3
votes
0answers
46 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
42 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
48 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 ...
5
votes
2answers
96 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
42 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
42 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
48 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
59 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 ...
0
votes
0answers
41 views

compute integral of dz/(z+1) on unit circle [closed]

I would like to compute the the following integral: $$ \oint_C\frac{dz}{z+1} $$ and $$ \int_C\frac{dz}{z+1} $$ where $C=\{z: ||z||=1\}$ thanks.
1
vote
1answer
41 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
110 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
17 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
71 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
49 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
28 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
78 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
56 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
66 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
75 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
39 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
90 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 ...
2
votes
0answers
34 views

Contour integration with 2 branch points

I need to compute a quite complicated Fourier transform, but I'm having problems due to the facts that I have two branch points. The integral I need to solve is $$\int_\infty^{-\infty} ...
0
votes
1answer
25 views

Contour Integrantion of a exponential function

I am trying to evaluate an integral of type $$ I = \int_{-\infty}^{\infty} \frac{e^{ikx}P(x)}{Q(x)} dx $$ Where ...
3
votes
2answers
72 views

erf(a+ib) error function separate into real and imaginary part

Is there an easy way to separate erf(a+ib) into real and imaginary part?
0
votes
1answer
32 views

Analytical Formula for Hilbert Transform of a Ricker Wavelet

I am attempting to validate some numerical code I have to compute Hilbert transforms. As I am interested in the Hilbert transforms of functions with rapid decay, I wanted to unit test my code with the ...
1
vote
0answers
62 views

On $\int_{-\infty}^{+\infty} {\frac{\tan(t-t_0)}{\cosh^2(t-t_0)} \cos(\omega t) \,\mathrm{d}t}$

How to count this? $$ \int_{-\infty}^{+\infty} {\frac{\tan(t-t_0)}{\cosh^2(t-t_0)} \cos(\omega t) \,\mathrm{d}t} $$ Can we use residue formula?
1
vote
1answer
79 views

Integral $ \int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx$

Hi I'm trying to solve this integral Fourier Transform $$ \int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx=\sqrt{2\pi|k|}(1+i) (-1+\text{sgn}(k)) $$ where sgn(k)$=1$ for k>1 and $-1$ for k<1. I am ...
0
votes
1answer
16 views

Beta function identity for $B(z,z)$

I would like to derive the identity $B(z,z)=2^{1-2z}B(z,\frac{1}{2})$ somehow. The Beta function is defined as $B(p,q)=\int_0^1 t^{p-1}(1-t)^{q-1}dt$ where $Re(p), Re(q)>0$ I used the ...