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

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Integral of ratio of complex polynomials

Let $p(z),q(z) \in \mathbb{C}[z]$ two polynomials with coefficients in $\mathbb{C}$ s.t. $deg(p) = m$, $deg(q) = n$ and $n \ge m +2$. I need to show that $$ \lim_{R \to \infty} \int_{|z| = R} ...
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1answer
18 views

Upper Bound on Complex Line Integral

I'm working through the second edition of Complex Variables by Stephen Fisher, and reached a proof involving the upper bound of line integrals, namely $$ \left| \int_\gamma u(z)\;dz \right| \leq ...
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2answers
39 views

Complex analysis proof about $|f(z)|$

I have to prove the following and have absolutely no idea where to start: If $f$ is holomorphic in $|z|>R$ and its limit at $\infty$ is $0$, then $\exists \; m \in \mathbb{N}$ such that $|f(z)| ...
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1answer
45 views

Contour integral of $\frac{1}{\sqrt z}$ with branch cut

I am a physicist who usually doesn't need to care about the fact that square root is not single-valued on the complex plane. But I would like to give a meaning to and compute the contour integrals : ...
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2answers
68 views

geometrical interpretation of a line integral issue

I was wondering : if the geometrical interpretation of a line integral is that the line integral gives the area under the function along a path, then why the line integral is equal to zero when the ...
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2answers
24 views

Prove that: $\oint_c{\dfrac{1}{z^{2n+1}}(z^2+1)^{2n}}=\binom {2n} {n} 2\pi i$

In a mathematical methods problem, where the $c$ is a the unit circle around the origin and in counterclockwise, I need to use a step that I'm not so sure about (Because I don't know how to develop ...
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0answers
21 views

How do you find the volume of a function rotated about the x axis along it's derivative? [closed]

So when you rotate a function, it is usually vertically. How do you rotate it around it's derivative, assuming that volume/area can overlap?
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1answer
107 views

Global Residue Theorem in CP^2.

Consider the following meromorphic form defined on $\mathbb{C}P^2$: Be $\phi_{1,2}$ the chart given by $\phi_{1,2}(Z_1,Z_2) = (1,Z_1,Z_2)$, in these coordinates define $\omega= \frac{1}{Z_1 Z_2} dZ_1 ...
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28 views

Proof of Sokhotski-Plemelj theorem

Sokhotski-Plemelj theorem states $$ \phi_i(z)=\frac{1}{2\pi i}\mathcal{P}\int_C\frac{\varphi(\zeta) \,d\zeta}{\zeta-z}+\frac{1}{2}\varphi(z), \, \\ \phi_e(z)=\frac{1}{2\pi ...
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2answers
43 views

how to solve this complex exponential integration ??

During exercising and example of Fourier Series , I encountered with an integration : $$ \frac{E\omega_o}{4\pi j}\int_{0}^{\frac{\pi}{\omega_o}}\Big[e^{-j\omega_o (n-1)t}-e^{-j\omega_o ...
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1answer
24 views

A question about complex integration

Let $z_0\in \Bbb C$ and $R>0$. Let $f:B(z_0,R)\to\Bbb C$ be a complex function such that $f=u+iv$ (where $B(z_0,R)$ is the open disc centered at $z_0$ with radius $R$). If $u$ and $v$ have ...
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3answers
60 views

Evaluatig: $\int_{0}^{\infty}{e^{ax^2}\cos(bx)dx}$

Evaluatig: $$\int_{0}^{\infty}{e^{ax^2}\cos(bx)dx}$$ Where $a, b\in \mathbb R^+$ What i have done: Because $\cos(bx)=\Re(e^{ibx})$, we can note that: ...
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1answer
21 views

Problem in computing complex integrals for fourier transform

This is from a problem set of open course 8.02 by MIT OCW. I am not able to understand how the integral was solved. I have basic knowledge of Fourier transformation, and the Dirac delta function ...
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54 views

Integrals of the type $f'(z)/f(z)$

I am having trouble understanding integrals of the form: $$\int_\gamma\frac{f'(z)}{f(z)}\,{\rm d}z$$I am aware that there are problems with the complex logarithm, and we have the formula: ...
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2answers
65 views

Evaluating: $\int_{-\infty}^{\infty}{\frac{1}{\cosh(kx)}dx}$

How can you integrate: $$\int_{-\infty}^{\infty}{\dfrac{1}{\cosh(kx)}dx}$$ I know that: ...
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2answers
75 views

Integrate along the vertical strip

I want to show that some integration with vertical line is bounded. function $f(\mu)$ is given by $$ f(\mu)=A^{-\sqrt{\mu}} \frac{(B_1-\sqrt\mu)}{(B_2-\sqrt\mu)(B_3+\sqrt\mu)} $$ where $f$ is defined ...
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24 views

Contour integration of logarithm: $g(\omega) \log[1 - \chi(q,\omega)]$

I'm trying to calculate the integral $$ \frac{1}{2\pi i} \int_\mathcal{C} g(\omega) \log[1 - \chi(q,\omega)], $$ where $g(\omega) = (e^{\beta \omega}-1)^{-1}$ has an infinite number of evenly spaced ...
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26 views

Complex integral difficulty

We need some sort of analytic expression for the integral: $$\int^\infty _{-\infty} \frac{\sqrt{\alpha^2 + 1}}{(\mathrm{i}\alpha)^{\frac{3}{4}}}\mathrm{e}^{\mathrm{i}\alpha \chi} \mathrm{d}\alpha$$ ...
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1answer
73 views

Using The Riemann Zeta Functional Equation

Riemann was able to establish the following link between the Riemann zeta function and the weighted prime counting function $J(x)$. $$\ln(\zeta(s))=s\int_1^\infty J(x)x^{-s-1}dx$$ Using the Mellin ...
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36 views

How to calculate this residue which has a pole of order n-r?

So I have this complex integral: $$ \oint \frac{dz}{2\pi}\frac{e^{iz(br-(n-r)a)}}{\left(1-(1-q)e^{-ik_{1}-iza}\right)^{n-r}}$$ b,r,q,a,n are all constants in this context. However I'm not entirely ...
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1answer
33 views

Integration along a keyhole

(H. Priestley complex Analysis Chapter 7 Exercise 9) Suppose $f$ is holomorphic inside and on $\gamma(0,1)$. By integration around the usual keyhole like this one : Integration of $\ln $ around a ...
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1answer
89 views

Verifying the complex integral: $\int_0^{\infty}\frac{\cos{ax}}{1+x^4}dx$

Verifying the integral: $$\int_0^{\infty}\dfrac{\cos{ax}}{1+x^4}dx$$ I started considering: $$\cos{x}=\dfrac{e^{ix}+e^{-ix}}{2}\implies \cos{ax}=\dfrac{e^{iax}+e^{-iax}}{2}?$$ So: ...
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1answer
52 views

Complex integral with Fourier

So, I've been scratching my head over this for the whole day. I'm trying to solve the following integral $$\int^\infty_{-\infty} \frac{e^{i \alpha (X-\xi)}}{\sqrt{\alpha^2+ \beta^2 }} \, d\alpha$$ ...
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2answers
44 views

Solving $\frac{df}{dt}=\frac{i\cdot f}{|f|}$ where $f: \mathbb{R^+} \mapsto \mathbb{C}$

I've never seen a complex DE before, so this is uncharted territory for me. But it's separable so it's easy to turn it into an integral: $$f(t) = \int_0^t\frac{i \cdot f}{|f|} dt$$ Can this be solved? ...
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38 views

Show the integration with a complex variable

I want to show that there exists inverse Laplace transform, $f(t)$ of the function $F(\lambda)$. In other word, given $F(\lambda)$, existence of function $f(t)$ such that $$ ...
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1answer
64 views

$\int_0^{\infty} \exp(i(t-\alpha)^2) dt$

It's fairly straight forward to show that $$ \int_0^{\infty} \exp(it^2) dt = \frac{\sqrt{\pi}}{2}\exp\left(i\frac{\pi}{4}\right) $$ via complex contour integration over a contour shaped like a piece ...
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1answer
53 views

Integral with complex variable

I want to compute $$ \int_{-\infty}^{\infty} \frac{1}{\sqrt{x+yi +2}} dy $$ where $i$ is the imaginary number. How to compute this integral??
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32 views

Moving limit inside a contour integral

I'm trying to compute this integral as part of a larger problem I'm working on. I'm trying to solve the integral $\int_0^\infty \frac{\sin(x)}{x}dx$ and to do it I'm using the method where you ...
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1answer
35 views

Complex integral using Residue Theorem with a regularised pole

I need to prove the following integral computation by applying the residue theorem: $$\int_{-\infty}^{+\infty}ds \ e^{-i\Omega ...
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1answer
36 views

Complex integration: normally on a closed contour?

I have been studying complex integration for a few months now, and it seems my textbook mostly considers integration on closed contours. Is there no interest in integration on non-closed contours ?
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3answers
55 views

Complex integration with trigonometric and logarithm

Show that $\int_0^{2\pi}\log\sin^22\theta dx=4\int_0^\pi\log\sin \theta d\theta=-4\pi \log2$ I did $$\int_0^{2\pi}\log\sin^22\theta d\theta=4\int_0^{\frac{\pi}{4}}\log\sin^22\theta d\theta$$ ...
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2answers
171 views

Complex Integration with trignometric function

Verify that $\int_0^{\frac{\pi}{2}}\frac{d\theta}{a+\sin^2\theta}=\frac{\pi}{2[(a(a+1)]^\frac{1}{2}}$ I know that $\sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{2}$ then I did ...
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2answers
41 views

Improper integral and residues

Evaluate $\int_0^\infty \frac{dx}{x^4+1}$ By the residue theorem $$\int_{-R}^Rf(x)dx+\int_{C_R}dz=2\pi i\sum Res(f,z_i)$$ but I have problems to evaluate it because $$z^4+1=0\Rightarrow ...
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1answer
67 views

Poles on the curve

Say I have this integral: $$\oint_\gamma f(z)\,{\rm d}z,$$and $f$ has a pole on $\gamma$. I understand that we "cut around" the pole with an arc of radius $\epsilon$ and then make $\epsilon \to 0$. ...
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2answers
38 views

Bounding a complex integral over a square

I'm solving the following exercise: Use the estimate lemma to prove that $$\left|\oint_\gamma \frac{z-2}{z-3}\,{\rm d}z\right| \leq 4\sqrt{10},$$where $\gamma$ is the square with vertices $\pm 1 ...
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1answer
30 views

Evaluate the improper integral with residues

Evaluate $\displaystyle\int_0^\infty\frac{dx}{x^2+1}$ I have that $z_0=i$ and $z_1=-i$ are singularity points but just $z_0=i$ is in the upper plane then ...
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1answer
19 views

Improper integrals and residues

I'm already read Conway, Churchill and Marsden but I'm still with doubts when it comes to improper integrals. Where come from this relation ...
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1answer
91 views

Calculation of an improper integral in the context of complex functions [duplicate]

I am facing the following improper integral: $$\int_0^\infty \frac{x^5\sin x}{(1+x^2)^3}dx.$$ Clearly the expression under the integral is a meromorphic function analytic on the nonnegative part of ...
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62 views

Using residue theorem find $\int_{\theta-\pi}^{\theta-2\pi}1/(r^2+1-2r\cos u)$

Find the value of $I$, using residue theorem. $$ I= \int_{\theta-\pi}^{\theta-2\pi} \frac{du}{r^2+1-2r\cos(u)} - \int_{\theta}^{\theta-\pi} \frac{du}{r^2+1-2r\cos(u)} $$ with $r<1$ and ...
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50 views

Integration using residue theorem

Can you find, using residue theorem, ($\epsilon >0$), the value of this integral ($I$)? \begin{equation} I=\lim_{\epsilon->0^{+}} \int_{-\infty}^{\infty} \frac{dw}{w+i\epsilon} \end{equation} ...
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1answer
62 views

How to calculate $\int_{-\infty}^{\infty} x^2 \cos(ax)e^{-x^2}dx$ using Cauchy's theorem

I want to calculate the integral $$\int_{-\infty}^{\infty} x^2 \cos(ax)e^{-x^2}dx$$ using complex analysis. I have a hint to look at the rectangle $(-R,0), (R,0), (R,h), (-R,h)$ for a certain ...
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3answers
60 views

Why $\lim_{R\to\infty}\int_{0}^{\pi}\sin(R^{2}e^{2i\theta})iRe^{i\theta}\:\mathrm{d}\theta = -\sqrt{\frac{\pi}{2}}$

This is a short question, but I'm simply not sure where to start, I know by Jordan's Lemma that the integral is not $0$, but I only know the below result due to Mathematica. ...
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2answers
219 views

When is this integral convergent?

Let $a \in \mathbb{C}$. Consider the integral $$\int_{-\infty}^{+\infty} \frac{e^{-ax}}{1 + e^x} dx,$$ for which values of $a$ is this convergent? Is it right to say that $a$ has to be purely ...
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1answer
27 views

Complex contour integral of fraction of polynomials

Let $n \in \mathbb{N}_0$ and set $p(z) = z^n + a_1 z^{n-1} + \cdots$ and $q(z) = z^{n+1} + b_1 z^{n} + \cdots$ to be two monic complex polynomials with no common zeros. I want to prove that ...
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1answer
26 views

Polar form of Taylor's theorem for complex analysis

(H.Priestley Exercise 5.7) Let $f \in H(D(0,R))$ and $f=\sum_{n=0}^{\infty} c_{n} z^n$ Using the integral formula for $c_n$ and the fact that $\int_\gamma f(z)z^{n-1}dz=0 \quad\forall n\ge1$ Show ...
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45 views

Integrate complex function over $\mathbb{C}^2$

I have a question in mind and I would appreciate your help. Usually in complex analysis we consider integrals of the form $\int_\gamma f(z) dz$ where $\gamma $ is a contour and ...
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2answers
43 views

Results for values of the residues in the Residue Theorem

If the sum of the residue is $0$, what can I conclude: that the value of the integral is $0$ or that the integral diverges? If the residue tends to $\infty$, should I conclude that the integral ...
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5answers
276 views

Evaluate the complex integral of function

Use the residue theorem to evaluate $\int_\gamma \frac{z^5}{1-z^3}dz$ where $\gamma$ is the circle $|z|=2$. I have that $z_0=1$ is a singularity point and taking $g(z)=z^5$ and $h(z)=1-z^3$ and ...
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1answer
56 views

Integrate $\int \log(1+2m\cos x+m^2) dx $

How do I integrate $\int \log(1+2m\cos x+m^2) dx $ ? I tried 2 things. First, I tried complex numbers. Putting $\cos x = \frac{e^{ix}+e^{-ix}}{2} $ which led me to $\int \log((me^{ix} ...
3
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2answers
80 views

How to integrate $e^{-\cos(\theta)}\cos(\theta + \sin(\theta))$

I am struggling to find a way to evaluate the following real integral: $$\int_{0}^{2\pi}e^{-\cos(\theta)}\cos(\theta + \sin(\theta))\:\mathrm{d}\theta$$ The exercise started by asking me to ...