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

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6
votes
3answers
413 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.
4
votes
1answer
210 views

Evaluate Complex Integral with $\frac{\Gamma(\frac{s}{2})} {\Gamma\big({\beta +1\over 2} - {s\over 2}\big)}$

I am proving this integral: $$ \int_{c\ -\ j\infty}^{c\ +\ j\infty} \left(\,x^{-1}\sigma\beta^{1 \over 2}\,\right)^{s}\ \Gamma\left(\,s \over 2\,\right) \Gamma\left(\,{\beta +1 \over 2} - {s \over ...
1
vote
1answer
484 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 ...
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 ...
21
votes
2answers
914 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 ...
10
votes
4answers
352 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
476 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 ...
2
votes
1answer
116 views

Some inequalities for an entire function $f$ [CSIR-NET-2014]

Let, $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ be an entire function and let $r$ be a positive real number. Then, which is(/are) correct? (a) $\sum_{n=0}^{\infty}|a_{n}|^{2}r^{2n}\le sup_{|z|=r} ...
6
votes
2answers
118 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 ...
1
vote
1answer
86 views

Evaluate Integral $\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma\,x^{-1}}\right)^s{\Gamma(\beta_1-1+s)\over \Gamma(\beta_1+\beta_2-1+s)}\,ds$

I am at this point of integration where: $$\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma\,x^{-1}}\right)^s{\Gamma(\beta_1-1+s)\over \Gamma(\beta_1+\beta_2-1+s)}\,ds$$ whereby $\beta_1$, ...
1
vote
1answer
101 views

Cauchy's Residue Theorem for Integral $\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma \over x}\right)^s{{1-\beta^{s+1}}\over s(s+1)}\,ds$

This is a similar problem to the one I posted here. I am at this point of integration where: $$\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma \over x}\right)^s{{1-\beta^{s+1}}\over s(s+1)}\,ds$$ ...
1
vote
4answers
98 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} ...
2
votes
2answers
84 views

Apply the Cauchy-Goursat theorem to show that $\int_C \operatorname{Log}(z+2)\, dz=0$ on a unit circle.

Cauchy-Goursat theorem. If a function $f$ is analytic at all points interior to and on a simple closed contour $C$, then $$\int_C f(z) \,dz=0.$$ This is a problem from Churchill's Complex Variables. ...
0
votes
1answer
75 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 ...
10
votes
5answers
540 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 $\displaystyle\int_{-a}^{a} \frac{\sqrt{a^2-x^2}}{1+x^2}dx = \pi\left(\sqrt{a^2+1}-1\right)$?
5
votes
2answers
226 views

Improper integrals with singularities on the REAL AXIS (Complex Variable)

I'm having some troubles when I try to solve improper integrals exercises that have singularities on the real axis. I have made a lot of exercises where singularities are inside a semicircle in the ...
3
votes
2answers
78 views

The value of $\sum_{n=-\infty}^{n=\infty}\frac{1}{n^2-z^2}$ on $\mathbb{C}\setminus\mathbb{Z}$

Show, for $z\not\in\mathbb{Z}$, that $$\sum_{n=-\infty}^{n=\infty}\frac{1}{n^2-z^2} = \frac{-\pi}{z\tan(\pi z)}$$ Hint: You may assume that there exists $C$ such that $|\pi\cot(\pi w)|\leq ...
3
votes
1answer
492 views

Complex antiderivative

I am confused on a couple things: 1.) Why is it that an integral of a complex valued function of a complex variable exists if f(z(t)) is piecewise continuous (and/or piecewise continuous on ...
2
votes
4answers
456 views

Cauchy integral formula

Can someone please help me answer this question as I cannot seem to get to the answer. Please note that the Cauchy integral formula must be used in order to solve it. Many thanks in advance! ...
2
votes
2answers
46 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
162 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
62 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
404 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
49 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 ...
3
votes
1answer
109 views

Function holomorphic except for real line and continuous everywhere is entire

1) I've already shown that if $f:\mathbb{C}\rightarrow\mathbb{C}$ is holomorphic everywhere except for a single point and if it continuous on whole $\mathbb{C}$ then it is entire. It was quite easy. ...
3
votes
2answers
425 views

$\int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$ with residue calculus

I'm trying to compute $\displaystyle \int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$, $(0<a<1)$ Let $f$ denote the integrand. I'm using the rectangular contour given by the following curves: ...
2
votes
1answer
108 views

Cauchy's Residue Theorem with Multiple Gamma Functions

I previously posted a similar problem here and here. This time however I am dealing with multiple gamma functions. This is the problem I am dealing with right now: $$ \int_{c\ -\ j\infty}^{c\ +\ ...
1
vote
1answer
58 views

one complex variables (integration)

how to prove $\int_{C_R}\frac{\log^3(z)}{(1+z^2)^2}\,dz$ goes to $0$ as $R$ goes to $\infty$, with $C_R=Re^{it}$ for $0<t<\pi$, and $R>0$
1
vote
3answers
596 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 ...
4
votes
1answer
155 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
232 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
2answers
85 views

Gaussian integral with a shift in the complex plane

$$ \int_{-\infty}^\infty e^{-(x+ia)^2} \text{d}x $$ where $a\in \mathbb{R}$. I don't know where to start but have reasons to believe the answer is $\sqrt{\pi}$. Namely $\int_{-\infty}^\infty ...
2
votes
3answers
81 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
156 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
1answer
42 views

What is the residue obtained from this integral?

Consider the following integral in two complex variables $z_1$ and $z_2$: $$\frac{1}{(2\pi i)^2}\oint_{{|z_1|=\epsilon}\atop{|z_2|=\epsilon}}dz_1 dz_2\frac{1}{z_1 ...
1
vote
2answers
62 views

Fresnel Integrals

I'm having trouble proving that the arc from $R$ to $Re^{i\pi/4}$ in the Fresnel contour goes to zero. Currently I have $\int_0^{\pi/4} ...
1
vote
1answer
86 views

An integral with the $\Gamma$ function: $\int_{c- i\infty}^{c+i\infty} u^{s}\:\Gamma(\beta +s-1) \:ds$

I previously posted a similar problem here and I have solved many of the problems from the answers given with explanations. This time however I am at this point of integration where: $$\int_{c\ -\ ...
1
vote
2answers
69 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
159 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
304 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
205 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
97 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 ...