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

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0
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1answer
10 views

find integral $\int_{1}^{-1} \sin\left(x^3\right) dx$

$$\int_{1}^{-1} \sin\left(x^3\right) dx$$ so I know the result is 0 since above function is odd. But how to compute this integral?
0
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0answers
13 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
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1answer
26 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 ...
0
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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= ...
-2
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1answer
33 views

Integrate $\int_0^{2 \pi } \frac{1}{(a+b \cos^2 (x))^2} \, \mathrm{d}x$ [on hold]

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
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1answer
38 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
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2answers
32 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}, ...
0
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0answers
11 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= ...
2
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0answers
94 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 ...
0
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0answers
22 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
20 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
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0answers
25 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
16 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
15 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
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0answers
32 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
51 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 ...
1
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2answers
29 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
99 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
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0answers
51 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
100 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: ...
6
votes
3answers
86 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
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0answers
18 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
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0answers
20 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
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0answers
25 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
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0answers
35 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
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0answers
35 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 ...
2
votes
2answers
51 views

Show that $\int_{0}^{\infty} \frac{\cosh(ax)}{\cosh(\pi x)} dx=\frac{1}{2}\sec(\frac{a}{2})$ using Residue Calculus

Show that the following expression is true $$\int_{0}^{\infty} \frac{\cosh(ax)}{\cosh(\pi x)} dx=\frac{1}{2}\sec(\frac{a}{2})$$ Edit: I forgot to mention that $|a|<\pi$ Specifically, using ...
0
votes
1answer
34 views

Using Residue Theory to evaluate $\int_{0}^{\infty} \frac{x^3sin(kx)}{x^4+a^4} dx$

I'm having difficulty evaluating the following integral using residue theory, and would love some advice on proceeding. Below I develop my approach to the problem: $$\int_{0}^{\infty} ...
2
votes
1answer
18 views

Coefficients of a certain Laurent series

If $$e^{t(z - 1/z)/2} = \sum_{n \in \mathbb{Z}} J_n(t)z^n$$ is the Laurent expansion in $\mathbb{C}^*$, how can I show that $$ J_{-n}(t) = (-1)^n J_n(t)$$ Any help would be appreciated.
1
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0answers
34 views

Theory of complex integration

I don't know if this is a pedestrian question or not, but here goes. I am working through Ahlfors' complex analysis, which has been a great (and challenging) text to get a grasp of the basics of the ...
0
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0answers
18 views

Complex Line Integral of $z^n$

I'm looking at an exercise in a complex analysis textbook and it asks the following: Calculate the integral $\qquad \qquad \int_{C} z^{n} dz$ where $C$ is the right half of the unit circle, in the ...
6
votes
1answer
75 views

Evaluating an indefinite integral using complex analysis

Using tools from complex analysis, I have to prove that $$ \int_0^{\infty} \frac{\ln x}{(x^2 + 1)^2}\,dx = - \frac{\pi}{4}.$$ But I'm not really sure where I should start. Any help would be ...
0
votes
1answer
29 views

Complex integral with singularity outside the circle

I try to solve for $z\in\mathbb{C}\setminus\bar{\mathbb{E}}$ the following integrals (without the residue theorem): a) $$\int\limits_{\partial\mathbb{E}}\frac{1}{\theta-z} d\theta$$ b) ...
-2
votes
1answer
40 views

Show that there is no nth roots in $U$.

Let $U\subseteq\mathbb{C}\setminus\left\{0\right\}$ be an open set and suppose that there is a path $\gamma$ in $U$ such that $\mbox{Ind}_{\gamma}(0)=1$. Show that there is no nth roots in $U$. ...
0
votes
0answers
37 views

Definite integral of error function times exponential and Gaussian

I am looking for the solution of the following integral $$\int_{-\infty}^\infty\text{d}x\,\text{erf}(x)e^{-a x^2-bx}$$ where $a$ is real but $b$ is in general complex. For the case of $b$ being real ...
0
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0answers
32 views

Let $R$ rational function. Show that $\int_0^\infty R(x)dx=-\sum_{w\in \widetilde{\mathbb{C}}}\mbox{Res}_{w}(R(z)\ln(z)).$

Let $R=\frac{P}{Q}$ where $P$ and $Q$ are polynomials such that $Q(x)\neq 0$ for all $x\geq 0$. Suppose that the degree of $Q$ exceeds that of $P$ be at least 2. Show that $$\int_0^\infty ...
1
vote
1answer
13 views

Differentiate $v(x,y)=-\int_0^x u_y(t,0)dt+\int_0^y u_x (x,t) dt$ w.r.t. $x,y$ to prove complex differentiability

The domain is an open unit box (if required) and $u$ is harmonic, $v$ is harmonic conjugate defined below. Prove Complex Differentiabilty Diffirentiate with respect to $x$ and $y$: ...
1
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1answer
21 views

Simplifying two complex integrals

I'm working through a complex analysis problem and I'm wondering whether there is a straightforward way to simplify these integrals any further (or compute them): $$ \int_0^1 ...
1
vote
1answer
64 views

Calculating a certain Laurent series (two parts)

I'm trying to solve the following problem: Let $t \in \mathbb{R}$ be fixed and let $$ e^{[\frac{t(z - 1/z)}{2}]} = \sum_{n \in \mathbb{Z}} J_n(t)z^n$$ be the Laurent expansion in $\mathbb{C}$*. ...
0
votes
0answers
46 views

Assistance with complex integration

I'm a bit stuck on a question on complex integrals I have no idea how to go about solving it and would be really grateful if someone could help. The question is: 2. Let $$\ F(x,y,z)=(2xy+4xz)i ...
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votes
0answers
24 views

Limit of a complex path integral

I cannot make any progress on the following problem: Let $R>\sqrt{2}$ and define $C_R :$ path parameterized by $z(t) = Re^{it}$ with $t \in [0,\pi]$ (The semi circle radius $R$ centred at 0). ...
2
votes
1answer
27 views

Showing a complex integral has zero real part

I need to show that $\int_{\varphi}^{\space} \bar{z} \space dz $ has zero real part, for all smooth closed paths $\varphi$. I've tested this with the example $\int_{C(0;1)}^{\space} \bar{z} \space dz ...
0
votes
1answer
24 views

How do I show the following modification of the Counting formula of zeros and poles?

Let $U\subset \mathbb{C}$ be an open and connected set, $g: U\rightarrow \mathbb{C}$ holomorphic function, $f$ meromorphic function in $U$ with zeros in $z_{1},z_{2},\ldots,z_{n}$ and poles in ...
0
votes
1answer
29 views

Integration of complex exponential function over $\mathbb C$

Find the limit $$\lim_{z \to \infty}\int_{\mathbb C}|w|e^{-|z-w|^2}dA(w) $$ where A is area measure such that dA=rdrd$\theta$ Please help me, I did four page computation by changing to polar ...
0
votes
1answer
18 views

$M\mathcal{l}$-inequality: Putting a bound on the integrand in $\int_{|z|=R}\frac{f(z)}{(z-a)(z-b)}dz$

This question is related to another question here regarding an alternate proof of Liouville's Theorem using Cauchy's Integral Formula. In trying to apply the $M\mathcal{l}-$inequality to the ...
0
votes
1answer
43 views

Show that Function is Path Dependent

Hi there here is a question that I have wrecking my head for quite sometime I hope that the experts could assist me! Show that I = $\int_c (x^2y dx + 2xy^2 dy)$ is path dependent in the xy-plane. I ...
2
votes
1answer
39 views

Show that the Cauchy principal value of $ \int_{-\infty}^{\infty}\frac{P(x)}{Q(x)}dx$ exists when $\mbox{deg}(Q)=\mbox{deg}(P)+1$.

Let $R=\frac{P}{Q}$ where $P$ and $Q$ are polynomials such that $Q$ has not zeros in $\mathbb{R}$ and $\mbox{deg}(Q)=\mbox{deg}(P)+1$. Show that the Cauchy principal value of $ ...
2
votes
2answers
38 views

Using Cauchy's Integral Formula on a simple closed contour

Hello I'm trying to evaluate the following two integrals where C is the unit circle centered at origin, but I encounter the same problem in both of them and can't think of what to do. $$1)\ \oint_{C} ...