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0answers
86 views

Complex integration from zero to infinity at different directions

The problem is computing the integral expression $$f(t)=\int_0^{\infty}\lambda^{t-1}e^{-\lambda z}d\lambda$$ where z is a complex variable with $Re(z)\ge 0$. Is it correct to substitute $w=\lambda z$ ...
1
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1answer
69 views

How can I use Cauchy formula to this Integral?

$$∫ \frac{5z^2 - 3z + 2}{(z-1)^3} dz$$ and the contour is any closed simple curve involving z=1 (sorry, I forgot to write this information) Need to solve it using Cauchy Integral formula Can anyone ...
0
votes
1answer
51 views

Prove that $f'(a) = \frac{1}{2\pi}\int_0^{2\pi}e^{-i\theta}f(a+e^{i\theta})d\theta$

I know this is to be derived from Gauss' Mean Value Theorem, but I can't get the $e^{-i\theta}$. Where am I going wrong? $f'(a) = \lim_{h \to 0}\frac{f(a+h) - f(a)}{h} = \lim_{h\to ...
0
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1answer
82 views

Find all functions $f(z)$ that are analytic everywhere in the entire complex plane that satisfy $f(2-i)=4i$

This question is apparently related to Cauchy's Integral Formula and related theorems, but I honestly don't know how to start, other than potentially saying that $f(z)$ is every function such that ...
0
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4answers
130 views

Evaluate $\oint_C\frac{dz}{z-2}$ around the circle $|z-2| = 4$

I don't completely understand how to approach these questions. I suppose the notation $\oint_C$ is something I'm not used to. So far, I have $\oint_C\frac{dz}{z-2} = \log(z-2)$. From here, I suppose ...
0
votes
1answer
98 views

Evaluate $\oint_C |z|^2 dz$ around the square with vertices at $(0,0), (1,0), (1,1), (0,1)$

I don't think I quite understand how to go about this. My solution so far: $\oint_C |z|^2 dz = \oint_C (x^2 + y^2)dz = \oint_C (x^2 + y^2) d(x+iy) = \oint_C x^2 + y^2 dx + i\oint_Cx^2+y^2dy$. ...
4
votes
2answers
203 views

Why do we use only upper half plane to do Residue Integration?

In the class of mathematics my professor showed me that "how to use the residue integration method?". And i doubt at almost the last step that professor did. he made a contour over the upper half ...
1
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0answers
91 views

Analytical Continuation

I have problem understanding the analytical continuation of the following complex integral. I even have the solution but can not figure out the intermediate steps. I would appreciate it if anyone ...
6
votes
3answers
132 views

Show that $\int_0^\infty e^{-x^2} \sin{2xb}\, dx =e^{-b^2}\int_0^b e^{x^2} \, dx, \, b>0, $?

Show that $$\int_0^\infty e^{-x^2} \sin{2xb}\, dx =e^{-b^2}\int_0^b e^{x^2} \, dx, \, b>0, $$ I need help. I did the following steps: Apply Cauchy's Theorem, being $\varphi (x) = e^{-z^2}$ analytic ...
0
votes
1answer
101 views

Complex double integral

I'm having trouble calculating following (complex) integral. $$\int_D z^n dA$$ where $D=\{ z \in \mathbb{C} \mid \lvert z \rvert \leq 1 \}$. I know how to calculate complex (line) integrals and real ...
1
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0answers
40 views

Integral of the determinant of the Jacobian in complex variables?

Let $f$ be a function that maps a disc $D$ onto a unit square $A$ such that $f(D) = A$. $f$ is holomorphic and one-to-one. Why does $\int_D|f'(x+iy)|^2 \, dx \, dy = 1$?
0
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0answers
22 views

Integration of error function and exponenial with none trivial integration limit

I would like to know the following integration $$\int_b^\infty \operatorname {erfc(x)}e^{x^2+iax}$$ which seems integrable as the integrand goes to $\frac{e^{ix}}{x}$ for large x. All reference ...
0
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1answer
81 views

A taylor series for an integral with a singularity

Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a smooth function with a single root we call $y_*$. Then define \begin{equation} F_{\delta}:=\int^{y_*+ \delta/2}_{y_*- \delta/2} 1/f(y)dy ...
2
votes
2answers
54 views

Does $\int_a^b f'(\gamma(t))\gamma'(t)\,dt$ depend on the path for meromorphic functions?

Given a meromorphic function $f:\mathbb C\to\mathbb C$ and a smooth curve $\gamma:[a,b]\to\Gamma\subset\mathbb C$ with $\gamma(a)\neq\gamma(b)$, I am tempted to think the fundamental theorem of ...
0
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1answer
101 views

Complex integral using residue theorem

I have $$\int_{|z|=1} z^m \sin\left(\frac{1}{z}\right)~dz,$$ for $m = 0,1,2,\dots$ I know that there is a singularity at $z=0$, and this singularity is within the curve, thus the residue theorem ...
5
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1answer
111 views

How to calculate this complex integral $\int_0^\infty \frac{1}{q+i}e^{-(q+b)^2}\text{d}q$? (Please Help)

I want to carry out the following integration $$\int_0^\infty \frac{1}{q+i}e^{-(q+b)^2}\text{d}q$$ which is trivial if calculated numerically with any value for b. But I really need to get an ...
4
votes
1answer
113 views

The substitution $y = ix$

Given $\int_0^\infty f(x) dx$, under what conditions would the substitution $y = ix$ not change the limits of integration, so to speak? Since setting $y = ix$ changes the range of integration to a ...
0
votes
2answers
64 views

Change of variable x=iy in improper integral

I'm trying to solve the following question: Let $I=\int_0^{\infty}\exp(-x^4)dx$. Take $x=i y$ to get $i\int_0^{\infty}\exp(-y^4)dy=i I$. Explain. This change of variable implies $I=iI$, with $I$ ...
0
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1answer
156 views

Complex integration problem

Im having a bit trouble getting started in complex integrals. Im integrating along a circle the integral $\int \frac{e^z}{z^2 + 1}$ where a) center O radius 1/2 b) center i radius 1 c) center i ...
4
votes
4answers
134 views

Intuitive reason for why many complex integrals vanish when the path is “blown-up”?

It is a standard trick for evaluating difficult integrals along the real line to consider a closed-contour and "blow-up" the complex part till it vanishes, leaving us with the residues picked up along ...
2
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0answers
408 views

Integrate complex conjugate

I'm doing some exercises on complex integration, and stumbled over this: $$\int_\gamma \bar{z} dz,$$ Where $\gamma$ is any closed $C^1$ curve which is the boundary of a bounded, connected, open $U ...
1
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1answer
315 views

Complex Analysis: Fundamental Theorem of Calculus

I was given the following question on a complex analysis exam but didn't answer it correctly: Evaluate $$\int \sqrt{z-1} dz$$ about the the unit disk $|z|=1$ using only the Fundamental Theorem of ...
0
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0answers
35 views

Complex Integration with two variables

Let $ z_1, z_2 \in \mathbb{C}^+ \equiv \{ \forall z \in \mathbb{C}^+, \Im(z)>0 \} $. Also define $ m_1, m_2 $ such that for $i=1,2$ the following is true $$ z_i=-\dfrac{1}{m_i} + ...
0
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1answer
84 views

Integral of $te^{-i\omega t}$ and Fubini's Theorem

I have a question about the value of the integral $\int_0^\infty t e^{-i \omega t} dt$. I stumbled upon it while trying to compute the Fourier transform of $|t|$. On one hand, we have that $|t e^{-i ...
4
votes
1answer
133 views

How do you integrate $\int_0^\infty \exp(it^k)\,\mathrm dt$ for $k \in \Bbb N$?

My problem is with the integral $$\int^\infty_0 e^{it^k}\,\mathrm dt$$ with $k\in\mathbb{N}$. Somehow it can be evaluated by use of Cauchy's theorem. But I don't see how. The best thing I can ...
0
votes
1answer
104 views

Evaluate this integral without using the residue theorem

I want to evaluate this integral $$ I = \int_c \tan z + \frac{\csc z}{z} dz $$ $$ c :|z| = 1 $$ apparently tanz is analytic in this region so its integral equals to zero now $$ I = ...
2
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0answers
32 views

$ g(u;s_1,s_2)=\int_{(\alpha+\beta)^{-1}}^{\alpha+\beta}b^{s_2}(u^2-b)^{-\dfrac{s_1+s_2}{2}}(u^2b-1)^{-\dfrac{s_1+s_2}{2}} \mathrm {d}b$

Do you have any idea how to compute this function? $$ \displaystyle g(u;s_1,s_2)=\int_{(\alpha+\beta)^{-1}}^{\alpha+\beta}b^{s_2}(u^2-b)^{-\dfrac{s_1+s_2}{2}}(u^2b-1)^{-\dfrac{s_1+s_2}{2}} \mathrm ...
3
votes
1answer
349 views

Change of variables in a complex integral

I want to evaluate this integral using Residue Theorem $$\int_C^\ \frac{4z} {z^4 +6z^2 +1} dz = $$ $$ C : |z| = 1 $$ so I substitute letting $$\ W = z ^ {2 } $$ $$ dw = 2z dz $$ and the ...
1
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1answer
149 views

Complex analysis contour integral

I am working on the integral $\displaystyle\int_0^{\infty}\frac{\log(x)}{x^2-1}$. I see it done here $\int_0^\infty\frac{\log x dx}{x^2-1}$ with a hint. but I am wondering if it is possible to ...
2
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1answer
85 views

Complex Integral of a meromorphic function

Please help with the following prelim problem. Thanks! Express the integral as a complex integral of a meromorphic function, where $\rho>0$ and $a$ is complex valued $$ \int_{|z|=\rho} ...
0
votes
1answer
89 views

Cauchy integral formula and holomorphic functions

I am stuck in a problem about holomorphic functions and using of Cauchy integral formula. I really have no idea how to start, so i would be glad if somebody could help me with it. Let $C=C(0,1)$ a ...
3
votes
1answer
53 views

Did I calculate this (simple) integral correctly?

Given the contour $C$: we are asked to calculate $\displaystyle\frac{1}{2\pi i}\oint \frac{ze^{z^2-4z}}{z^2-1}dz$. I wrote it as such: $$\frac{1}{2}\left(\frac{1}{2\pi i}\oint ...
1
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0answers
211 views

Laplace transform of a product of functions

While trying to compute the Laplace transform of a certain product, part of the calculation leaves me with a Bromwich integral which has the form: ...
0
votes
2answers
2k 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 ...
0
votes
6answers
97 views

Integration issue

I am trying to solve $\int^{+\infty}_{-\infty}\frac{1}{x}dx $. I read that it is a contour integral along the semi-circle of large radius in the lower complex plane. First, is there any justification ...
0
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1answer
117 views

How to do complex integration. E.g. $ \int_\frac{\pi}{2}^{\frac{\pi}{2} + i} \cos(2z) \; \mathrm{d}z $

For my homework assignment I've been given a number of complex integrals to solve. I've already asked for help on a specific example here, but I was somewhat dissatisfied with the answers. The answers ...
1
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1answer
314 views

How to solve using Cauchy Integral formula?

Let $C$ be the positively oriented boundary of the square whose sides lie along the lines $x=+/-2$ and $y=+/-2$. I am supposed to use the Cauchy Integral formula to evaluate $$\int_C ...
1
vote
1answer
132 views

Need help integrating $\tan x$ and $\tan^n x$ using reduction

I have tried to use integration by parts taking $u$ as $\tan x$ and $v$ as $1$: $$\int \tan x \,dx = \int \tan x \cdot 1\; dx = \tan x \cdot x - \int \sec^2 x \cdot x\; dx$$ then by taking $u$ as ...
0
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1answer
167 views

Integrating $z^n$ and $(\overline{z})^n$ along a line segment in the complex plane

Let $z_1$ and $z_2$ be distinct points of $\mathbb{C}$. Let $[z_1,z_2]$ denote the oriented line segment starting at $z_1$ and ending at $z_2$. Evaluate the integral of $z^n$ and $(\overline{z})^n$ ...
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2answers
207 views

Is an integral in the complex plane an integral over a single number?

A recent question from Juan Saloman reminded me of something that has nagged me for years, and I have never understood and never heard explained. (or maybe I just don't remember, but anyway ...) In ...
1
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1answer
148 views

Finding the integral $\int_0^\pi\dfrac{d\theta}{(2+\cos\theta)^2}$ by complex analysis

Trying to find the integral $\int_0^\pi\dfrac{d\theta}{(2+\cos\theta)^2}$ by complex analysis, I let $z = \exp(i\theta)$, $dz = i \exp(i\theta)d\theta$, so $ d\theta=\dfrac{dz}{iz}$. I am trying ...
0
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1answer
112 views

Calculating Residues

I want to calculate this integral $$I:=\int dk^{0}\frac{e^{-ik^{0}(x^{0}-x'^{0})}}{\left(\left(k^{0}\right)^{2}-|\vec{k}|^{2}\right)} $$ for that I recall the Residue Theorem: $$I=2\pi i \left\{ ...
21
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2answers
544 views

Summation using residues

In reference to this question about showing that the following interesting series takes on the value $$\sum_{n=0}^\infty \frac{1}{(2n+1)\operatorname{sinh}((2n+1)\pi)}=\frac{\log(2)}{8}$$ I tried ...
0
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1answer
182 views

Problems in interpreting an integral that should be solved with residue method

Usually, when I solve an integral using residue method, I find real functions as integrands. I am not able to provide an interpretation for the following complex integral $$ \int_{-\infty}^{\infty} ...
10
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4answers
286 views

contour integration of logarithm

I must compute the following integral $$\displaystyle\int_{0}^{+\infty}\frac{\log x}{1+x^3}dx$$ Can someone suggest me the right circuit in the complex plane over which to do the integration? I ...
1
vote
1answer
238 views

finding $\int_0^\infty \dfrac{dx}{1+x^4}$ through complex analysis

I am trying to find $\int_0^\infty \dfrac{dx}{1+x^4}$ by setting it equal to $\dfrac{1}{2}\oint_C \dfrac{dz}{1+z^4}$ and solving that. By a computer program I've calculated it to be $\approx 1.11072$; ...
1
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2answers
103 views

Computing with Cauchy Residue theorem

how do I calculate $$\operatorname{Res}\left(\frac{1}{z^2 \cdot \sin(z))}, 0\right)$$ What is the order of the pole? $3$?
0
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2answers
194 views

a question about Cauchy integral formula

I'm new in the complex analysis and I'm stuck with this integral : $I=\displaystyle \int_{|z|=4} \frac{\mathrm{d}z}{(z^2+9)(z+9)} $ the exercise is about Cauchy integral, I don't want the whole ...
0
votes
1answer
65 views

Is this OK: $\int_a^b \!\mathrm{d}x \,\,f(x) =^? \int_{\mathrm{i}\,a}^{{\mathrm{i}\,b}} \!\mathrm{d} (\mathrm{-i}y)\,\,f(\mathrm{-i}y).$ Any proof?

This is related to Wick rotation in QFT but it is not exactly it. I'll take a 2-dimensional spacetime to be brief but usually there are more. I've checked with a few functions and with finite ...
2
votes
1answer
234 views

How to evaluate this complex integral !?

We have the following complex integral : $$\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}e^{-\frac{\pi}{2}\cot\left(\frac{\pi}{s}\right)}\frac{x^{s}}{s}ds$$ Where $x\in\mathbb{R}:x>1$. i tried closing ...