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

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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 ...
2
votes
0answers
17 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
vote
1answer
21 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 ...
0
votes
0answers
49 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
vote
1answer
23 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
votes
2answers
96 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
vote
0answers
22 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 = \frac{1-i}...
3
votes
1answer
47 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
vote
2answers
63 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 ...
1
vote
0answers
15 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
votes
1answer
41 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 $a+...
0
votes
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
47 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 ...
3
votes
1answer
44 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 ...
2
votes
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 ...
0
votes
0answers
25 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
votes
2answers
54 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} \frac{...
0
votes
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 ...
0
votes
0answers
59 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
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 $p > 0$ ...
0
votes
1answer
24 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
votes
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
vote
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 ...
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 $\Re(f'(\gamma(t))\bar{f(\...
1
vote
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 $f$...
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= \frac{1}...
-2
votes
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.
0
votes
0answers
106 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
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[...
0
votes
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^{...
2
votes
0answers
97 views

Difficulty evaluating complex integral

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 ...
3
votes
1answer
118 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
votes
1answer
23 views

Computing Integral without the Residue Theorem

I have to evaluate this integral, $\int_{C} \frac{z\cos(z) dz}{(z^2 + 5)(z+2)(z-2)},$ where $C$ is $x^2 + y^2 = 3$. I discovered that none of the singularities are in $C$. Therefore, I conclude $$\...
1
vote
0answers
30 views

Determine the number of zeros using the Argument Principle

I'm tasked with finding the zeros of $f(z)=z^3+1$ that lie inside the first quadrant using the Argument Principle, which I have simplified below: $$N=\frac{1}{2\pi}[arg(f(z))]_C$$ where N represents ...
0
votes
1answer
20 views

Evaluating a complex integral on a circle

I have the function $ f(z) = \frac{z^3}{z^2+i} $ and I'm trying to calculate the integral: $ \int_{C(0;2)} f(z)dz$ where $C$ is the circle centered at the origin with radius $2$. Could someone ...
0
votes
1answer
20 views

Can we multiply by -1 when flipping limits for integration over the complex plane?

I have this integral that goes from $ci$ to $0$, along the imaginary axis, and I'm finding that I seem to get different answers if I (1) Parametrize and evaluate it. (2) Multiply the whole thing by ...
1
vote
0answers
33 views

Integrating a complex function

I have the integral $ \int_{C(0;2)} \frac{e^z}{z^3+9}dz $ I was told I could use Cauchy's Integral formula but I'm still stuck, I'm not sure how to apply it. Any help would be great!
3
votes
3answers
43 views

Improper integral with complex limits

I would like to compute an integral of the form ($a,b \neq 0$) $$\int_{-\infty}^{\infty} e^{-(ax+ib)^2} dx = \frac{1}{a} \int_{-\infty+ib}^{\infty+ib} e^{-z^2} dz$$ where we made the substitution $z ...
2
votes
1answer
58 views

What do $\int_{-1}^1\frac{dx}{2x+1-2i}$ and $\frac12\log(2x+1-2i)$ mean?

Suppose we want to evaluate $$I=\oint_C\frac{dz}{z+\frac12}$$ where $C$ is the unit square with diagonal corners at $-1-i$ and $1+i$. If we let $z:=re^{it}-\frac12$, then $$I=\int_0^{2\pi}\frac{2rie^{...
1
vote
2answers
31 views

Determining possible values of a complex integral

I'm looking at the complex integral: $ \frac{1}{2 \pi i} \int_{C(0;r)}\frac{cos(z)}{z^{n+1}}dz $ and I'm wondering how I'd go about determining all the possible values of this for $ r>0 $ and $ ...
2
votes
2answers
105 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 ...
1
vote
0answers
23 views

Complex Integral evaluation.

I am completely stuck with the evaluation of the following integral : $I = \int_{-\infty}^{\infty} \frac{\sinh(x)}{\sinh(ax)}dx, a>1.$ I am supposed to use a rectangle, such that the bounds are $-...
1
vote
0answers
58 views

Can I switch to polar coordinates if my function has complex poles?

You can think this of the following as a 3d QFT where we try to calculate the self-energy of two fields. $I$ is a this external self-energy and let us assume it does not depend on the loop momenta ...
5
votes
3answers
109 views

Prove that $\oint_{|z|=r} {dz \over P(z)} = 0$

I got stuck on this problem, hope anyone can give me some hints to go on solving this: P is a polynomial with degree greater than 1 and all the roots of $P$ in complex plane are in the disk B: $|z|...
6
votes
3answers
122 views

If $f$ is entire and $\lim_{z\to\infty} \frac{f(z)}{z} = 0$ show that $f$ is constant

I'm learning about complex analysis and need to verify my work to this problem since my textbook does not provide any solution: If $f$ is entire and $\lim_{z\to\infty} \frac{f(z)}{z} = 0$ show ...
1
vote
0answers
26 views

Hyperelliptic Curves and Period Integrals

I've been thinking about the period integrals on a hyperelliptic curve of the form $y^{2}=F(x)$, where $F(x)$ is a polynomial of degree-$2n$ with complex coefficients. Of course, a hyperelliptic ...
1
vote
0answers
23 views

Integral around a square in the complex plane

Let $f(z)$ be any continuous function defined in the complex plane with the property that $$\bigg|\int_{R_n}f(w)dw\bigg|\leq n^2\log(n),$$ for any $n>1$ and any square $R_n$ with side length $n$. ...
0
votes
0answers
28 views

Line integration of $\tanh(z)$

I have to evaluate the following integral $\qquad \qquad \int_C \tanh (z)dz $ where $C$ is the unit circle in the anti-clockwise direction. Is there any substitution that will work here, I have ...
1
vote
0answers
44 views

Show that $\int_{C} \frac{e^{az}}{\sin(\pi z)} \,dz \rightarrow 0 \quad as \quad R \rightarrow \infty$

Given the straight line in the complex plane: $b+iR$ to $b+1+iR$ where $0<b<1$ and $|Im(a)|<\pi$, show the following: $$\int_{C} \frac{e^{az}}{\sin(\pi z)} \,dz \rightarrow 0 \quad as \quad ...
0
votes
0answers
51 views

How to resolve the integral of exponential function with rational power

I want to resolve the integral of exponential function which has the following form $$\int\limits_a^\infty {{t^{\nu - 1}}{e^{ - \frac{\beta }{t} - \gamma t}}dt}$$ where, $a$ is a constant. I found ...