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

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1answer
19 views

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

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
16 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, ...
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1answer
31 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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1answer
53 views

Let f be an entire function. Prove that $f(z) = e^{z^2}$ for all $z \in \mathbb{C}$. [on hold]

Let $f$ be an entire function. Assume that $|f(z)|$ $> \frac{1}{3}$ $|e^{z^2}|$ for all $z$ in the complex plane and assume that $f(0)=1$ Prove that $f(z)=e^{z^2}$ for all z in the complex ...
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2answers
46 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
29 views

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

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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92 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
30 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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42 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 ...
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1answer
37 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
37 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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20 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
52 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
29 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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52 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
48 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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116 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
32 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 ...
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1answer
131 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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92 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
17 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 ...
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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 ...
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1answer
15 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 ...
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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
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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 ...
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13 views

Use this parametrization to compute the following integral.

Let $$\Gamma$$ be the circumference centered at 1-i of radius 5 and transversed once in the counterclockwise direction. Parametrize the contour $$\Gamma$$. Use this parametrization to compute the ...
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1answer
19 views

Parametrize the contours of integration where Gamma is arc of the circle of radius…

Parametrize the contours of integration and write the integrals in terms of the parametrizations. Do not calculate them. $$\int\frac{\bar(z)}{z^3}dz$$ where $$\Gamma$$ is the arc of the circle of ...
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0answers
78 views

Complex line integral for closed homotopic curve.

I try to show that the following analogous result for $\textbf{closed curve}$ : Let $\gamma_0, \gamma_1$ be continuous paths in a domain $D \subseteq \mathbb{C}$ and $\omega$ be a closed form on $D$. ...
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1answer
54 views

Are there exists an analytic function satisfying the following condition

Let, $D=\{z\in \mathbb C:|z|<1\}$. Then there exists a non-constant analytic function$f$ on $D$ such that for all $n=2,3,4,...$ (a) $f\left(\frac{i}{n}\right)=0$. (b) ...
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1answer
34 views

Parametrize the contours of integration

I am having a difficult time figuring this problem out: Parametrize the contours of integration and write the integrals in terms of the parametrizations. $$\int_{\Gamma} (3\bar{z}^2+2z^3)\,dz$$ ...
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1answer
21 views

Definite integral of absolute value complex function

Seems pretty straight forward but absolute values have always given me headaches $$\int_0^1 |1 -t + it|^2$$ Now usually I get roots and split up the intervals for when the function is greater or ...
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Need help integrating exp(A*cos(x - k1)…

Hi Guys so i need some help integrating this function: $$\mathcal I = \int_0^{2\pi} e^{A\cos({\psi - \theta}) + B\cos(\psi - \phi)} d\psi$$ where $\theta$ and $\phi$ are independent of each other ...
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30 views

Evaluating this contour integral.

I was reading a paper that had the following integral $\displaystyle\prod_{n=1}^L\oint_{C_n}\frac{dx_n}{2\pi i}\prod_{k<l}^L(x_k-x_l)\prod_{m=1}^L\frac{Q_w(x_m)}{Q^+_\theta(x_m)Q^-_\theta(x_m)}$ ...
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2answers
112 views

Improper integrals with singularities on the REAL AXIS (Complex Variable)

I'm having some troubles when I try to solve improper integrals exercises that have singularities on the real axis. I have made a lot of exercises where singularities are inside a semicircle in the ...
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2answers
41 views

Fresnel Integrals

I'm having trouble proving that the arc from $R$ to $Re^{i\pi/4}$ in the Fresnel contour goes to zero. Currently I have $\int_0^{\pi/4} ...
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53 views

Why is $f(s)=\int^{b}_{a}\frac{1}{t^s} dt$ holomorphic?

In complex analysis, let $a, b>0$ in $\mathbb R$, $f(s)=\int^{b}_{a}1/t^s dt$, then $f$ is holomorphic for $Re(s)>0$. If $s\neq 1$, then $f(s)=\frac{a^{1-s}}{(1-s)}-\frac{b^{1-s}}{(1-s)}$, ...
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1answer
40 views

Complex integration identity

I'm trying to prove VI.8.4 from Sarson's Complex Functions Theory: Let $f \in C^1$ be a complex-valued function defined and continuous on the disk $|z-z_0| < R$. For $0 < r < R $ let $C_r$ ...
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1answer
34 views

Are these two elliptic integral evaluations identical?

I'm reading a paper on the Schwarz D minimal surface, and I'm wondering whether the authors have made a mistake. They evaluate the integral $$ \int_0^z \frac{2t\;\mathrm{d}t}{\sqrt{t^8-14t^2+1}}, $$ ...
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1answer
63 views

Proof of Laurent series co-efficients in Complex Residue

Am trying to see if there is any proof available for coefficients in Laurent series with regards to Residue in Complex Integration. The laurent series for a complex function is given by $$ f(z) = ...
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1answer
25 views

Writing same equation in different forms

I am working with a unit circle with imaginary integration. I know from experience that this can be written as $f(\theta)=\cos t+ i \sin t$ or $e^{i \theta } $ My question would be if i have a circle ...
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1answer
48 views

Proof some 2 D Fourier transforms

Here are several Fourier transforms I used, I would like to prove those identity. I took some times to figure out how they are derived, I tried the residue theorem and other methods, but I failed, ...
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1answer
31 views

Complex var. integral: $\oint_{|z|=1} \frac{z^2\ dz}{\sin^3{z}\cos{z}}$

Integrate $\displaystyle\oint_C \dfrac{z^2\ dz}{\sin^3{z}\cos{z}}$; $C \rightarrow |z|=1$ I already know that $|z|=1$ is a circumference with $r=1$ and center at $(0,0)$. I also know there are ...
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0answers
45 views

Complex integration misconception?

Playing around with the complex integretion I encountered the following: Consider a holomorphic function $f(z)$ on $\Omega$. Let's say this holomorphic function has a primitve $F(z)$ such that $F'(z) ...
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1answer
60 views

Some inequalities for an entire function $f$

Let, $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ be an entire function and let $r$ be a positive real number. Then, which is(/are) correct? (a) $\sum_{n=0}^{\infty}|a_{n}|^{2}r^{2n}\le sup_{|z|=r} ...
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1answer
93 views

Cauchy Residue Theorem and Cauchy integral formula

Is it true that you can use the Cauchy Residue Theorem and the Cauchy integral formula interchangeably? I believe that the functions that satisfy the conditions of one, will indeed satisfy the ...
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1answer
80 views

Function holomorphic except for real line and continuous everywhere is entire

1) I've already shown that if $f:\mathbb{C}\rightarrow\mathbb{C}$ is holomorphic everywhere except for a single point and if it continuous on whole $\mathbb{C}$ then it is entire. It was quite easy. ...
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0answers
31 views

How can I solve $\int \left| \sum_{i=0}^{N} f_i(x) \right|^2 d x $ for complex functions $f_i$?

The $f_i(x)$ are complex functions and I know that $\int_{-\infty}^{\infty} f_i (x ) \, d x=c_i$. How can solve this integral? $$\int_{-\infty}^{\infty} \left| \sum_{i=0}^{N} f_i(x) \right|^2 d x $$ ...
1
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1answer
44 views

A complex integration arround the boundary of a rectangular region

Let, $u(x,y)$ be thereal part of an entire function $f(z)=u(x,y)+iv(x,y)$ for $z=x+iy\in \mathbb C$. If $C$ is the positively oriented boundary of a rectangular region $R$ in $\mathbb R^{2}$ then the ...