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

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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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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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47 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
44 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
51 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
39 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
71 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
40 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
30 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
73 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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59 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 ...
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134 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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96 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 ...
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1answer
47 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
50 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
45 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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41 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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22 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
64 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
37 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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59 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
58 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
124 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
36 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
141 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
109 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
23 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
24 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
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 ...
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1answer
20 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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20 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 ...
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19 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
30 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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100 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
66 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
40 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
30 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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34 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
159 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
51 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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54 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 $f(s)=\frac{a^{1-s}}{(1-s)}-\frac{b^{1-s}}{(1-s)}$, ...
3
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1answer
41 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
37 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
68 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
26 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
49 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, ...
2
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1answer
33 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 ...