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countour

† ˈcountour, -or Obs. [An earlier form of counter n.2, AF. countour, as an official title.] 1. Eng. Hist. An accountant; an officer who appears to have assisted in early times in collecting or auditing the county dues.[1292 Britton ii. xxi. §3 Ou seignurs, ou counseillers, ou countours.] 1297 R. Glo... Oxford English Dictionary

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countrel

† countrel Obs. rare. [Cf. countour.] = accountant.1479 Paston Lett. No. 839 III. 254 Lete my countrelle doo what hym liste. Oxford English Dictionary

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Countour integral $\int {{{(\overline z )}^2}dz} $ > Evaluate $\int {{{(\overline z )}^2}dz} $ along the straight line segment from $z=0$ to $z=2+i$. My attempt to this question is I change z into $x+iy$ and do the i...

Note that: $$z=0\to2+i\implies z(t)=(2+i)t\quad t\in[0,1],t\in\mathbb R\\\\\int\bar z^2dz=\int_0^1 \overline{(2+i)}^2t^2(2+i)dt=(2-i)^2(2+i)/3=5(2-i)/3=10/3-5i/3$$

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countour integral > Let $ f: \mathbb{C} \to \mathbb{C} $ be defined on the complex plane by $$ f(z) = \begin{cases} f(z)=z & \textrm{if}\ \operatorname{Re}(z) \ge 0 \\\ f(z)=z^{2} & \textrm{if}\ \operatorname{Re}(z) <...

Your assumption is not correct. $f(z)$ is not continuous on the imaginary line and therefore is not holomorphic on the domain defined by $|z|<1$. This mean the integral is not necessarily $0$. We can integrate the "regular" way (with parametrization). Let $z = e^{it}$ then $$ \int_C f(z)\ dz = \int_...

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Countour integration + partial fraction decompositon + poles Source: José Figueroa-O'Farrill's Mathematical Techniques III Lecture Notes Trying to evaluate the following integral: $\oint_{\text{ }\Gamma}\frac{2z+1}{...

Consider, for instance, $\frac1z$, whose domain is $\mathbb C\setminus\\{0\\}$. Then the loop $\Gamma_1$ is homotopically null (that is, it is homotopic to a _constant_ loop) in $\mathbb C\setminus\\{0\\}$, and therefore $\int_{\Gamma_1}\frac{\mathrm dz}z=0$. So$$\int_{\Gamma_1}\frac{\mathrm dz}{(z-...

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Software or website for live 2D countour plots? I'm looking for a software tool to plot live contour plots. With _live_ I mean something like Desmos, where you can enter formulas with parameters, and then adjust ...

You can probably do it easily in Mathematica and turn it into an interactive demonstration that can be played with the free CDF player. For examples, see the Wolfram Demonstrations Project, which includes source code for many examples. The example below should get you started: * Graph and Contour Pl...

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Prove $\int_0^\infty \frac{dx}{x^{1/\alpha}(x+1)}$ goes to zero along countour circling branch point. Background: This is part of the countour integration problem where you show that the circle IV goes to zero as $\ep...

The key is to have $$ \lim_{r\to0}\frac{2\pi r}{r^{1/a}}=0 $$ so you need $1-\frac1a\gt0$. That would be $a\lt0$ or $a\gt1$. The $2\pi r$ is the circumference of the small circle and $r^{-1/a}$ is the absolute value of $\frac1{x^{1/a}}$ on the circle. The factor of $\frac1{x+1}$ is near $1$ close to...

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Integral countour with Mangoldt function In an analytic proof for prime number theorem I found a passage I could not understand: > $$A=\sum_{n\leq x} \frac{1}{2 \pi i} \int_{c-\infty i}^{c+\infty i} \frac{\Lambda(n) ...

The reason is that the integral $$\int_{c-i\infty}^{c+i\infty}\frac{x^s}{s(s+1)}\,ds$$ vanishes when $x<1$. To prove that, consider the contour: ![contour]( (image is from Apostol's _Introduction to Analytic Number Theory_ , where you can also find the details of the proof). Elementary estimates sho...

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Using Cauchy's integral formula to evaluate a function This problem is from Brown/Churchill Complex Variables and Applications, $8$th edition $2009$. Section $52$, exercise $2$, subsection (a) How do I show that th...

$$\oint\limits_{|z-i|=2}\frac{1}{z^2+4}dz=\oint\limits_{|z-i|=2}\frac{\frac{1}{z+2i}}{z-2i}dz=\left.2\pi i\left(\frac{1}{z+2i}\right)\right|_{z=2i}=2\pi i\frac{1}{4i}=\frac{\pi}{2}$$ Now, **why** did I do the above the way I did? Draw a picture of the circle $\,|z-i|=2\,$ and try to locate the poles...

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frayer

† ˈfrayer Obs. [f. fray v.1 + -er1.] a. One who frightens away. b. One who makes a disturbance; a fighter, rioter.1494 Fabyan Chron. vii. 583 Both frayers were taken & brought vnto the countour in the Pultry. 1543 Becon Policy War Wks. 1564 I. 143 They be the aungels of God..the exhorters vnto vertu... Oxford English Dictionary

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contour plot in multiple linear regression I have recently saw some examples about contour plots and multiple linear regression, for what I know a countour plot is obtained for having a graphical view of how the weigh...

You have a model $F(x_1,x_2)$ and you need to build the contour plot. So, you have , for a given level $c$, $$F(x_1,x_2)=c$$ which gives an equation of $x_2$ as a function of $x_1$. Let us consider the first case : it writes $$50+10x_1+7x_2=c$$ so $$x_2=\frac{1}{7} (c-10 x_1-50)$$ and these are para...

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Evaluate complex integral using deformation theorem It's given that, $\gamma$ is a circle of radius $r$ and centre $z=a$ inside $\Gamma$. I need to use the deformation theorem to evaluate this: $$\oint_\Gamma \ (z-a)^...

Think of $z=a+r e^{i \theta}$. Then $dz=i r e^{i \theta}$ and the contour integral becomes $$i r^{n+1} \int_0^{2 \pi} d\theta \, e^{i (n+1) \theta}$$ So long as $n$ is an integer, the above integral vanishes...except when $n=-1$, when it is equal to $i 2 \pi$. Yes, even when $n$ is less than $-1$, t...

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Semicircle contour for integrating $t^2/(t^2+a^2)^3$ > Let $a\in\mathbb{R}$. Evaluate $$\int_0^{\infty}\dfrac{t^2}{(t^2+a^2)^3}dt$$ The function is even, so the value of the integral is half of $\int_{-\infty}^{\inft...

Standard calculus result. If $p(x),q(x)$ are two polynomials with $\deg(q(x))>\deg(p(x))$, then $$\lim_{x\rightarrow\pm \infty} \frac{p(x)}{q(x)} = 0.$$ You can prove it by dividing top and bottom by the highest power of $x$ appearing in $p$. In your particular case, the numerator is degree 3 in $R$...

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Complex integral computation with $\sinh$ I need to prove the following integral computation by applying the residue theorem: $$\int_{-\infty}^{+\infty}ds\frac{e^{-i\Omega s}}{(\sinh{[\frac{a}{2}s-i\epsilon]})^2}=-8\p...

We assume that $\Omega$ is real-valued and positive. The poles are located at $s_n = \frac{2i}{a}(n\pi+\epsilon)$. If we analyze the integrand, we see that $$\frac{e^{-i\Omega s}}{\sinh^2(\frac{a}{2}(s-\frac{2i\epsilon}{a}))}=\frac{e^{-i\Omega s_n}\left(1-i\Omega(s-s_n)+O\left(s-s_n\right)^2\right) ...

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Limit of the function $V(x,y)=x^4-x^2+2xy+y^2$ Let $$V(x,y)=x^4-x^2+2xy+y^2$$ Consider the coupled d.e.'s:$$\frac {\mathrm d x} {\mathrm d t} = - \frac {\partial V} {\partial x}, \qquad \frac {\mathrm d y} {\mathrm d...

The limit must be one of the three critical points. At these points you have $\frac{dx}{dt} = \frac{dy}{dt} = 0$. You've calculated these to be (-1, 1), (0,0), and (1,-1). The point (0,0) is a repellor, the other two points are attractors. Looking at the contour plot you made (better yet, use a dire...

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countour

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countour

countrel

Countour integral $\int {{{(\overline z )}^2}dz} $ > Evaluate $\int {{{(\overline z )}^2}dz} $ along the straight line segment from $z=0$ to $z=2+i$. My attempt to this question is I change z into $x+iy$ and do the i...

countour integral > Let $ f: \mathbb{C} \to \mathbb{C} $ be defined on the complex plane by $$ f(z) = \begin{cases} f(z)=z & \textrm{if}\ \operatorname{Re}(z) \ge 0 \\\ f(z)=z^{2} & \textrm{if}\ \operatorname{Re}(z) <...

Countour integration + partial fraction decompositon + poles Source: José Figueroa-O'Farrill's Mathematical Techniques III Lecture Notes Trying to evaluate the following integral: $\oint_{\text{ }\Gamma}\frac{2z+1}{...

Software or website for live 2D countour plots? I'm looking for a software tool to plot live contour plots. With _live_ I mean something like Desmos, where you can enter formulas with parameters, and then adjust ...

Prove $\int_0^\infty \frac{dx}{x^{1/\alpha}(x+1)}$ goes to zero along countour circling branch point. Background: This is part of the countour integration problem where you show that the circle IV goes to zero as $\ep...

Integral countour with Mangoldt function In an analytic proof for prime number theorem I found a passage I could not understand: > $$A=\sum_{n\leq x} \frac{1}{2 \pi i} \int_{c-\infty i}^{c+\infty i} \frac{\Lambda(n) ...

Using Cauchy's integral formula to evaluate a function This problem is from Brown/Churchill Complex Variables and Applications, $8$th edition $2009$. Section $52$, exercise $2$, subsection (a) How do I show that th...

frayer

contour plot in multiple linear regression I have recently saw some examples about contour plots and multiple linear regression, for what I know a countour plot is obtained for having a graphical view of how the weigh...

Evaluate complex integral using deformation theorem It's given that, $\gamma$ is a circle of radius $r$ and centre $z=a$ inside $\Gamma$. I need to use the deformation theorem to evaluate this: $$\oint_\Gamma \ (z-a)^...

Semicircle contour for integrating $t^2/(t^2+a^2)^3$ > Let $a\in\mathbb{R}$. Evaluate $$\int_0^{\infty}\dfrac{t^2}{(t^2+a^2)^3}dt$$ The function is even, so the value of the integral is half of $\int_{-\infty}^{\inft...

Complex integral computation with $\sinh$ I need to prove the following integral computation by applying the residue theorem: $$\int_{-\infty}^{+\infty}ds\frac{e^{-i\Omega s}}{(\sinh{[\frac{a}{2}s-i\epsilon]})^2}=-8\p...

Limit of the function $V(x,y)=x^4-x^2+2xy+y^2$ Let $$V(x,y)=x^4-x^2+2xy+y^2$$ Consider the coupled d.e.'s:$$\frac {\mathrm d x} {\mathrm d t} = - \frac {\partial V} {\partial x}, \qquad \frac {\mathrm d y} {\mathrm d...