Tagged Questions

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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What is wrong in this use of Cauchy residue theorem?

Consider $F(z)$ a function such that $\overline{F(z)}=F(\overline{z})$, with no pole, decreasing faster than any power $\frac{1}{z}$ when z is imaginary going to $_{-}^{+}i \infty$. I define the ...
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Compute $\mathrm{Res}(\frac{e^{iz}}{z(z^2+1)^2},i)$

I have to compute $\mathrm{Res}(\frac{e^{iz}}{z(z^2+1)^2},i)$. Do I have to use the result from $Res[f/g,z_0]=\frac{f(z_0)}{g'(z_0)}$ -Proof (I think I have a pole of order $2$)? Otherwise, how ...
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Prove that $\mathrm{Res}(\frac{f(z)}{g(z)}, z_0) = \frac{f(z_0)}{g'(z_0)}$ [duplicate]

Show that if $f$ and $g$ are analytic on a neighborhood of $z_0$ with $f(z_0)\not= 0$ and $z_0$ is a simple zero of $g$, then we have $\mathrm{Res}(\frac{f(z)}{g(z)}, z_0) = \frac{f(z_0)}{g'(z_0)}$...
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Compute $\mathrm{Res}(\frac{1+2z+3z^2}{1+z+z^2-3z^3},1)$

I have to compute $\mathrm{Res}(\frac{1+2z+3z^2}{1+z+z^2-3z^3},1)$. I know that $\mathrm{Res}(f,z_0)+a_{-1}= \int_{C_p} \frac{f(z)dz}{z-z_0}$, where $C_p$ is simply the circle at $z_0$ with radius ...
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How to solve the laplace transform of $f_m(t_m)$ = $f_1(t)$ $\int_{0}^{\alpha} f_2(\tau) d\tau$ + $f_2(t)$ $\int_{0}^{\alpha} f_1(\tau) d\tau$.

Could you please help me to solve the following : if $t_m$ = min($t_1$,$t_2$) The probability density function $t_1$ is $f_1(t_1)$ and $t_2$ is$f_2(t_2)$ then $f_m(t_m)$ is the probability density ...
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Residue for $\frac{\zeta(s)}{\zeta(2s)}$ at zeros of $\zeta(2s).$

I want to calculate residue at the poles for $\frac{\zeta(s)}{\zeta(2s)}.$ For pole of numerator $s=1$ I have calculated the residue. I am having trouble at the zeros of denominator. Basically I am ...
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Write $\mbox{Res}(g, 0)$ in terms of the $\mbox{Res}(f, z)$ where $g(z)=\frac{1}{z^{2}}f\left(\frac{1}{z}\right)$.

Let $M\subseteq \mathbb{C}$ finite set and $f:\: \mathbb{C} \setminus M \longrightarrow \mathbb{C}$ be a holomorphic function. Consider $g(z)=\frac{1}{z^{2}}f\left(\frac{1}{z}\right)$. The quiestion ...
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Show that the numbers $3$,$3^2$,$3^3$,$3^4$,$3^5$,$3^6$ for a reduced residue system modulo 7.

Show that the numbers $3$,$3^2$,$3^3$,$3^4$,$3^5$,$3^6$ for a reduced residue system modulo 7. -A bit lost with this question, we just started a section on reduced residue sets and only covered ...
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Cauchy Principal Value of $\int_0^\infty \frac{x}{(x^2 + a^2) \, \sin(\mu x)} dx$

The problem here is to evaluate $$\int_0^\infty \frac{x}{(x^2 + a^2) \, \sin(\mu x)} dx$$ for $a,\mu >0.$ Clearly this integral doesn't converge in the usual sense, but we can calculate its ...
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Inverse Laplace Transform of $e^{\frac{1}{s}-s}$

doing some work on a PDE system I have stumbled across a Laplace transform which I'm not sure how to invert: $$F(s) = e^{\frac{1}{s}-s}$$ I can't find it in any table and the strong singular growth ...
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How can $\sin x = e^{iz}$?

This is probably a trivial question but I just don't see it. I'm solving the integral $$\int_{0}^{\infty}\dfrac{x \sin x }{x^2 + 5x + 4}dx$$ using the residue theorem. The thing is that they're ...
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Show that the numbers $-13, -9, - 4, -1, 9, 18, 21$ form a complete residue system modulo 7

Show that the numbers $-13, -9, - 4, -1, 9, 18, 21$ form a complete residue system modulo 7 We have just started he section on modular arithmetic so I am new to a residue system, we did a similar ...
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Cauchy Residue Theorem Integral

I have been given the integral $$\int_0^ {2\pi} \frac{sin^2\theta} {2 - cos\theta} d\theta$$ I have use the substitutions $z=e^{i\theta}$ |$d\theta = \frac{1}{iz}dz$ and a lot of algebra to transform ...
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Integrate $I=\int_e^\infty\left(\frac{\log\log y}{y(y+1)}\right)^2dy$ using residue calculus?

Can this integral be done using the residue calculus? $$I=\int_e^\infty\left(\frac{\log\log y}{y(y+1)}\right)^2dy$$ ? My (empirical) investigative attempts have been to use a keyhole contour centred ...
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Complex integral of the function $f(z)=\dfrac{1}{z^4+1}$

I must calculate this integral $$\int_C\dfrac{dz}{z^4+1}$$ , where $C$ is the circle $x^2+y^2=2x$. My result is $\int_C\dfrac{dz}{z^4+1}=-\dfrac{\pi}{\sqrt{2}}$ , but my book "A collection of problems ...
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Improper integral using residue theorem

I am meant to use the residue theorem to show that $\int\limits_{-\infty}^\infty \frac{\cos t}{(t^2+1)^2}dt=\frac{\pi}{e}$. So far I have deduced that I should take a contour over $\alpha$ the path ...
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Compute the series $\sum_{n=1}^{+\infty} \frac{1}{n^3\sin(n\pi\sqrt{2})}.$

I need to compute $$\sum_{n=1}^{+\infty} \frac{1}{n^3\sin(n\pi\sqrt{2})}.$$ This an exercice of "Amar and Matheron, complex analysis". I proved the convergence and now to compute the sum, I follow the ...
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Cauchy's Residue Theorem and Cauchy's Theorem

Cauchy's theorem in short says for a holomorphic function $f$ which is holomorphic on and inside a path $\gamma$ the path integral is $0$ I have calculated a path integral around a path where there ...
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Integral involving power of trigonometric functions

I'm having a technical problem evaluating the following integral: $$\int_{r=0}^1\int_{\theta=0}^{\pi \over2} \cos^{2\epsilon -1}\theta \sin^{\epsilon-1}\theta e^{-ikr\sin^\epsilon\theta}d\theta dr$$ ...
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Complex integration with infinitely many poles on imaginary axis

I'm trying to integrate with a closed contour on the upper-half of the complex plane. $I = \displaystyle\int_{-\infty}^\infty \dfrac{z\,\mathrm{sech(z)}}{[(z-a)^2+b^2][(z+a)^2+b^2]} dz$ There are ...
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Complex roots in order to apply residue theorem

$$\int_{0}^{2\pi}\frac{d\theta}{(4 + 2\sin\theta)^2}$$ $$\sin\theta = \frac{z - z^{-1}}{2i}$$ $$d\theta = \frac{dz}{iz}$$ $$\oint_c\frac{dz}{iz\left(4 + \frac{z - z^{-1}}{i}\right) ^2}$$ ending up ...
$$\int\frac{d\theta}{a + b\cos\theta}$$ Given that $$\cos\theta = \frac{z + z^{-1}}{2}$$ $$d\theta = \frac{dz}{iz}$$ We have $$\oint_c \frac{dz}{iz\left(a + b\frac{z +z^{-1}}{2}\right)}$$ \...
If $f:G\longrightarrow\mathbb{C}$ is analytic except for the isolated singularities and has infinitely many singularities, why then the singularities only can accumulate on boundary of $G$? Which is ...