The tag has no wiki summary.

learn more… | top users | synonyms

0
votes
2answers
26 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 ...
3
votes
2answers
69 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
58 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 $$\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$$ ...
1
vote
2answers
36 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 ...
6
votes
2answers
87 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
69 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. $$ ...
4
votes
1answer
63 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
31 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
143 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 ...
2
votes
3answers
61 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: ...
3
votes
1answer
162 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 ...
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 ...
0
votes
1answer
36 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
24 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 ...
0
votes
0answers
13 views

Cauchy integral formula using a point in the border of contour

I have this integral to solve $\displaystyle{\int_C\frac{1}{(z+i)(z-i)}}\,dz$, with $C$ given by $\|z+i\|=1$. I need a point inside $C$, that is a circunference centered in $-i$, to apply the Cauchy ...
1
vote
1answer
23 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
1answer
36 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
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
25 views

Complex integrals with certain conditions

Let $r \in \mathbb R_{\geq 0}, a, b \in \mathbb C$ such that $|b-a|\neq r$ and $\gamma:[0,2\pi] \to \mathbb C$ given by $\gamma(t)=a+re^{it}$. Calculate $\int_{\gamma} (z-b)^ndz$ if $n \neq -1$. ...
1
vote
1answer
53 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
25 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
52 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
2answers
50 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
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 ...
1
vote
1answer
42 views

Cauchy Integral Formula Example

I am a beginner at calculating Cauchy Integrals but this one didn't look familiar to examples I have found. I think it is a linear integral. Could anyone give ideas how to solve it? Thank you in ...
2
votes
1answer
25 views

Basic complex integral property

I am trying to learn complex analysis using Alfohrs textbook and I have doubts about the proof of this property: When $a\leq b$, the fundamental inequality (1) $|\int_a^b f(t)dt| \leq \int_a^b ...
2
votes
2answers
54 views

A Contour Integral I

What is the value of the integral \begin{align} \int_{-a}^{c} \sqrt{ \frac{a+x}{c-x} } \ \frac{dx}{(d-x)(x-b)} \end{align}
1
vote
1answer
49 views

Complex integration Question - Contour Method [duplicate]

I'm asked to find: $$\int_{-\infty}^\infty \frac{\ln(x^2+1)}{1+x^2} dx $$ Attempt Considering $$ \oint \frac{\ln(z^2+1)}{(z+i)(z-i)} dz $$ So first I find the branch points of the function. This ...
3
votes
3answers
74 views

How can an improper integral have multiple values?

Integrals like this are said to dependend on the contour of integration: $$\int^{\infty}_{-\infty}\frac{x\sin x}{x^2-\sigma^2}dx=\pi e^{i\sigma}\space \mathrm{or}\quad \pi \cos\sigma $$ How is it ...
0
votes
0answers
47 views

A question on particular functions in $L^\infty$

Let D be the open unit disk in the complex plane, $\mu$ be the arc length measure, and $f, g\in L^{\infty}(\partial D,\mu)$ satisfying the following equations: $$ \int_{\partial D} ...
3
votes
1answer
22 views

Analytic function with values at specific points

Problem: Show that there is no function $f$, holomorphic on a neighbourhood of $z=0$, such that its value on $z=\frac{1}{n}$ is $(-1)^n (\frac{1}{n})$. To contrast, find a function holomorphic ...
0
votes
1answer
36 views

Quick question on complex integral

For $f(z) = (1-z^2)^{\frac{1}{2}}$, how do I show that the integral of $f(z)$ from $0$ to $\pi$ is $O(R^{-2})$? $$\int f(z) dz = \int \frac{1}{z(1-z^2)^{\frac{1}{2}}} dz $$
1
vote
1answer
34 views

Is Cauchy's integral theorem affected by integral direction?

hello,everyone,I hava a exam question aboat the integraion~ as shown below I know 1/(Z^2-1)=1/2(1/[z-1]+1/[z+1]) the integrantion around 1 should be 2*pi*i, but i am confused if the integrantion ...
0
votes
1answer
85 views

an amazing complex integral along unit circle

Let $f$ be an entire function on $\mathbb{C}$. Let $z_0\in\mathbb{C}$. Let $C=\{z\in\mathbb{C}\mid |z-z_0|=1\}$. Suppose $f(z)\neq f(z_0)$ for any $|z-z_0|\leq 1$, $z\neq z_0$. Given $f'(z_0)=2$, ...
-1
votes
2answers
33 views

Complex integration around a singularity [duplicate]

I am trying to integrate the function $f(z)=$$\frac{5}{z^2}$ from -3 to 3 and I am supposed to develop a closed region that avoids the origin and use the analyticity of the function in this region to ...
0
votes
1answer
41 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 ...
2
votes
1answer
54 views

Integration of complex function with respect to complex variable

I was given as homework to calculate the complex integral limit $$\lim_{T\rightarrow \infty} \frac {1}{2\pi i}\int_{c-iT}^{c+iT}\frac {x^s}{s^{k+1}}ds $$ where $c>0$ and $k\geq1$ is an integer. ...
0
votes
1answer
29 views

Complex Integral - exponential divided by a monomial

How does one solve integrals like this- $$I=\int^\beta_0 dx \frac{\exp(i\omega_nx)}{x-a}$$ where $\omega_n=\frac{\pi n}{\beta} $. EDIT: $\beta$ is a finite, real ...
1
vote
0answers
21 views

Second Derivative of Holomorphic function According to Cauchy's

I am asked to prove that, supposing f is holomorphic on the region G, w $\in$ G and $\gamma$ is a positively oriented, simple, closed, smooth, G-contractiblecurve such that w is inside $\gamma$ , we ...
1
vote
1answer
39 views

Integrating the function Im(z) on a variety of contours.

I've been asked to evaluate $\int_C Im(z) dz$ for a variety of contours, which I've had no issue in doing. For the sake of clarity, these contours included the upper and lower halves of the circle ...
0
votes
2answers
36 views

Complex Analysis, Integral over a Square

Given that $C$ is the boundary of the square with corners at $\pm4 \pm4i$ (sorry my formatting always seems to be stubborn, but that is plus or minus 4 plus or minus 4i, I am asked to compute $$\int_C ...
1
vote
1answer
32 views

A problem with Cauchy Theorem

I want to resolve the folowing contour integral, using the Cauchy theorem: $$ \oint_C \cot(\pi z)\,dz $$ where $C$ is rectangle defined by $x=\frac{1}{2},x=\pi, y=-1, y=1 $ I do understand that ...
0
votes
0answers
28 views

Contour integral going to zero on a limit

I've been asked to prove the following; $$\lim_{R \rightarrow \infty}\int_{C_R} \frac{z^2 + 8z + 42}{(z^2+4)(z^2-4z+5)}dz=0$$ Given that $C_R$ is a circle of radius $R$ centered at $0$. I thought ...
0
votes
1answer
29 views

Complex Integration-Computing winding number of a curve

I need to compute the winding number of $\alpha$ with respect to the point p=(1/2,0) Where $\alpha: [0,2\pi] \to \mathbb R^2$ $\alpha(t)=((2 Cos[t] - 1)*Cos[t], (2 Cos[t] - 1)*Sin[t])$ The winding ...
0
votes
1answer
50 views

Laplace's Method Integration

Consider the integral \begin{equation} I_n(x)=\int^2_1 (\log_{e}t) e^{-x(t-1)^{n}} \, dt \end{equation} Use Laplace's Method to show that \begin{equation} I_n(x) \sim \frac{1}{nx^\frac{2}{n}} ...
5
votes
2answers
110 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 ...
5
votes
1answer
70 views

Any general methods to calculate integral of $P(x)/Q(x)$ from $0$ to $\infty$?

In complex analysis, we have general formula for $P(x)/Q(x)$ [$P$ and $Q$ are polynomials] from minus infinity to infinity, if $ \deg Q - \deg P > 2$. Is it possible to have a general formula for ...
1
vote
3answers
54 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 ...
4
votes
1answer
117 views

Residue theorem with exponential and trig functions

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