# Tagged Questions

For questions about integration methods that use results from complex analysis and their applications

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### Need help with $\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$

I'm having trouble trying to evaluate this definite integral. Mathematica didn't help much. $$\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$$ where $a$, $b$ $\in \Bbb R^+$. Is it possible ...
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### Closed contour integral: $\int_{\mathbb{c}}\frac{ z}{2z^{2}+1} dz$ where the contour is the unit circle

first and foremost please excuse my English. given $∫_c \frac{{z}}{2z^{2}+1}dz$ where the contour is the unit circle. so c = $e^{it}$ from 0 to $2\pi$. since the contour is the unit circle we can ...
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### Choice of branch cut

I'm trying to rewrite an integral of the form $$\int_{-∞}^∞ (x^2+k^2)^{-s/2}e^{i(xc_1+kc_2)}dx$$ ($s>0,c_1,c_2,k\in\mathbb R$) in such a way that it is positive (for some $s$). To do so I'd like ...
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### The value of the integral $\int_{C} \dfrac{z^2+1 dz}{(z+1)(z+2)}$ where $C$ is $|z|=\dfrac{3}{2}$

The value of the integral $\int_{C} \dfrac{z^2+1 dz}{(z+1)(z+2)}$ where $C$ is $|z|=\dfrac{3}{2}$ is: a) $0$ b) $\pi i$ c)$2\pi i$ d) $4\pi i$ I have tried: Resolving them into factors by partial ...
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### Basic contour integral with Cauchy's Residue Theorem

So I'm looking at doing a basic contour integral using Cauchy's Residue Theorem. I feel I understand how to do this, and have gone over my work numerous times, yet the webwork system I'm doing this ...
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### Suppose that for each $a\in \mathbb{C}$ at least one coefficient of the Taylor's series $f$ about $a$ is zero. Show that $f$ is a polynomial.

Let $f:\mathbb{C}\rightarrow\mathbb{C}$ be a holomorphic function. Suppose that for each $a\in \mathbb{C}$ at least one coefficient of the Taylor's series $f$ about $a$ is zero. Show that $f$ is a ...
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### Complex contour integral Problem

Show that $$\oint_{|z|=1} \dfrac {(z+w)(z^{n-1})} {z-w}dz=0$$ using Residue calculus, where $n<0$ and $|w|<1$.
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### How to integrate |z| dz?

As the title says, how do we integrate $|z|dz$ on a straight line on the complex plane? Suppose that I've already known the parametrization. If it were on reals, we would break the integral down to ...
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### Evaluate $\int_{ - \infty}^{\infty} \frac{dx}{1+x^2}$ using complex integration

I'm trying to evaluate the real integral $$\int_{ - \infty}^{\infty} \frac{dx}{1+x^2}$$ Denote $\Gamma_{1}=\left[-R,R\right]\ \Gamma_{2}=Re^{it}$, for $t\in\left[0,\pi\right]$, and let $\gamma$ be a ...
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### VERIFICATION: Prove that $\int_{-\infty}^{\infty}\frac{1-b+x^{2}}{\left(1-b+x^{2}\right)^{2}+4bx^{2}}dx=\pi$ for $0<b<1$

I need some reassurance that what I did here actually shows what need to be shown. Please correct me if I'm wrong. In Donald Sarason's "Notes on complex function theory", this question appears at ...
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### Computing complex line integrals with antiderivatives

This question is a about complex line integrals. So far, I know that the following theorem is often useful: $\textbf{Theorem.}$ Assume $D \subseteq \mathbb{C}$ is open and $f:D \to \mathbb{C}$ has an ...
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I want to integrate the following :$$\int_{|z|=2} \frac{dz}{z^{2}-1}$$ in the positive direction. So my idea is two split the integral into a sum of two integral , something like $$\int_{|z|=2} \frac{... 1answer 29 views ### Solving an integral by Cauchy Formula I want to solve the integral$$\oint_{|z|=\frac{1}{2}}{\frac{e^{1-z}}{z^3(1-z)}dz}$$Its a long time ago that I solved such integrals. Is it just by definition of the line integral? Maybe someone can ... 0answers 61 views ### Find all possible values of the integral Find all possible values of \displaystyle I= \int_C \frac{dz}{1+z^2}, where C is a curve with initial point 0 and final point 1 that does not meet the poles of \dfrac{1} {1+z^2}. It looks ... 0answers 47 views ### Evaluating Improper Integrals with Residues - don't think I'm calculating the residues properly I have to evaluate the integrals \displaystyle \int_{-\infty}^{\infty}\frac{dx}{x^{2}+p^{2}}, for p > 0, and \displaystyle \int_{-\infty}^{\infty} \frac{dx}{(x^{2}+p^{2})^{2}}, for p > 0 ... 1answer 25 views ### Is integrating e^{iz^{2}}, along the real axis in the complex plane the same as integrating the riemann integral of e^{x^2}? In the title, z\in \mathbb{C} and x\in\mathbb{R}. More specific to my problem, I am hoping that \int_{0}^{R}e^{iz^{2}}dz=\int_{0}^{R}e^{x^{2}}dx. Maybe this is obvious but I want to make sure. 1answer 15 views ### Lower bound on the distance fom a point to the border of a region. I'm trying to prove the following result in complex analysis: If f  is an analytic and bijective function from the unit disc to an open connected region A then the distance from f(0) to the ... 2answers 33 views ### Mistake while evaluating the gaussian integral with imaginary term in exponent I am trying to evaluate the integral I=\int_0^\infty e^{-ix^2}\,dx as one component of evaluating a contour integral but I am dropping a factor of 1/2 and after checking my work many times, I ... 0answers 14 views ### Find \frac{d}{dt}[\bar{f(\gamma(t))}] in the context of of finding \frac{d}{dt}[|f(\gamma(t)|^2] I am trying to prove this exact problem, but more rigorously and without referencing analyticity: Ahlfors complex integration. I think the way to proceed is to try and find \Re(f'(\gamma(t))\bar{f(\... 1answer 34 views ### If f has a pole of order m at z_0 find the order of the pole of g(z) = \frac{f'(z)}{f(z)} at z_0. If f has a pole of order m at z_0 find the order of the pole of g(z) = \frac{f'(z)}{f(z)} at z_0. What is the coefficient of (z-z_o)^{-1} in the Laurent expansion for g(z). M Since f... 0answers 19 views ### Complex integral evaluation; I get the right answer, but one of my steps is a little fishy The integral is \int_{\gamma}\frac{1}{z^{2}-1}dz along the path \gamma(t)=2e^{ti},\;t\in[0,2\pi] Which I attempt to do by parts: \begin{equation*} \int_{\gamma}\frac{1}{z^{2}-1}dz= \frac{1}... 1answer 36 views ### Integrate \int_0^{2 \pi } \frac{1}{(a+b \cos^2 (x))^2} \, \mathrm{d}x [closed] I'm having a trouble with this integral expression:$$\int_0^{2 \pi } \frac{1}{(a+b \cos^2 (x))^2} \, \mathrm{d}x$$I want to solve to using residue but it seems hard. 0answers 108 views ### Schwarz's Lemma application Need help with this problem. Let f be an entire function such that |f'(z)| \leq |z| for all z. Show that f(z) = A+Bz^2, with |B| \leq \frac{1}{2}. My attempt: What I think is the way ... 2answers 33 views ### Prove this is a metric, what else should I consider? Let C_b(\mathbb{R}) be the space of the bounded continuous functions with values in \mathbb{C} defined in \mathbb{R} (f:\mathbb{R}\rightarrow\mathbb{C}) prove that: with x\in \mathbb{R}, h\in[... 0answers 12 views ### Equality involving holomorphic function and its series coefficients [duplicate] Function f(z)=a_0 + a_1z +a_2z^2+... convergences on \left\{z:|z|<R\right\}. Prove that for any 0<r<R$$\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{it})|^{2}dt= \sum_{n=0}^{\infty}|a_n|^{2}r^{...
The integral along the path $\gamma(t)=e^{2ti},\;t\in[0,2\pi]$ is $\begin{equation*} \int_{\gamma}\frac{1}{z^{2}-1}dz \end{equation*}$. I approached this like a real integral in the hopes things ...