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

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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
21 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
24 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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20 views

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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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
55 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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32 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
23 views

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

How to compute the following real integrals using the residue theorem: $$\int_{-\infty}^{\infty} \frac{1}{(x^2+p^2)(x^2+q^2)} dx$$ $$\int_{0}^{2\pi} \frac{sin^2(\theta)}{5+4cos(\theta)} d\theta$$ ...
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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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20 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
46 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
71 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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28 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
33 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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21 views

Given two smooth contours, $C_1$ and $C_2$, that respectively lie on the upper and lower half plane compute the integral of $f(z)dz$ over each

Let $a$ be a fixed real positive number. Then, let $C_1$ be a smooth oriented path from $a$ to $-a$, which lies entirely on the upper half-plane, and let $C_2$ be the smooth oriented path from $-a$ to ...
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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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34 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
47 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
65 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
37 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 $2\pi i$?
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1answer
21 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
63 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
49 views

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 antiderivative on its entire domain. I feel as if i have to use a lot of topology to prove these facts and I am not as proficient in that area as I ...
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0answers
121 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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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
35 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 ...
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1answer
45 views

Why is $\frac{1}{n!}=\frac{1}{2\pi r^n}\int_{0}^{2\pi}e^{re^{it}}e^{-int}dt$

How to prove that $\displaystyle\frac{1}{n!}=\frac{1}{2\pi r^n}\int_{0}^{2\pi}e^{re^{it}}e^{-int}dt$ for any natural number $n$ and poisitive real number $r$ I got with $f(z)=e^z$ and Cauchy's ...
3
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1answer
42 views

Cauchy's Integral parametric conjugate

By considering the conjugate of its parametric form, evaluate $$\frac{1}{2\pi i}\int_{\gamma(0;1)}\frac{\overline{f(z)}}{z-a}dz$$ when $|a|<1$ and $|a|>1$, where $f$ is holomorphic in in the ...
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1answer
42 views

Deriving the Poisson Integral Formula from the Cauchy Integral Formula

If $f$ is analytic inside and on the unit circle $\gamma$, show that for $0<|z|<1$, $$2\pi if(z)=\int_\gamma \frac{f(w)}{w-z}dw-\int_\gamma \frac{f(w)}{w-1/\bar{z}}dw$$ and then derive the ...
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3answers
37 views

Applying Cauchy's Integral Theorem to $\int_{C_R} z^n \ dz$

First, Cauchy's Integral Theorem: If $f$ is a continuous function on $U$ admitting a holomorphic primitive $g$, and $\gamma$ is a closed path in $U$, then \begin{equation} \int_\gamma f = 0 ...
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21 views

Is this computation using Cauchy's integral formula correct?

I need to compute the integral $ \int_\gamma \frac {dz}{z^3}$, where $\gamma$ is the square with vertices $-1-i, 1-i,1+i, -1+i$. I used Cauchy's integral formula for derivatives the function $f(z)=1$ ...
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2answers
48 views

Integrals of fractions(Complex)

I'm a bit clueless about some (presumably basic) complex integrals. How would I integrate (over a circle centered at the origin, let's say of radius 2) things like $\frac{1}{z^2+z+1}$ or $\frac ...
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1answer
58 views

How to integrate $e^z/z^2$?

This may be a very basic question. How to compute the integral $ \int_\gamma \frac{e^z}{z^2} \, dz$, where $\gamma$ is the unit circle? I did it with Cauchy's integral formula for $\int_\gamma ...
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1answer
30 views

Complex Line Integral $\int_{i}^{i+1}{z\>dz}$ along a straight line parallel to the $x$ axis.

PROBLEM Integrate $\int_{i}^{i+1}{zdz}$ along a straight line parallel to the $x$ axis. The definition of a complex line integral states let $f(z)$ be a continuous complex-valued function of a ...
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53 views

Winding number of composition of curve

Let $f$ be analytic in a "simply connected domain" D $\subseteq \mathbb{C}$ and let $\gamma$ be a closed piecewise differentiable path in $D$. Set $\beta = f \circ \gamma.$ Show that $\frac{1}{2\pi ...
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2answers
53 views

Evaluating an integral using Cauchy's Integral Formula

I am having a little bit of trouble with the following: $$\int_{\gamma}\frac{z^2-1}{z^2+1}dz$$ where $\gamma$ is a circle of radius $2$ centered at 0. I am trying to separate this or simplify it into ...
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2answers
117 views

Complicated contour integral to be solved.

Anyone can help to solve the following integral? $$I=\int_{0}^{\infty} dp p^{-1}e^{-2p^{2}M^{-2}}\sin(pr)\frac{M^2}{M^2+p^2}$$ at this stage I am able to write the integral as ...
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1answer
34 views

Complex analysis ~ Binomial theorem

Given the identity $ \binom {2n} {n} = \frac{1}{2\pi i} \int_{C_r} \frac{(1+z)^{2n}}{z^{n+1}}dz,$ with $C_r$ the unit circle, prove that $\forall n \in \mathbb{N}$: $\binom {2n} {n} \leq 4 ...
3
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1answer
133 views

Show that $ \int^1_0 x^3 \sqrt{x} \sqrt{1-x} dx = \frac{\pi}{5!} \frac{1.3.5.7}{2^5} $

I'm trying to show the following. $$ \int^1_0 x^3 \sqrt{x} \sqrt{1-x} dx = \frac{\pi}{5!} \frac{1\cdot3\cdot5\cdot7}{2^5} $$ This is a problem regarding contour integration. My complex analysis ...
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2answers
95 views

Compute the integral $\int_C (z^2-1)^\frac{1}{2} dz$ where $R>1$

Let C be the circle of Radius $R>1$, centered at the origin, in the complex plane. Compute the integral $\int_C (z^2-1)^\frac{1}{2} dz$ where we employ a branch of the integrand defined by a ...
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1answer
18 views

Upper bound on complex integral

If $f(z)=\sum_{n=0}^{\infty}c_nz^n$ and we know $$c_k=\frac{1}{2\pi i}\int_\gamma \frac{f(z)}{z^{k+1}}dz$$ for $\gamma$ a circle of radius r centred at the origin, traversed once in the positive ...
0
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1answer
23 views

Complex Integration with Power Series

Let $f(z)=\sum_{n=0}^{\infty}c_nz^n$ have radius of convergence $R>0.$ Use the fact that $$\sum\limits_{n=0}^{\infty}\int_\gamma c_n z^ndz=\int_\gamma ...
0
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1answer
17 views

Complex Integration parametrisation

I'm trying to integrate $\int_\gamma (z^2-2)dz$ where $\gamma$ is a spiral that loops 3 times and ends at (3,0) on the Argand diagram. I have found the parametric equations for this contour to be ...
0
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1answer
18 views

Complex contour integration of a branch (Not even sure what it's asking)

$f(z)$ is the branch $z^{-1 + i} = e^{(-1 +i)\ln{z}}$ such that $|z| > 0$ and $0 < arg(z) < 2\pi$. I'm to integrate $f(z)$ over the contour $e^{i\theta}$ (just the unit circle). ... I have ...
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0answers
19 views

Prove that z(t) and z~(t) are admissible parametrizations of the same smooth curve

Prove that $$z(t)=t+it^2, 0\leq t \leq1 $$ and $$\tilde{z}(t)=tan\Gamma+itan\Gamma, 0 \leq \Gamma \leq \frac{\pi}{4}$$ are admissible parametrizatiions of the same smooth curve. Do the above ...