pi i$. But I have no idea how to expand this.
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1 answer
65 views

Use Residues to find the inverse Laplace transform $F(s)=\frac{2s^3}{(s^2-4)}$

Use Residues to find the inverse Laplace transform $F(s)=\frac{2s^3}{(s^2-4)}$. The answer from the text book is $f(t)=\cosh^2(t)+\cos^2(t)$. But my result is $2\cos^2(t)\cdot \cosh^2(t)$. Which is ...
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6 votes
3 answers
121 views

Fourier transform of $\frac{x}{\sinh(x)}$

I was given to calculate the Fourier Transform of $\frac{x}{\sinh(x)}$. So, the problem is to calculate the integral $$ \int_\mathbb{R} \frac{x}{\sinh(x)}e^{-i \omega x} dx $$ I know such an integrals ...
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3 votes
2 answers
217 views

Calculate an integral using contour integration

I have to calculate the following integral using contour integration: $$\int_0^1 \frac{dx}{(x+2)\sqrt[3]{x^2(x-1)}}$$ I've tried to solve this using the residue theorem, but I don't know how to ...
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1 vote
1 answer
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All possible values of a complex integral

I'm struggling with the following task: Find all possible values of the complex integral $\int_0^1 \frac{dz}{z^2 + 1}$ if we are integrating it along all possible curves from $0$ to $1$. The official ...
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0 votes
1 answer
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How to deal with numerator in contour integration

Consider the integral $$ \oint dx\, \frac{g(x)}{x - x_{0}} $$ where $x_{0} \in \mathbb{C}$ is the position of a pole for the integrand that we assume to be integrated along a closed path encircling ...
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  • 713
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1 answer
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Interesting result in a complex integration

Studying for my complex-analysis exam I found an interesting integral. The activity consisted of calculating a complex integral around a curve using the residue theorem. The integral had the form $$ ...
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  • 315
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Undefined residue at infinity

I am trying to find the residue of $\frac{(cos(sin(z)))}{z^2}$ at $\infty$. What I did was $Res(f(x),\infty) = Res (f(\frac{1}{z}), z=0)$. What I got was $Res(z=0)=-z^2cos(sin(\frac{1}{z}))$ which is ...
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Why is the residue $\mathop{\mathrm{Res}}_{z=\pi/2} \frac{z}{\cos z}$ not $0$?

I'm supposed to get the residue of the function $\dfrac{z}{\cos z}$ at $z = \pi/2$. Here is my solution: \begin{align*} \mathop{\mathrm{Res}}_{z=\pi/2} \frac{z}{\cos z} &= \frac{1}{(m-1)!} \lim_{z ...
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Issues with calculating integral $\int_{0}^{+\infty} \frac{x\sin x}{(x^{2} + a^{2})^{2}} dx$ using residues.

I have this integral $I = \int_{0}^{+\infty} \frac{x\sin x}{(x^{2} + a^{2})^{2}} dx$. I tried to calculate it myself, but apparently I'm using the wrong residues formula, the answer doesn't come out. $...
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  • 1
0 votes
1 answer
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Classification of singularities of a complex function

I'm trying to classify in $\mathbb C\cup\{\infty\}$, the singularities of the function $$f(z)=\frac{e^{z^2}\sin(1/z)}{z}.$$ Clearly the function has only one singularity in $z=0$ and in order to tudy ...
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  • 342
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0 answers
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Solve $\int_{-\pi}^{\pi}{\frac{d\phi}{a+b\cos \phi}}$ using residues.

I do a variable substitution by adding $\pi$ to $\phi$ and the sign of $\cos$ changes. $\int_0^{2\pi}{\dfrac{d\phi}{a-b\cos \phi}} = 2\pi \sum_{k=1}^n {res \frac{1}{z} \big [(a-b \frac{1}{2}(z + \frac{...
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  • 523
0 votes
2 answers
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Find residue of $f(z) = \frac{\sin z}{(z^2+1)^2}$ at $z = \infty$

Find residue of $f(z) = \dfrac{\sin z}{(z^2+1)^2}$ at $z = \infty$. Then this is the same as finding the residue at $z=0$ for $\dfrac{-1}{z^2}f(1/z)= \dfrac{-z^2 \sin 1/z}{(z^2+1)^2}$ $z = 0$ is a ...
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  • 523
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0 answers
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Find residue of $f(z) = \dfrac{\cos z}{(z^2 + 1)^2}$ at $z=\infty$

Find residue of $f(z) = \dfrac{\cos z}{(z^2 + 1)^2}$ at $z=\infty$. I am new to this topic. First of all I can consider the function $f(1/z)$ and thus find its residue on $z=0$. Then, I can see that $...
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  • 523
1 vote
1 answer
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Calculating the integral using residues.

How can this integral be calculated using Cauchy's basic residue theorem? I tried to represent $cos(\alpha x)$ as $Re[e^{i\alpha x}]$, and try to calculate the general integral in this form: $\int\...
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1 vote
2 answers
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A quick way to calculate residues of logarithmic derivatives

Assume that $U\subset \mathbb{C}$ is open an let $f(s)$ be a meropmorphic function on $U$. Consider the logarithmic derivative of $f(s)$ where I mean the meromorphic function $\frac{f'(s)}{f(s)}$. I ...
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1 vote
2 answers
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Сalculate the integral PV $\int_{0}^{\infty} \frac{dx}{x^\alpha(x-a)}dx.$

Сalculate the integral $$ \mathrm{PV}\hspace{-0.5ex}\int_{0}^{\infty} \frac{dx}{x^\alpha(x-a)}, $$ where $0<\alpha <1$ and $a>0$. So we have simple poles $z = 0$ and $z = a$; We can build ...
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  • 55
0 votes
1 answer
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Confusion regarding pole of a complex function.

I am a graduate student.I am studying complex analysis.I encountered the following problem in a lecture: Find the residue of $f(z)=\frac{z-\sinh(z)}{z^2\sinh(z)} $ at $z=\pi i$. Now,this problem is ...
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0 answers
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How to count the total zeros of a complex polynomial outside a closed curve?

Set up Suppose $\gamma$ a simple closed curve, oriented in a counterclockwise direction. $f(z)$ is a complex polynomial $$ f(z)=a_n(z-z_n)^n+a_{n-1}(z-z_{n-1})^{n-1}+\cdots+a_0. $$ We already know ...
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  • 33
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1 answer
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Showing the integral $\int_{C_N} \frac{1}{(2z-1)\sin{\pi z}}dz$ converges to zero as $N \to \infty$

I have a question about the 10th question part b of Chapter 11 of the Complex Analysis by Bak & Newman. The question says that Show that $1-1/3+1/5-1/7+...=\pi/4$ by using the integral of $\frac{1}...
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1 answer
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Evaluate $PV \int_{-\infty}^{\infty} \frac{1-e^{iax}}{x^2}dx. a>0$

Evaluate $PV \int_{-\infty}^{\infty} \frac{1-e^{iax}}{x^2}dx. a>0$ using residues. So I have a theory how to calculate $PV \int_{-\infty}^{\infty} f(x)e^{iax}dx$ a>0, but I don’t know how to ...
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  • 55
0 votes
0 answers
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Evaluate $\int_0^{\infty}\frac{\log( x)}{x^2+a^2} \,dx$ using contour integration; Re a > 0

Evaluate $\int_0^{\infty}\frac{\log( x)}{x^2+a^2} \,dx$ using contour integration; $Re (a) > 0$ I found two questions where a > 0 but in my case I have the following condition: Re a > 0 (It ...
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1 vote
0 answers
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Using multivector residue theorem to evaluate multiple integrals

I recently started learning about Geometric Algebra and Geometric Calculus. Since the residue theorem can be generalized for multivector functions I wondered if one could use it to evaluate certain ...
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2 votes
2 answers
67 views

Calculating residue of $f(z)=\frac{z^2+z^3}{({\sin z})^3}$ at its singularities

So the singularities would be at $n\pi$, $n\in \mathbb{Z}$. The residue in $z=0$, I have already calculated (through shifting the series) and it equals $1$ however for the others $n\pi$ I am stuck. ...
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  • 443
3 votes
1 answer
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Why do we use exponentials while integrating trigonometric functions in complex analysis

Let p(x) be some polynomial function. Now, we have an integral of the form : $$I=\int_{-\infty}^{\infty} \frac{\cos(x)}{p(x)}dx$$ What is usually done is that, we define this integral as : $$I'=\int_{-...
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0 votes
0 answers
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Convert Bessel function contour integral definition to the imaginary line

I would like to solve the following integral $$I_{\nu}(x) = \frac{1}{{2\pi i}}\int_{-i\infty}^{i\infty} d\lambda \frac{e^{\frac{x}{2}(\lambda - 1/\lambda)}}{\lambda^{\nu+1}}.$$ The integrand is ...
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Regularization of divergent integral with Resiue theorem

I am currently trying to calculate the following integral $$ \int_{0}^{\infty} d k \frac{k}{k^{2}-(a+i \varepsilon)^2} $$ where $a$ and $\varepsilon$ are real parameters. For this, I have found the ...
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  • 181
0 votes
0 answers
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Is it possible to count the highest order zero of a complex polynomial function by using some integral?

Set up Suppose $\gamma$ a simple closed curve, oriented in a counterclockwise direction. $f(z)$ analytic on and inside $\gamma$. Let $z_1,z_2, ..., z_n$ be the zeros of $f$ inside $\gamma$. Write ...
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  • 33
2 votes
0 answers
60 views

Integrate $\int_0^\infty e^{-\sqrt{x}} \mathrm{d} x$ using complex analysis

I'm trying to compute the integral $\int_0^\infty e^{-\sqrt{x}} \mathrm{d} x$ using complex analysis. I'm working on a problem for a complex analysis class that asks me to do so, but I'm struggling ...
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0 votes
1 answer
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Use residues to evaluate $\int_0^\infty \frac{x^{1/3}}{1+x^2} \, dx$

I need to use residues to evaluate $\int_0^\infty \frac{x^{1/3}}{1+x^2} \, dx$. So first, $$\int_0^\infty \frac{x^{1/3}}{1+x^2} \, dx = \int_{C_R} \frac{z^{1/3}}{1+z^2} + \int_{-R}^R \frac{x^{1/3}}{1+...
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0 votes
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45 views

Why I can not calculate the residual of $\frac{\cos(x)}{x^2+1}$ like this to solve this integral?

I have the following integral to solve: $$I =\int_{-\infty}^{+\infty}\frac{\cos(x)}{x^2+1}$$ and after applying the big circle lemma and the residue theorem, I run into this expression: $$I = \pi i \...
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5 votes
3 answers
397 views

Application of residue at infinity

I am trying to figure out how to find a contour to solve for this integral $$ \int_{-\infty}^{\infty}\frac{2x}{x^2+x+1}dx = -\frac{2\pi}{\sqrt{3}} $$ using the residue theorem and the residue at ...
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2 votes
1 answer
56 views

How to calculate the correct residue?

I am trying to calculate the residue of the function $$f(z)=\frac{2}{3z^2+8iz-3}$$ so as to evaluate the integral $$I=\int_{0}^{2\pi}\frac{1}{3\sin(\theta)+4}d\theta$$ I have found that $f$ has ...
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17 votes
2 answers
395 views

How to prove $\int_0^\infty\frac {\tanh(x)-x\exp(-x)}{x^2}dx=\frac{\zeta'(0)}{\zeta(0)}-\frac{\zeta'(2)}{\zeta(2)}+\gamma-\frac73\log(2)$?

By educated guessing, inspired by this solution of $\int_0^\infty\frac {\tanh^3(x)}{x^2}dx$, I have found numerically: $$\int\limits_0^\infty\frac {\tanh(x)-x\exp(-x)}{x^2}dx=\frac{\zeta'(0)}{\zeta(0)}...
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  • 884
0 votes
1 answer
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Residues with Inequalities

If $f$ is an entire function such that $|f(z)|\leq A|z|$ for all $z\in\mathbb{C}$ for some fixed $A>0$, then can I write $$\left|\frac{1}{2\pi i}\int_\gamma\frac{f(z)dz}{(z-c)^n}\right|\leq\frac{1}{...
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