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

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1answer
26 views

Help With Bromwich Inversion Formula Proof

To prove(copied from handwritten notes so possibly wrong): Bromwich Inversion Formula. Fix $x_0∈ℝ $. If $F$ is complex analytic on $\{z:\Re z > x_0\} $ and for every $x>x_0$, $y↦ F(x + iy )$ ...
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1answer
27 views

Improper integral (using methods in complex variables) [on hold]

Let $0<a<1$. Evaluate the integral $$\int_0^\infty \frac{x^{a-1}}{1+x} dx.$$
2
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1answer
29 views

Complex integration on circle

Calculate the integral of $g(z)$ along the closed path $|z-i|=2$ in the positive direction when i)$g(z)=\frac{1}{z^2+4}$ ii)$g(z)=\frac{1}{(z^2+4)^2}$ First I checked the described area ...
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1answer
24 views

Complex integration and theorems

If $C$ is a closed path oriented in the positive direction and $$g(z_0)=\int_C \frac{z^3+2z}{(z-z_0)^3}$$ show that $g(z_0)=6\pi iz_0$ when $z_0$ is in interior of $C$ and $g(z_0)=0$ when $z_0$ is out ...
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0answers
18 views

when is $\int_{t=-x}^{x} e^{iat}\operatorname{sinc}(t) \, dt = 0$, with $x \in \mathbb{R}$?

For what value of $x \in \mathbb{R}$ is $\int_{t=-x}^{x} e^{iat}\operatorname{sinc}(t)\,dt = 0$, where $a$ is some constant?
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1answer
37 views

Why is $\left| \int_{|z|=1} \frac{f(z)}{z}dz\right| \leq \int_{|z|=1} \frac{|f(z)|}{|z|}|dz|$?

Why is $$\left| \int_{|z|=1} \frac{f(z)}{z}dz\right| \leq \int_{|z|=1} \frac{|f(z)|}{|z|}|dz|$$ and not $$\left| \int_{|z|=1} \frac{f(z)}{z}dz\right| = \int_{|z|=1} \frac{|f(z)|}{|z|}|dz|?$$
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18 views

Inverse $z$ transform - contour integration

Here is my task: Find inverse $z$ transform of $$X(z)=\frac{1}{2-3z}$$if $$|z|>\frac{2}{3}$$ using definition formula. I found that $$x(n)=\dfrac{1}{3}\left (\dfrac{2}{3}\right ...
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0answers
28 views

Lebesgue integration in $\mathbb{C}$

I'm confused as to how we are supposed to integrate $$\frac{1}{\pi}\int_U\left[\frac{d}{dz}\left( \frac{z-\alpha}{1-\bar\alpha z }\right)\right]^2 \, dm$$ where $U$ is the unit disc, ...
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1answer
22 views

Cauchy integral formula for rational function, help with step

I have $P(\lambda) = (i\lambda)^m + O(\lambda^{m-1})$ a polynomial in $\lambda$, and $\Gamma$ a contour counterclockwise around the roots of $P$. I need to prove: ...
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1answer
23 views

Integrals with complex functions: integration by parts and conjugate

I am working with integrals of complex functions. I assume all terms are well-defined. If $u=u(x,t):\mathbb{R}^n\times\mathbb{R}_+ \to \mathbb{R}$ (a real function), I have \begin{equation} ...
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1answer
52 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 $R>0$
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1answer
52 views

How can I solve this integral with complex number?

$n$ here is a complex number such that $n=n_r+in_i$ How can I solve this integral? $$\int_{0}^{\infty}\frac{x^4}{|x^2-n^2|^2} d x=? $$
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2answers
81 views

$\int\limits_{\gamma} \frac{z}{(z-1)(z-2)}dz$, $\gamma(\theta) = re^{i\theta}$, $2 < r < \infty$

For $0 < r < 2$, we can use Cauchy's integral formula and choose our holomorphic function to be $f(z) = \frac{z}{z - 2}$ since $z = 1$ is the only pole, but if $r > 2$, then both poles $z = ...
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3answers
44 views

Fourier transform of $e^{-4\pi ^2 x^2}$

How do you prove $$\int_{-\infty}^{\infty}e^{-(2\pi x + i\xi/2)^2}dx=\int_{-\infty}^{\infty}e^{-(2\pi x)^2}dx$$ for $\xi \in \mathbb{R}$. The Question arises from calculating the Fourier Transform ...
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0answers
24 views

Contour integration along a contour containing two branch points

I need to compute following contour integrations: $$f(u)=\oint_\alpha dz \sqrt{z^3+z+u} \qquad ; \qquad g(u)=\oint_\beta dz \sqrt{z^3+z+u}$$ In which $\alpha$ and $\beta$ are two contours in ...
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0answers
30 views

Integration everywhere [duplicate]

Find this $\displaystyle\int\frac{e^ x}{x}\;dx$
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21 views

How do I evaluate a integral complex

a. I definitely like starting the function of Part A b.
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1answer
33 views

Use Cauchy's Integral Formula to evaluate the following integrals.

Use Cauchy's Integral Formula to evaluate the following integral: $$\int\limits_\Gamma \frac{1}{{(z-1)^3}{(z-2)^2}}dz$$ where $$\Gamma$$is a circumference of radius $4$ centered at $-2+i$ and ...
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1answer
27 views

Calculate complex integral $\int_\Gamma\frac{\ln(z+5)}{z^3+iz^2+6z}$

$\Gamma$ is a circle of radius 2 around the point $1+i$. I've parametrized the circle as $\gamma(t)=2e^{it}+1+i$ substituting $z$ in te integral for that expression gets really ugly really quickly. ...
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1answer
19 views

Characteristic Function and Density Function

Consider a random variable $X$ with density function $f(x)$, moment generating function $M(t):= \int e^{tx}f(x) dx$ (existing in an interval containing $0$), cumulant generating function $K(t):=\log ...
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1answer
27 views

How to use Cauchy's integral formula with more than one pole?

$\int\limits_{\gamma} \frac{z^2}{z(z-2)}$ $\gamma(\theta) = 3e^{i\theta}$, $0 \leq \theta \leq 2\pi$ Cauchy's integral formula is given by: $$\int\limits_{\gamma} \frac{f(z)}{(z-a)^{n+1}} = ...
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0answers
35 views

Contour integration with a branch cut. Parameterizing f(z) properly

I have a contour integral of a function of the form $(z^6-P)^\alpha z^\beta$ Here $\alpha\in R$, $\beta\in N$ and $P$ is some constant. I therefore have branch points at the sixth roots of $P$. The ...
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complex equations proving

I have a problem with the below equations, I can't prove it. Suppose that F[f(x)]=F(alfa) then prove equations 1 and 2.
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1answer
58 views

show that $\int_0^{\infty}\ln(2x)\frac{ax-u(1-e^{-2ax})}{\sinh^2(ax)}x^{2u-1}dx=\frac{1}{2u}(\frac{\pi}{a})^{2u}|B_{2u}|:u \in N$

$$\int_0^{\infty}\ln(2x)\frac{ax-u(1-e^{-2ax})}{\sinh^2(ax)}x^{2u-1}dx=\frac{1}{2u}(\frac{\pi}{a})^{2u}|B_{2u}|:u \in N$$ using real or complexe analysis where $B_{2u}$ is bernoulli number
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1answer
68 views

Are the integrals of the following function path independent in the following domain?

Are the integrals of the function: $$f(z)=\frac{1}{z+1}+\frac{1}{(z+1)^2}+e^{\frac{1}{z}}$$ path independent in the following domain: $$D= \{Re z >0\}\setminus\{1\}$$ My thoughts on the ...
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0answers
34 views

Substitution in complex integration

Let $\gamma$ be the circle $|z| = r$, $0 < r < \frac{\pi}{2}$, taken positively. Find $$ \int _{\gamma} \frac{1}{\text{tan}^{17}(z)} dz ~\text{and} \int _{\gamma} \frac{1}{\text{sin}^{15}(z)} dz ...
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1answer
44 views

Trying to understand a proof in Rudin concerning winding number

In the proof of theorem 10.10 in Real and complex analysis Rudin states that if we will differentiate $$\phi(t) = \exp \left\{\int_a^t \frac{\gamma'(s)}{\gamma(s)-z} \,\textrm{d}s\right\}, \textrm{we ...
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1answer
26 views

How to compute the following real integrals using the residue theorem?

How to compute the following real integrals using the residue theorem: $$\int_{-\infty}^{\infty} \frac{1}{(x^2+p^2)(x^2+q^2)} dx$$ $$\int_{0}^{2\pi} \frac{sin^2(\theta)}{5+4cos(\theta)} d\theta$$ ...
2
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1answer
44 views

Evaluate the integral $\int_0^\infty\frac{x^a\ln{x}}{x+b} \, dx$

Integrate $\dfrac{x^a\ln{x}}{x+b}$ from 0 to infinity where $b > 0$ and $-1 < a < 0 $ I'm having trouble deciding how to approach the problem! Thank you!
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0answers
23 views

Calculate the integral $f(z)=\frac{e^{iz}}{z(z-\pi/2)^2}$ over $|z+1|=2$

Calculate the integral $f(z)=\frac{e^{iz}}{z(z-\pi/2)^2}$ over $|z+1|=2$. Since the singularity at $z=0$ is in the given contour, I integrated using Cauchy's theorem to get $$2\pi i \left[ ...
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2answers
50 views

How do I use residue theorem to evaluate this improper integral to get a good looking solution?

The problem is $\int_{0}^{\infty} \frac{\sqrt{x}}{x^2+2x+5}dx$ I replace x with z, and did some algebra, but the solution was rather nasty. it contains exponential and arctan such and such. However, ...
3
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1answer
43 views

Any idea how to evaluate this equation?

I'm trying to evaluate(approximate) the following integral $$ F(x,t;q) = \int_{-\infty}^{\infty}\frac{q}{q+2ik} e^{i(kx +8k^3 t)}\; dk $$ It's similar to the Airy function but I can't get rid of ...
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1answer
40 views

Equality concerning a certain complex integration

I would like to verify the following statement. However, I do not know what should I do. Generally, I think that, since there is \$1/2\pi i$ part, Winding number, Residue theorem and Argument ...
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2answers
79 views

Contour Integral of $\log(z)/(1+z^a)$ where $a\gt1$

I am asked to prove that: $$ \int_{0}^{+\infty}\frac{\log z}{1+z^{\alpha}}\,dz = -\frac{\pi^2}{\alpha^2}\cdot\frac{\cos\frac{\pi}{\alpha}}{\sin^2\frac{\pi}{\alpha}},$$ provided that $\alpha > 1$, ...
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0answers
30 views

(Complex) integral over a half circle

I read a book and find one example which I do not understand : Let $f(z) = \frac{e^{iaz}-e^{ibz}}{z^2}$ on $B(0,1)\setminus\{0\}$. Let $\gamma$ be the half circle over x-axis : $\gamma (t) = ...
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1answer
35 views

Complex entire functions without taking values on a segment are constant!

Let $a,b$ be two distinct complex numbers and $f$ be an entire complex function, i.e. a complex function which is analytic on the whole complex plane, and $$R(f)\subset\mathbb C-\{\lambda ...
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23 views

Compute the integrals $\int_{C_1}f(z)dz$ and $\int_{C_2}f(z)\,dz$.

Given two smooth contours, $C_1$ and $C_2$, that respectively lie on the upper and lower half plane compute $\int_{C_1}f(z)dz$ and $\int_{C_2}f(z)\,dz$. Let $a$ be a fixed real positive number. ...
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1answer
22 views

Bound of a complex function

I have the complex function $\frac{C^{1+iz}}{iz(1+iz)}$ for $C$ a positive constant and $z$ a complex variable for which $\Im(z)>1$. I am looking for a bound of this function. Any suggestions? I ...
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38 views

Parametrizing a Rectangle for a Path Integral- Complex Analysis

Okay the problem I'm trying to solve is: I'm farily certain I can solve this, once I can figure out how to parametrize the rectangle. I read somewhere on here for another question that I can ...
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2answers
42 views

How does this integration pass?

Can someone explain the passage in the red-box for me? I am getting \begin{align} & -\frac{1}{2\pi}\int_{0}^{2\pi} \log r \;d\theta - \frac{1}{2\pi} \int \log \| e^{i\theta_1} - ...
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1answer
48 views

Find the Taylor Series expansion of the given analytic function

Find the Taylor Series expansion of the given analytic function $f(z)$, centered at point $z_0$; find the disk of convergence. a) $f(z)=\frac{1}{-2+3i-z}$ $z_0=3$ b) $f(z)=(2-z)\cos{(3z^2)}$ ...
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1answer
34 views

Find the Taylor series expansion (centered at $z_{0}=0$) of the function $f(z)=\sin(z^3)$.

Find the Taylor series expansion (centered at $z_{0}=0$) of the function $f(z)=\sin(z^3)$. Use this expansion: a) to find $f^{69}(0)$; b) to compute the integral transversed once in the positive ...
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2answers
67 views

Let f be an analytic function in the disk … Prove that

Let $f$ be a function analytic in the disk $D={z:|z-1-i| \leq 4}$ that satisfies $|f(z)| \leq 1$ for all $z$ in $D$. Prove that $|f^{(3)} (1-i)| \leq \frac{3}{4}$. Hint: apply Cauchy's Integral ...
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2answers
38 views

Can I use Cauchy's integral formula to solve this integral? [closed]

Can I use Cauchy's integral formula to solve this integral? $$\int_C \frac{\cos(z)^2}{z^3} dz $$ where $C$ is the contour of a circle centre $0$ and radius $1$. i.e is the solution $2\pi i$?
2
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1answer
25 views

contour integral complex conjugate

I'm having trouble trying to find this integral, where $C$ is the semicircle, centre $z = 1$, of radius $1$, lying in the upper half-plane $$ \int_C \bar{z}\ {dz} $$ Currently I have that, ...
3
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1answer
67 views

Use Cauchy's Integral Formula to evaluate the following integral.

$$\int \frac{e^z\sin z}{(2z+5i)^2} \, dz,$$ where $$\gamma$$ is a circumference of radius $5$ centered at $-4$ and traversed once in the negative (with respect to its interior) direction. I've been ...
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2answers
56 views

$f(z)=\frac{1}{z}$ has an antiderivative on any simply connected domain

Prove that the function $f(z)=\frac{1}{z}$ has an antiderivative on any simply connected domain of $\Bbb C$ which does not contain zero. Also prove that this function does not have an ...
6
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0answers
125 views

Complex contour integral: How does the stationary point method used in this case?

I was reading a paper which has the following integral in order to do the inverse Laplace transformation: $$ I=\frac{1}{2\pi i}\int_{-i\infty+\gamma}^{i\infty+\gamma} ...
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0answers
95 views

Using the “appropriate” formula

I am asked to solve $$\int_{C}\frac{1}{z+i}dz$$ where $C$ is parametrized $z(t) = 2+e^{it}$ for $t \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ by finding the antiderivative $F(z)$ of $f(z)$ and then ...
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1answer
36 views

Explaining information on Contour

I wish to compute the following line integral $$\int_{C}(x-iy)dz$$ where $z(t)=(e^t- 1,t)$, $t \in [0,2]$ $dz = dx + idy = (e^t - 1 + i)dt$ We then have $\int_{C}(e^{t} - 1 - it)(e^t - 1 + i)dt$ I'm ...