1
vote
2answers
21 views

An integral involving two variables and the floor function

Let $N$ be some fixed positive integer. I have the following function $$ g(z) = z \int_1^N [t] e^{2 \pi i t z} \ dt. $$ How would one compute $$ \int_0^1 g(z) \ dz ? $$ Thanks!
0
votes
1answer
29 views

Show that for $|f(z)| \leq C (|z| + 1)\log(|z| + 1)$, there is an $a$ such that $f(z) = az$

Let $f: \mathbb{C} \to \mathbb{C}$ be analytic and suppose a $C \geq 0$ exists such that \begin{align*} |f(z)| \leq C(|z| + 1) \log(|z| + 1) \end{align*} for all $z \in \mathbb{C}$, where $\log: ...
2
votes
2answers
27 views

Computing $\int_{\gamma} {dz \over (z-3)(z)}$

Compute, using the Cauchy Integral Formula, $$ \int_{\gamma} {dz \over (z-3)(z)} $$ where $\gamma$ is the circle of radius $2$ centered at the origin, oriented counterclockwise. ...
8
votes
1answer
208 views

Cool Integral = $\pi/2$ !!

I am trying to calculate the integral $$ I_n=\int \limits_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}dx. $$ (I have literature on this, if people want). Note, we can write the ...
2
votes
1answer
49 views

The $\frac{1}{x+i\varepsilon}$ distribution.

I read that the distribution defined as: $$ \lim_{\varepsilon \rightarrow 0}\frac{1}{x+i\varepsilon}$$ is equal to $$p.v. \frac{1}{x} -i\pi \delta(x)$$ So that for any test function $f$, ...
4
votes
0answers
93 views

Beautiful Closed form $\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$

Hi I am trying to calculate $$ I:=\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$$ Note, the closed form is beautiful and is given by $$ I=−\frac{3}{8}\zeta_2\zeta_3 -\frac{2}{3}\zeta_2\ln^3 2 ...
2
votes
1answer
48 views

$\sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}$

Hi I am trying to calculate the sum given by $$ \sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}=\ = \sqrt{\frac{\pi}{\alpha}} e^{\beta^2/(4\alpha)} ...
5
votes
0answers
73 views

Beauty$\sum_{j,k=1}^\infty \frac{H_j(H_{k+1}-1)}{jk(k+1)(j+k)}=-\zeta(2)-2\zeta(3)+4\zeta(2)\zeta(3)+2\zeta(5)$

Hi I am trying to calculate the infinite double sum $$ S:=\sum_{j,k=1}^\infty \frac{H_j(H_{k+1}-1)}{jk(k+1)(j+k)}=-\zeta(2)-2\zeta(3)+4\zeta(2)\zeta(3)+2\zeta(5),\quad H_n:=\sum_{k=1}^n\frac{1}{k}\ \ ...
3
votes
0answers
53 views

Saddle point method: a rigorous proof?

I am trying to prove in a fully rigorous way the Saddle Point method for holomorphic functions of 1 complex variable. In books I find only complicated general statements or non-rigorous proofs. Hence ...
1
vote
0answers
43 views

Cauchy Integral Theorem problem (lack of understanding)

First of all i was asked to evaluate this integral $\int_\gamma \frac{2z}{(z-1)(z-3)} dz$ where $\gamma (t) = 2e^{it}$ for $0\leq t \leq 2\pi$. Now I thought I would have to calculate this ...
2
votes
1answer
52 views

Integral $\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt$

I am looking for a closed form expression for the logarithmic trigonometric integral $$ I_{n,p}=\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt \quad (n\geq 0, p\geq 0). $$ Closed form expression ...
6
votes
2answers
102 views

Integrate $I=\int_0^1\frac{\ln x}{x^n-1}dx$

Hi I am trying to obtain a closed form for$$ I_n=\int_0^1\frac{\ln x}{x^n-1}dx, \quad n\geq 1. $$ This integral is quite nice and generates many other known closed form results such as $$ ...
2
votes
1answer
22 views

Convergence Question:

This is related to the Dirichlet eta function. Does $$\int_1^\infty \frac{dx}{x^z}$$ converge for $Re(z)>1$? Just wondering. If so, then does $$\int_1^2 \frac{dx}{x^z}+\int_3^4 ...
2
votes
1answer
66 views

Question on the Prime Number Theorem (the Tchebychev Function) [duplicate]

This has been giving me nothing but a headache: Let the Tchebychev Function, $\psi (x)$ be defined: $$\psi (x) = \sum_{p^m \le x}\log p \space \space \space , \space \space \space p \in \mathbb P$$ ...
1
vote
1answer
41 views

Find the integral in the complex plane

I'm having some trouble computing these integrals, they're on the practice final, but no solutions given. I'm hoping to get some help here. Calculate the following Integral of $(z \cdot ...
1
vote
2answers
49 views

Proving that the line integral $\int_{\gamma_{2}} e^{ix^2}\:\mathrm{d}x$ tends to zero

Let $f(z) = e^{iz^2}$ and $\gamma_2 = \{ z : z = Re^{i\theta}, 0 \leq \theta \leq \frac{\pi}{4} \} $. All the sources I have found online, says that the line integral $$ \left| \int_{\gamma_2} ...
2
votes
1answer
34 views

I want to compute $\int_0^\infty \frac{x^t}{1+x^2}dx \; \forall t \in (-1,1)$ using residue theroem.

I want to compute $$\int_0^\infty \frac{x^t}{1+x^2}dx \qquad \forall t \in (-1,1)$$ using residue theroem. I consider $$f(z) = \frac{z^t}{1+z^2}$$ I find two pole of order 1 in $z=i$ and $z=-i$ with ...
2
votes
2answers
45 views

compute the integral using residue theory

I am trying to compute an integral in an example in my complex analysis textbook: $$\int_{-\infty}^\infty {xsinx\over x^4+1}dx$$ The book gives some startup hints, but I don't quite follow, I set ...
0
votes
0answers
19 views

computing integral using residue theory [duplicate]

I want to compute the integral $\int_{-\infty}^\infty {x^4\over {1+x^8}}dx$ by using residue theory. I find the zero of $Q(x)$ is $i^{1/4}$. Do I have to factor the denominator into 8 different ...
2
votes
2answers
50 views

Find $\int_\Gamma\frac{2z+j}{z^3(z^2+1)}\mathrm{d}z$ where $Γ:|z-1-i| = 2$

pls, some ideas for integral solution (residue theory)? $$\int_\Gamma\dfrac{2z+j}{z^3(z^2+1)}\mathrm{d}z$$ Where $Γ:|z-1-i| = 2$ is positively oriented circle. Thx, for help!
1
vote
1answer
62 views

Integrating this complex function, using Residue Theorem [duplicate]

I am having a massive amount of trouble integrating this, I really have no clue how to get the answer in the book: $$\int_{-\infty}^{\infty} \frac{x^4}{1+x^8}dx$$ I know I need to find the poles ...
1
vote
1answer
42 views

Complex integration, help

I need help integrating $\int_{-\infty}^{\infty}\frac{z \sin (z)}{\left(z^2+1\right) \left(z^2+2\right)} dz$. I calculated the integral over the closed upper half circle in the complex plane which is ...
2
votes
2answers
46 views

Integral along $\Gamma_c := \{c + i t \mid c>0 , -\infty < t < \infty\}$

I have a Complex Analysis homework problem which I've been working on for some time, and have become stuck. I am asked to compute $$ I \equiv {1 \over 2\pi{\rm i}}\int _{\Gamma_c}{a^{s} \over ...
2
votes
0answers
42 views

Integral of Difference of Logs

I get the expansion of $h$ to be $$ h(z) = {1 \over z } \sum_{r=1}^{\infty}{1 \over r}{(-{\alpha \over z}})^r $$ $$ \Rightarrow h(z) = \sum_{r=-2}^{-\infty}{{(-\alpha)^{r+1} \over -(r+1)} z^{r}} $$ ...
1
vote
2answers
112 views

integral of sin(x) to the power 2014

For a course in Complex Analysis we're tasked to find the integral of \begin{align*} \int_0^{2 \pi} (\sin\theta)^{2014} d \theta \end{align*} but I'm a bit stumped so far on how to do this. What I've ...
0
votes
2answers
14 views

Parametric equations in complex analysis

I am trying to find $\int_C (1+i-2z')dz$ where$z'$ is the conjugate of $z$ and where C is the parabola $y=x^2$ from $z_1=0$ to $z_2=1+i$. How do I write the parametric equations for this?
0
votes
0answers
18 views

Showing $\int_{\gamma}f(z)dz = \int_{\gamma_1}f(z)dz + \int_{\gamma_2}f(z)dz$ with non analytic points.

Suppose $f$ is analytic on the complex plane except at $z_1,z_2$, that $\gamma_1$ and $\gamma_2$ are simple closed curves with $z_1,z_2$ in their interiors and $\gamma_1$ and $\gamma_2$ are in the ...
0
votes
4answers
63 views

Integral of a function $f:\mathbb{R}\rightarrow \mathbb{C}$

My real analysis book defines derivatives and integrals only for a function $f:A\rightarrow \mathbb{R}$, where $A\subset \mathbb{R}$. But, when talking about Fourier series, it comes out an integral ...
0
votes
0answers
26 views

Question concerning the gamma function in relation to other holomorphic functions when $Re(\xi) > 0$

Let $f$ be indefinitely differentiable on $\mathbb R$ that has compact support. $\implies f$ belongs to the Schwartz space. Consider: $$I(\xi) = \frac1{\Gamma(\xi)} \int_0^\infty f(x)x^{-1+\xi}dx$$ ...
1
vote
2answers
72 views

How does the integral $\int_{D_C} e^{ia z}P(z)/Q/(z)\,\mathrm{d}z$ blow up.

In my book I have a theorem that goes something like the following Let $P(x)$ be $Q(x)$ polynomials such that $\deg(Q) \geq \deg(P) + 2$. Then \begin{align*} \int_{-\infty}^{\infty} ...
0
votes
0answers
54 views

Find the countour integral of $\int_{γ} \sqrt{z} dz$ where $γ=C(2,1)^+$ or $γ=C(1,1)^{+}$ or $γ=C(0,1)^{+}$

Find the countour integral of $\int_{γ} \sqrt{z} dz$ where $γ=C(2,1)^+$ or $γ=C(1,1)^{+}$ or $γ=C(0,1)^{+}$ With $\sqrt{z}$ I mean the branch with the non-positive real axis as branch cut. With ...
0
votes
1answer
51 views

How do I find contour integral with no poles?

I would like to know if the contour integral have no poles how do I solve it? Please explain with workings. Thank you. $\displaystyle\oint_C z^5 \sin\left(\dfrac{1}{z^2}\right) \space dz$
1
vote
1answer
29 views

Integrating a complex function with Cauchy formula

We have I =$\oint_{C}^{} \frac{(z-1)\sin(z)}{z^2 - 2z - 3}$, C is a circle for which $|z-2| = 2$. I wrote $I = \oint_{C}^{} \frac{(z-1)\sin(z)}{4(z-3)} - \oint_{C}^{} \frac{(z-1)\sin(z)}{4(z+1)}$ ...
0
votes
2answers
79 views

A simple yet complex path integral

Let L be an elipse arc with parametrization $z = 2 \cos(t) + 4i \sin(t)$, $t \in [0, \frac{\pi}{2}]$. How would one solve $\int_{L}^{} z^{-1} dz$?
2
votes
1answer
57 views

Morera's Theorem on Integrals

Use Morera's theorem and an interchange of the order of integration to show that the following function is analytic on the indicated domain; find a power-series expansion for the function by using the ...
2
votes
1answer
67 views

Integrating $\frac{\sin x}{x} dx$ , why do we choose $e^{iz} / z$?

I am studying the famous integral of $\dfrac{\sin x}{x}$ in complex analysis. My lecturer integrated $e^{iz}/z$ over a semi-circle with $0$ (the origin), taken out by a small semi-circle. He asked ...
2
votes
1answer
76 views

$f(z):=\int_{\mathbb{R}} \frac{1}{t-z} d\mu(t)$ show $\lim_{y\rightarrow 0}iyf(iy)=-\mu(\lbrace 0 \rbrace)$

I have some trouble with part b) Let $\mu$ be a finite Borel measure (i.e finite measure on the $\sigma$-algebra of Borel sets on $\mathbb{R}$). Define the function $$f(z):=\int_{\mathbb{R}} ...
0
votes
1answer
59 views

Trigonometric integral on complex plane

I am trying to take the integral of $$\int_0 ^{2\pi}{d\theta\over {3+\sin\theta+\cos\theta}}$$ I did this multiple times, and get an answer of $24\pi/(1+i)$, I am really not sure about this one and ...
1
vote
0answers
20 views

Trigonometry integration with a bound

So, I want to integrate $\int_\gamma sinz\; dz$ where $\gamma$ is any curve joining $i\to \pi$. Can I say that it is beacause $\int sinz=-cosz$, and $-cosz$ is analytic on the domain containing ...
-1
votes
1answer
83 views

Help ! bizzare integral [closed]

How to integrate $$ I_1=\underbrace{\int\frac{x^2}{\sqrt{9x^4+4x^2+1}}dx}_{I_1} $$ and $$I_2=\underbrace{\int\frac{dx}{\sqrt{9x^4+4x^2+1}}}_{I_2}$$
1
vote
2answers
54 views

evaluate integral by complex method

Can you guys help me how to evaluate this integral by complex analysis method?
1
vote
0answers
32 views

Contour Integral of a function

The contour integral $$\int_{C(π/5, π/4)} cot(5x) dx$$ This is what I did: $$cot (5x) = sin(5x)/cos(5x) = 1/5$$ $$2 \pi i \cdot 1.5 =2/5 \pi i$$ Is this how you do it?
0
votes
0answers
30 views

General method of integration when poles on contour

Is there a general method for calculating a contour integral when you have a pole on the contour? For example, how do I integrate, $\frac{1}{z-1}$ over the unit circle centred at the origin?
5
votes
4answers
127 views

Integrating $ \int \limits_{-\infty}^{\infty} \dfrac{\sin^2(x)}{x^2} \operatorname d\!x $

I'm trying to evaluate $\displaystyle \int \limits_{-\infty}^{\infty} \dfrac{\sin^2(x)}{x^2} \operatorname d\!x $. My first though was to use residue calculus, since we've got the pole of order 2 ...
1
vote
3answers
73 views

Integrating $\int _0^\pi \frac{1}{1+\sin^2(\theta)}$ using Cauchy's formula

I need to evaluate $\displaystyle \int _0^\pi \frac{d\theta}{1+\sin^2(\theta)}$ by using Cauchy's integral formula, and the substitution $z = e^{i\theta}.$ So far, I have that $$d\theta = ...
3
votes
1answer
41 views

Evaluating trigonometric integral and Cauchy's Theorem

I am trying to evaluate the following integral: $\int_0 ^\pi {d\theta\over{1+\sin^2\theta}}$ I tried using the substitution of $\sin\theta={1\over 2i}(z-1/z)$, where $z=e^{i\theta}$, and ...
2
votes
0answers
67 views

Question on the Fourier Transform, specifically concerning polynomials

Suppose $P$ is a polynomial of degree $\ge 2$ with distinct roots, none lying on the real axis. Calculate: $$\int_{- \infty}^{\infty}\frac{e^{-2 \pi i x \xi}}{P(x)}dx,\space \space \space\xi \in ...
0
votes
1answer
71 views

Please help me solve this complex integral.

$$\oint_C \frac{\cos(z-a)}{(z-a)}\mathrm{d}z$$ Such that $a\in \Bbb R^2$ and $C$ is a single closed curved defined by $|z-a|=\frac{|a|}{2}$ Here $z=x+iy$ is a complex number. Please solve the above ...
2
votes
1answer
92 views

Integrating $ \int_{0}^{2\pi}\frac{d\theta}{3 + \sin\theta + \cos\theta}$

I want to do the following integral in my complex analysis class: $$ \int_{0}^{2\pi}\frac{d\theta}{3 + \sin\theta + \cos\theta}$$ I don't have the solution (in the textbook) but I checked via ...
1
vote
0answers
30 views

Show that $\int_{-\infty}^\infty f(t)dt=0$ where $f\in H^\infty(\mathbb{H})$

The problem is stated as follows: Let $\mathbb{H}$ denote the open upper half plane. Let $f \in H^{\infty}(\mathbb{H})$ Suppose $f$ can be extended to be continuous on $\overline{\mathbb{H}}$ with ...