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

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0
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2answers
27 views

Contour integration with logarithms

I'm having trouble calculating the below integral to get the right answer: $$\frac{1}{2\pi i}\int_\gamma \frac{3}{z-2}\; dz$$ where $\gamma$ is parametrised by $\gamma(t)=3e^{it}, t\in [0,2\pi]$. So ...
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1answer
18 views

Recommendations for tutorials specifically devoted to real integration using contour integral techniques.

Complex analysis, and in particular contour integrals and the residue theory have proved a very powerful tool in computing a large class of real function integrals which would be quite troublesome to ...
8
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3answers
137 views

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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4answers
56 views

Computing the residue of a rational function

The real integral I am trying to compute with residues/contour integration is $\int_{-\infty}^{\infty}\frac{x^2}{(x^2+a^2)^3} \,dx$ For $a$ positive and by using the complex integral ...
1
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0answers
36 views

Exchange series and integral in a complex context.

Let $a_{n},b_{n},c_{n}$ complex sequences and let $k>0$ a real parameter. Assuming that $$\sum_{n\geq1}\sum_{m\geq1}\left|\frac{a_{m}b_{n}}{c_{m+n+k}}\right|<\infty\tag{1} $$ if $k>1/2 $ ...
1
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1answer
24 views

Complex line integral over a square

Evaluate following complex line integral.Let $c=\{z|\max\{|\text{Re}(z)|,|\text{Im}(z)|\}=1\}$ be the square with orientation $+1$. Calculate $$\int_c \frac{z\ dz}{\cos(z)-1}$$ with ...
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0answers
24 views

Find the complex integral over a path

I have to integrate the complex function $$z+1/z$$ which is parameterized by $\gamma(t), 0 \le t\le 1$ and satisfies $Im\gamma(t) > 0$, $\gamma(0) = -4+i$ and $\gamma(1) = 6+2i$. Can I assume the ...
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0answers
41 views

complex analysis2 [closed]

let $f=u+iv$ be an entire function, and suppose that its imaginary part $v$ is non negative in the upper half plane but is equal to zero at all points of the real axis. (a) prove that ...
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1answer
45 views

Solution to nonlinear ODE with square root

How do I solve the following equation? $\dot{x}=\sqrt{x^{2}-\frac{2}{3}x^{3}}$ with $x(0)=0$? I'm guessing I have to work with $dt=\frac{dx}{\sqrt{x^{2}-\frac{2}{3}x^{3}}}$ and integrate in [0,t'] ...
2
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1answer
16 views

How do you compute a complex exterior derivative?

The context is deriving cauchy riemann equations using green's/stoke's theorem. The function is the complex function $f(x,y)=u(x,y)+iv(x,y)$ with associated one form $u(x,y)dx+iv(x,y)dy$. Here is my ...
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0answers
54 views

Showing that if $\int_C f(z) \, dz=0$ for every circle $C$ in $\mathbb{C}$, then $f$ is holomorphic.

I understand that this is the difficult direction of Morera's proof, applied to disks, rather than triangles. However, the trick of defining $$F(z)=\int_{\gamma(t)}f(w)\,dw, \text{ with } ...
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0answers
18 views

How to find integrals of the form $\int_{-r}^{+r}\frac{\gamma^{2}-1-y^{2}}{y-z\gamma} dy$

I'm trying to solve the above integral which will give a dispersion relation. Note: $\gamma$ and y are real but z is a complex variable.
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0answers
22 views

Integral problem with branch point from Physics

The question come from a Summation like this $${ \sum _{ { z=i\omega }_{ n } } { \frac { -\alpha E\pi }{ 4{ z }^{ 3 }\sqrt { -\alpha -z } } } }$$ I can use Cauchy theorem to transform it to a ...
3
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2answers
48 views

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 ...
6
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1answer
157 views

The complex function $\log(1+e^{iz})$

G.H. Hardy states the following: The function of the complex variable $z$ $$ e^{i p z} \log(1 \pm e^{iz})\frac{1}{z^{2} \pm \theta^{2}} = f(z), \ (-1 < p <1, \theta >0),$$ ...
2
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2answers
58 views

Using residues to compute complex integrals

Let $\phi:\mathbb{C}\backslash\{\pm i,\pm 2i\}\rightarrow\mathbb{C}$ with $\phi(z)=\frac{e^{iz}}{(z^2+1)^2(z^2+4)}$. How can I find $$\int_{-\infty}^\infty\frac{\cos x}{(x^2+1)^2(x^2+4)}dx$$ ...
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2answers
34 views

Prove the integral of $Re(z)$ around a simple closed curve is imaginary

If $θ$ is any simple closed curve in $\mathbb{C}$, prove that $\int_{θ}Re(z)dz$ is pure imaginary I have shown if $θ$ is the unit circle then the answer is $\frac{i}{2}$ but I'm struggling with a ...
4
votes
1answer
39 views

Show that if $ |f( \frac{1}{n}) | \leq \frac{1}{n!}$ then $0$ is an essential singularity

Given holomorphic non-constant function $f:D(0,1) \smallsetminus \{0\} \rightarrow \mathbb{C}$ so $\forall n=2,3,...:\ |f(\frac{1}{n})| \leq \frac{1}{n!}$ I need do show that $0$ is an essential ...
0
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1answer
17 views

complex integrating on half-unit circle

$\int_{|z| = \frac{1}{2}} \frac{e^{1/z}}{1-z}$ Singularities: $1-z = 1$ implies $z=1$ is a simple pole. It is not in $|z| = \frac{1}{2}$. Whereas, $e^{1/z}$ has an essential singularity inside $|z| ...
2
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0answers
14 views

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 ...
1
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1answer
20 views

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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0answers
45 views

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 ...
1
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1answer
19 views

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 ...
6
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2answers
90 views

Why is $\frac{1}{2\pi i}\int_{\gamma}\frac{f'(z)}{f(z)}\text{d}z$ an integer?

Let $f$ be holomorphic in an open set $\Omega \subset \mathbb{C}$ and $\gamma$ a closed curve in the interior of $\Omega$, at which $f$ never vanishes. Are these hypotheses enough to conclude that ...
1
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0answers
21 views

Evaluating a complex integral using Cauchys Integral Formula

I need to evaluate the following complex integral: $ \int_{\phi}\frac{z^3}{z^2+i} dz$ where $\phi$ is the circle centered at $0$ with radius $2$ I know that there is a singularity at $z = ...
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3answers
9k views

How to integrate complex exponential??

Consider $$\int^{\frac{1}{2}}_{-\frac{1}{2} } e^{i2\pi f} \,df = \int^{\frac{1}{2} }_{-\frac{1}{2} } \cos(2 \pi f)\, df$$ Why do we only look at the real part? What about the imaginary part ...
3
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1answer
46 views

How to integrate $\int_{0}^{1}\text{log}(\text{sin}(\pi x))\text{d}x$ using complex analysis

Here is exercise 9, chapter 3 from Stein & Shakarchi's Complex Analysis II: Show that: $$\int_{0}^{1}\text{log}(\text{sin}(\pi x))\text{d}x=-\text{log(2)}$$ [Hint: use a contour through the set ...
1
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2answers
59 views

Integrate $\int_{0}^{2\pi}\log|1-e^{i\theta}| d\theta$

Here is exercise 11, chapter 3 from Stein & Shakarchi's Complex Analysis II: Show that if $|a|<1$, then: $$\int_{0}^{2\pi}\log|1-ae^{i\theta}|\,d\theta = 0$$ Then, prove that the above ...
3
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1answer
116 views

$\prod_{n=1}^\infty\frac{|a_n|}{a_n}\frac{z-a_n}{\overline{a}_n z-1}$ convergence

If $a_n\in\mathbb{C}$ are complex number such that $|a_n|<1$ and $\sum_{n}(1-|a_n|)<\infty$, then I know that following Blaschke product define an analytic function on the open unit disk ...
0
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1answer
39 views

integral of $z^{-1}$

I know something similar might been asked, but I learn by example. The assignment is to find the integral of $z^{-1}$ in the square with points $(1+i) , (1-i), (-1-i), (-1+i)$ Do we rewrite $z$ as ...
1
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0answers
12 views

Cauchy-Riemann equation analogue but for the quaternions

given a function over the quaternions $$ U(x,y,z,t)+iV(x,y,z,t)+jW(x,y,z,t)+kR(x,y,z,t)=f(x,y,z,t) $$ what are the analogues of the Cauchy Riemann equation for the quaternionic plane so the function ...
0
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1answer
19 views

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 $|w|<1$.
3
votes
1answer
43 views

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 ...
2
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1answer
71 views

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 ...
3
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1answer
43 views

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 ...
0
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0answers
23 views

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 ...
0
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2answers
51 views

Complex integration problem via Cauchy's integral formula

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} ...
0
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1answer
28 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 ...
0
votes
0answers
55 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 ...
2
votes
0answers
45 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 $p > 0$ ...
0
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1answer
23 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. ...
0
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1answer
14 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 ...
1
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2answers
30 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 ...
1
vote
1answer
579 views

Proof of Laurent series co-efficients in Complex Residue

Am trying to see if there is any proof available for coefficients in Laurent series with regards to Residue in Complex Integration. The laurent series for a complex function is given by $$ f(z) = ...
2
votes
2answers
104 views

Let $U=B_1(0)$ and $ f(z)=\sum_{n=1}^{\infty}2^{-n^2}z^{2^n}$. Show that $f$ has not analytic extensions to any open set $G$ with $U\subsetneq G$.

Let $U=B_1(0)$ and $$f:U \rightarrow \mathbb{C},\qquad f(z)=\sum_{n=1}^{\infty}2^{-n^2}z^{2^n}.$$ Show that $f$ has not analytic extensions to any open set $G$ with $U\subsetneq G$. Remark: Suposse ...
0
votes
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 ...
1
vote
1answer
33 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 ...
4
votes
2answers
432 views

Gaussian integral with a shift in the complex plane

$$ \int_{-\infty}^\infty e^{-(x+ia)^2} \text{d}x $$ where $a\in \mathbb{R}$. I don't know where to start but have reasons to believe the answer is $\sqrt{\pi}$. Namely $\int_{-\infty}^\infty ...
0
votes
1answer
57 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 ...
0
votes
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= ...