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

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1
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1answer
364 views

Integrating squared absolute value of a complex sequence

I was reading through my book in complex analysis and i encountered this problem. Given, $F=\sum_{n=0}^{\infty} a_nX^n$ is a convergent power series with radius of convergence R. We are asked to show ...
21
votes
2answers
710 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 ...
15
votes
7answers
2k views

Integration by means of complex analysis

Dear all: this time I have the integral $$\int_0^\infty\frac{1-\cos x}{x^2(x^2+1)}\,dx$$and we must try to solve it using complex integration, residues, Cauchy's Theorem and the whole lot. (BTW, does ...
5
votes
3answers
321 views

Improper integration involving complex analytic arguments

I am trying to evaluate the following: $\displaystyle \int_{0}^{\infty} \frac{1}{1+x^a}dx$, where $a>1$ and $a \in \mathbb{R}$ Any help will be much appreciated.
10
votes
4answers
319 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 ...
4
votes
2answers
414 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 ...
0
votes
1answer
54 views

Contour integral of analytic function with singularity

I am supposed to integrate $f(z)=$$\frac{5}{z}$ from -3 t0 3 but I am having trouble understanding how to do this. I've done the integration the "hard" way by using parametrizations but now I need to ...
9
votes
5answers
509 views

Showing that $\displaystyle\int_{-a}^{a} \frac{\sqrt{a^2-x^2}}{1+x^2}dx = \pi\left (\sqrt{a^2+1}-1\right)$.

How can I show that $\displaystyle\int_{-a}^{a} \frac{\sqrt{a^2-x^2}}{1+x^2}dx = \pi\left(\sqrt{a^2+1}-1\right)$?
2
votes
2answers
44 views

question related with cauch'ys inequality

We have a statement that if f(z) is analytic in a domain D and $C = \{z : |z-a| = R\}$ is contained in D. Then, $|f^n(a)|\leq \frac{n!M_R}{R^n}$ where $M_R = \max|f(z)|$ on C. What if the given D ...
6
votes
3answers
152 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 ...
5
votes
1answer
51 views

Evaluate $\int_{|z|=r}^{}\frac{\log\ z}{z^2+1}dz$ where $r>0$

I tried to evaluate $$\int_{|z|=r}^{}\frac{\log\ z}{z^2+1}dz$$ when $r>0$. Clearly $\log z$ is not continuous at $z=-r$. So is this integral meaningful then? Since the function is continuous and ...
5
votes
1answer
273 views

Residue theorem with exponential and trig functions

The following integral should be doable using the residue theorum: $$\frac1{2\pi}\int_{0}^{2\pi}e^{\cos\theta}\cos(n\theta) \,d\theta$$
4
votes
2answers
46 views

Show that $\int_{|z|=1} \frac{e^z}{z^k} dz = \frac{2\pi i}{(k-1)!}$

I'm supposed to show that $$\int_{|z|=1} \frac{e^z}{z^k} dz = \frac{2\pi i}{(k-1)!}$$ where $|z|=1$ is traversed counterclockwise and $k>0$. We can parametrize this path as $\gamma(t)=e^{it}$ for ...
1
vote
4answers
63 views

Poisson Integral is equal to 1

Show $$ \int_{-\pi}^{\pi}P(r, \theta)d\theta = 1 $$ Let $\alpha(r) = \frac{r^2 - 1}{2r}$ and $\gamma(r) = -\big(\frac{r^2 + 1}{2r}\big)$. Then $$ \frac{1}{2\pi} ...
6
votes
2answers
100 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 ...
4
votes
1answer
150 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 ...
3
votes
2answers
218 views

Evaluate $\int_{C}\frac{e^{\alpha z}}{z}dz$ where $\alpha \in \mathbb R$ and C is the circle $\gamma(t)=e^{it}$…

Let $\alpha \in \mathbb R$ and C be the circle $\gamma(t)=e^\alpha t$, $-\pi\le t \le \pi$ Evaluate $$\int_{C}\frac{e^{\alpha z}}{z}dz.$$ Use the above, to show that $$\int_{0}^{\pi}e^{\alpha ...
2
votes
3answers
70 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: ...
2
votes
2answers
103 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 ...
1
vote
2answers
55 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} ...
1
vote
0answers
119 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
vote
4answers
295 views

Integrate $\int_0^\infty \frac{\sqrt{x}}{x^{2}+1}\, \mbox{d} x$

I've been trying to integrate the following $$\int_{0}^{\infty} \frac{\sqrt{x}}{x^{2}+1} \mbox{d} x$$ on half an annulus in the upper half plane. I keep getting $\frac{\pi}{\sqrt{2}}\ i$, which ...
0
votes
3answers
103 views

Geometrical Interpretation of the Cauchy-Goursat Theorem?

The Cauchy-Goursat theorem is really non-intuitive and is very astounding. Can someone geometrically explain to me why its true? I'm specifically talking about this version of the theorem: For ...
0
votes
1answer
86 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 ...