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5
votes
2answers
53 views

Integral with rational functions of powers and exponentials

Any ideas how to solve: \begin{equation} \int_0^\infty x^{n+\frac{1}{2}} (e^{a x }-1)^{-\frac{1}{2}} e^{i x t} dx \end{equation} where $a$ and $t$ are real, positive constants; $n$ is a positive ...
0
votes
0answers
33 views

What is the Riemann surface of the exponential integral?

I have recently encountered a differential equation whose solution has a term \begin{equation} \frac{1}{2}e^{-\frac{1}{2 \varepsilon} e^{i \tau}} \int_{\tau_0}^\tau e^{\frac{1}{2 \varepsilon} e^{i ...
5
votes
3answers
126 views

Find the Fourier Transform of $\dfrac{x}{x^4+4}$

I have a problem here that becomes quite difficult to manage. I have to find the fourier transform of: $$f(x)=\frac{x}{x^4+4}$$ I'm sure there will be many ways to do this and I'll post my method ...
1
vote
2answers
42 views

How to integrate $x\times \frac{\sin(x)}{x^2+a^2}$ from zero to infinity

I am trying to evaluate $\int_0^\infty\frac{x \sin(x)}{x^2+a^2} dx$. I get $\frac{\pi}{4} \sin(ia)$ using residue theorem. I integrated over the path that goes from -R to R along the real axis and ...
1
vote
1answer
33 views

complex integration-how to solve the given problem

how do we calculate the value $\frac{1}{2\pi i}\int\frac{\sum_{n=0}^{15}z^n}{(z-i)^3}dz$ in $C$:|z-i|=2 ? the answer for this is 1+15i.. how to get it? can someone please explain?
1
vote
1answer
29 views

Estimate of complex integral

Prove that $$ \left|\int_c (2-\frac{e^z}{z-\log 2}) dz \right| <\frac{2}{3} $$ when C is the part of circle $\left| \frac{z}{\pi} -1 \right|^2 =2$ where $Re(z)\geq 0$. ($\log$ means natural ...
2
votes
1answer
39 views

Can use residue theorem for this integral

I need to compute $$I=\int_C \dfrac{e^{\sqrt{1+u}}\cdot\sqrt[4]{1+u}}{\sqrt{u}} \,\mathrm {d}u$$ where $C$ is the unit circle. I am confused about whether I can use the residue theorem to compute it? ...
0
votes
2answers
169 views

Prove a function has a removable singularity at $z=0$.

Let $f$ be a holomorphic function on $\mathbb{C}\smallsetminus \{0\}$. Suppose $\int_{|z|=1}z^nf(z)\,dz=0$ for any $n=0,1,2,\ldots$. Prove that $f$ has a removable singularity at $z=0$. How to prove? ...
3
votes
2answers
147 views

Behavior of $f(z)=\int_0^1\mathrm{e}^{\alpha t^2}\sin(tz)\,dt$ when $\alpha <0$

Define $$f(z)=\int_0^1\mathrm{e}^{\alpha t^2}\sin(tz)\,dt,$$ where $\alpha \in \mathbb{R}$. If $\alpha >0$ then $f(z)$ has infinitely many real zeros and at most a finite number of complex ...
1
vote
2answers
44 views

Contour expression explanation

$$ \begin{align} & \int_\gamma zdz \end{align} $$ $$\\ \gamma = [e,1]+[1,-1+\sqrt3] $$ contour $\gamma$ is defined as above and I can't understand it. Could someone please explain it to me and ...
3
votes
1answer
38 views

Is this Integral calculation correct?

Can someone confirm if my solution is right or if I have done something that is not permitted $$ \begin{align} & \int_\gamma e^{\pi z}=\int_\gamma \left( \frac{ e^{\pi z}}{\pi}\right)' \, dz ...
0
votes
1answer
42 views

Image of a parametrised curve and its geometrical meaning

$\gamma_1(t)= t^2 + i\, t^4 , t\in [ -1, 1]$ which is the Image of this curve and / or what is the geometrical description of the set of points $ z : z=\gamma_1(t)$ I am helping a friend study ...
1
vote
3answers
244 views

Possible values of $\int \frac{dz}{\sqrt{1-z^2}}$ over a closed curve in a region?

This is related to Ahlfors' problem #5 following section 4.4.7. Let $\sigma$ be a path in $\mathbb{C}$ starting at $-1$ and ending at $+1$. Let $\gamma$ be a closed curve in $\mathbb{C}$ which does ...
0
votes
1answer
44 views

How to find the area where $\frac{1}{z^2-4}$, $z \in \mathbb{C}$ is holomorphic?

Suppose that you are given a problem of finding the following complex integral: $$\int_\tau \frac{1}{z^2-4} dz$$ where $\tau = \{z \in \mathbb{C}: |z|=4 \}$. My question is (in the context of this ...
1
vote
2answers
42 views

Contour integral of complex logarithm

Evaluate $$\int_C Log(z) dz$$ where $Log(z)$ is the principle branch of the complex logarithm (Arg$(z)\in(-\pi,\pi)$) and $C$ is the contour given by the horizontal line connecting $z=i$ to $z=i+1$, ...
3
votes
2answers
592 views

integral of complex logarithm

Consider the integral $$I=\int_0^{2\pi}\log\left|re^{it}-a\right|\,dt$$ where $a$ is a complex number and $0<r<|a|$. We have ...
2
votes
1answer
87 views

Complex integral involving logarithm

I've been working on this integral for quite a while and I think I've been able to progress but now I'm stuck. So I have to prove that $$\int_C f(z)\ dz =\int_C\frac{2z}{(1+z^2)\log(2+z^2)}dz =\pi ...
2
votes
1answer
132 views

How do you integrate Gaussian integral with contour integration method?

How do you integrate $$\int^{\infty}_{-\infty} e^{-x^2} dx$$ with contour integration method? I do not even know how to setup the problem.
2
votes
1answer
39 views

can't follow the steps of a specific complex integration

Hi: I already asked this question on the complex analysis tag but nobody answered it so then I found this complex-integration tag and was hoping that someone might be able to answer it here. It is ...
0
votes
0answers
64 views

What does this complex contour integral represent?

How would one evaluate the following complex contour integral in "Integral and Series Representations of Riemann’s Zeta function, Dirichelet’s Eta Function and a Medley of Related Results." The ...
1
vote
4answers
85 views

Help with Complex integration

I have to calculate the following integral $$\int_{-\infty}^{\infty} \frac{\cos(x)}{e^x+e^{-x}} dx$$ Anyone can give me an idea about what complex function or what path I should choose to calculate ...
1
vote
1answer
38 views

Mean value theorem for harmonic

In Problems and Solution in Mathematics by Ta-Tsien, exercise 5123, the mean value theorem is used as: \begin{equation} \text{log} |F(0)| = \frac{1}{2 \pi} \int_0^{2\pi} \text{log}|F(re^{i\theta})| ...
1
vote
2answers
51 views

The value of the itegral $\int_{\gamma} \dfrac{dz}{z-a}$ is a multiple of $2\pi i$

I am reading Ahlfors' proof of the lemma: Lemma If the piecewise differentiable closed curve $\gamma$ does not pass through the point $a$, then the value of the integral $$\int_{\gamma} ...
0
votes
2answers
29 views

Cauchy integral formula or something else?

I need to determine the function $\;f(z)$ if $$f''(z)=\oint_{\partial C_1(0)}{\sin^2\xi \over\left(\xi-z\right)^3}\mathbb{d}\xi$$ with $C_1(0):\left|z\right|<1$ positive. Additionally ...
1
vote
2answers
40 views

Show that $\sum_{k=1}^{n}a_ke^{2 \pi ikx}$ has a root in $\left[ 0,1 \right]$

Let $a_1, \dots ,a_n$ be arbitrary complex numbers. Define: $f \left( x \right)=\sum_{k=1}^{n}a_ke^{2 \pi ikx}$. I wish to show that there exists an $x\in\left[ 0,1 \right]$ s.t. $f \left( x ...
3
votes
2answers
75 views

How to calculate $\int_{\partial B_2(0)}\frac{2z^2+7z+11}{z^3+4z^2-z-4}\;dz$?

I want to calculate $$\displaystyle\int_{\partial B_2(0)}\underbrace{\frac{2z^2+7z+11}{z^3+4z^2-z-4}}_{=:f(z)}\;dz\tag{0}$$ Partial fraction decomposition yields ...
3
votes
1answer
67 views

If $f$ has pole of order $m$, then $\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$

Statement: Let $$f(z):=\sum_{k=-\infty}^\infty a_kz^k$$ have a pole of order $m$ at $z_0$. Then $$\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$$ ...
4
votes
2answers
393 views

Evaluating real integral using residue calculus: why different results?

I have to evaluate the real integral $$ I = \int_0^{\infty} \frac{\log^2 x}{x^2+1}. $$ using residue calculus. Its value is $\frac{\pi^3}{8}$, as you can verify (for example) introducing the function ...
6
votes
2answers
92 views

The integral $\int_0^\infty \dfrac{x \sin(x)}{x^2+1} dx$

How to calculate: $$\int_0^\infty \frac{x \sin(x)}{x^2+1} dx$$ I thought I should find the integral on the path $[-R,R] \cup \{Re^{i \phi} : 0 \leq \phi \leq \pi\}$. I can easily take the residue in ...
0
votes
0answers
40 views

Solving an ordinary differential equation with two functions

Please I am trying to solve this differential equation but I do not seem to get the correct answer. Please help thank you. My equation is; $\frac{dc(x)}{dx} = 1+ b\frac{y(x)-y(x-1)}{y(x-1)}$ where b ...
1
vote
2answers
77 views

Finding $\int_0^{2\pi}\frac{\sin t + 4}{\cos t + \frac{5}3} dt$

I'd like to ask something about the following integral: $$ \int_0^{2\pi}\frac{\sin t + 4}{\cos t + \frac{5}3} dt $$ I rewrote and took another variable. $$ ...
2
votes
2answers
84 views

Liouville's Criterion and Integration of $\sin(z)/z$ using elementary functions

Liouville's Criterion: $\int fe^g$ is elementary iff there is rational function $q$ in $\mathbb{C}(z)$ such that $$f=q'+qg'$$ where $f\neq0$ and $g$ is a non constant function in $\mathbb{C}(z)$. Now ...
4
votes
1answer
75 views

Complex Integration: theorem of Residues

I am trying to prove a theorem that is doing my head in a bit. I have tried to simplify the problem as much as possible and leave out the details, even though it might look a bit too big. The ...
0
votes
1answer
34 views

Complex exponent integral - prove $\int_a^b{e^{\lambda x} \text{dx}}=\frac{1}{\lambda}\left(e^{\lambda b}-e^{\lambda a}\right) $

How to prove the exponent integration rule: $$\int_a^b{e^{\lambda x} \text{dx}}=\frac{1}{\lambda}\left(e^{\lambda b}-e^{\lambda a}\right) $$ In the complex version of it - that is, when $\lambda \neq ...
4
votes
2answers
146 views

Complex integration, any ideas?

I need to solve the following integral of a function analytic on an open environmemt of the unit circle, but i dont know how to get that answers $$\frac{1}{2\pi ...
3
votes
1answer
172 views

How to find $\int_0^{\pi}\frac{\sin n\theta}{\cos\theta-\cos\alpha}d\theta$

I was doing some work in physics and I came up with a definite integral. I tried everything I could but couldn't solve the integral. The integral is $$ \int_0^\pi {\sin\left(n\theta\right)\over ...
2
votes
3answers
62 views

Help with equality of complex integrals

I need to prove this equality of integrals...but i dont know how to begin, so if anyone can give an idea... Let f a continuous function on $\overline{D}=\{z : |z|\leq 1\}$. Then: ...
0
votes
1answer
39 views

understanding a particular step in proof of cauchy's theorem for triangles

"Hi: I am reading "complex variables" by Ash and Novinger and they prove "cauchy's theorem for triangles early in the book". Unfortunately, there's a step in their proof that I don't follow. ...
1
vote
1answer
39 views

$\int _{C^{+}(0,3)} \frac {dz}{2-\sin z}$

$$\int _{C^{+}(0,3)} \frac {dz}{2-\sin z},$$ $z$ is complex. I have no idea how to solve $2-\sin z$. I will be really grateful for any help
1
vote
1answer
61 views

Help with improper integral [duplicate]

I need help solving this integral: $$\int_0 ^\infty \frac{\sin(x)}{x} dx$$ I have a help that says that try to calculate the integral of $$\frac{e^{iz}}{z}$$ for a "proper path"... but I don't know ...
1
vote
1answer
26 views

Complex integral with different contours

If I have a complex integral to solve using the Cauchy Integral formula with the same point but with different contours, in which the point used is inside both contours, is the result the same? Say ...
1
vote
1answer
26 views

Parametric form of curve $\vert z+i\vert = 1$

I need to integrate a complex function through the curve $\vert z+i\vert = 1$. As far as I know I need the parametric form of this curve. I know that when I have $\vert z\vert = 1$, the parametric ...
1
vote
3answers
58 views

How to handle the complex integration of this function around a branch point

I have this complex integral to which I don't know if it's possible to assign a value: The integral is on a small circle around the origin. The function is $\frac{1}{(z-1)\sqrt{z}}$. The fact is ...
1
vote
1answer
56 views

Complex integral inequality

Statement Let $\gamma$ be the curve that goes through the upper unit circle counterclockwise (positive orientation). Prove that $$\left|\int_{\gamma} \dfrac{\sin(z)}{z^2}dz\right|\leq ...
1
vote
0answers
22 views

Explicit computation of integrals along loops

I am learning a bit about integration on one-dimensional complex tori. It's exciting stuff, but I have some trouble making things a bit explicit. Let's consider the elliptic curve $E = \mathbb ...
1
vote
0answers
26 views

Find natural number to satisfy inequality given with integral

Find the number $N\in \mathbb{N} $ for which the following inequality is true, if $\sigma(t)=|z|=1 $, with $0\leq t\leq \pi/2$ and $\left | \int_{1}^{i} e^{z^{-1}} \right | \leq N \frac{\pi}{2}$. ...
1
vote
0answers
57 views

Calculating this integral: $\int_{\partial_{10}(1-2i)}\frac{39!\require{cancel}\cancel{z^z}z^z}{(z+4)^{42}}dz$

Please take a look at $$\int_{\partial B_{10}(1-2i)}\frac{39!\require{cancel}\cancel{z^z}ze^z}{(z+4)^{42}}dz$$ At a first glance, this looks like a case for Cauchy's differentiation formula, which ...
1
vote
0answers
22 views

Evaluate the bound of the following complex integral

Let $q$ be a complex analytic function such that $\left|q\right|\neq0$ everywhere in the domain, except at the boundaries ($\left|q\right|\rightarrow0$ as $z\rightarrow\pm\infty$). Is it possible ...
2
votes
1answer
343 views

Complex integral over circle using Cauchy's formula

I have to integrate the complex function $$ \frac{e^z-1}{z^5} $$ over the curve $\gamma(t)=1+re^{-5it}$ where $t \in [0,2\pi]$. The curve has winding number -5 with respect to a point inside the disc ...
0
votes
2answers
126 views

Cauchy's argument principle, trouble working simple contour integral

I'm trying to teach myself Cauchy's argument principle by doing a simple example, but apparently I'm missing something, because every time I try to do the contour integral I get 0. Cauchy's argument ...