Questions on the evaluation of integrals along a locus in the complex plane.

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Effective Branch cut

I have problems understanding how to get to the "effective Brach cut" in the top answer of this post: Dog Bone Contour Integral ? The answer says that one has a Branch cut for ...
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2answers
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Integral on the real line between 0 and infinity using contour integration

For part (a) I have that the singularity is at $(1+i)/root2$ and it is a simple pole? For part (b) I have that the residue at $f(z)$ at that point is $-(1+i)/4root2$ For part (c) I used the ML ...
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2answers
66 views

Integrals on the real line using contour integration

I know I am supposed to split it up like this and gamma(R) tends to zero and the other tends to my integral as R tends to infinity? I compute the residue at $2i$ which I think is $sin(2i)/2$ ? ...
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2answers
38 views

Contour integral, f(z)=$ze^{z^2}$

For part $(a)$ is the answer just $0$? Using Cauchy-Goursat theorem? For part $(b)$ I am confused. Do I use ? It seems very complicated. Am I missing a trick?
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1answer
49 views

A real integral (may be requires contour integration)?

The integral I have in mind is $$\int^\infty_0 x^{r}(x + \lambda)^{-1}dx$$ where $r \in (-1, 0)$, and $\lambda$ is a non-negative constant. I apologize if this is really easy and I am missing some ...
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1answer
15 views

Question about a certain step in Rudin's General Cauchy Theorem proof

I am having trouble seeing a certain claim that Rudin makes in proving his "Global Cauchy's Theorem": $\textbf{Cauchy's Theorem.}$ Suppose $f$ is holomorphic in $\Omega$, which is an open set in ...
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27 views

Contour integration, cos(z)sin(z)

Evaluate \begin{equation*} \int_{\Gamma}\cos(z)\sin(z)dz,~\Gamma:\gamma(t):=\pi t+(1-t)i,~0\leq t\leq 1. \end{equation*} I think I should do it using this \begin{equation*} ...
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2answers
67 views

$\int \frac{\exp (z)(\sin(3z)}{(z^2-2)(z^2)} dz$ on $|z|=1$

So I need to calculate \begin{equation*} \int \frac{\exp(z) \sin(3z)}{(z^2-2)z^2} \, dz~\text{on}~|z|=1. \end{equation*} So I have found the singularities and residues and observed that the ...
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27 views

Contour integral from first principles

What does it mean by 'evaluate from first principles? Does it mean use ? For part (a) do I parametrise as $\gamma(t)=a+2e^{it}$ with $t$ between $0$ and $2\pi$? Doing this I end up with the ...
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23 views

Counting poles that are shared between $f$ and $g$

Suppose I have a meromorphic function $f(z)$ with poles at $f_i$ and $\mathcal{Res}(f,f_i)=1$, and $g(z)$ with poles at $g_i$ and $\mathcal{Res}(g,g_i)=1$. I would like to construct a function ...
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1answer
44 views

Contour integral $|z-i|=1/9$

Calculate \begin{equation*} \int_{\Gamma}\frac{1}{z^4+16}dz, \end{equation*} where $\Gamma :|z-i|=\frac{1}{9}$. I have asked I similar question to this but I still do not understand.... when I find ...
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1answer
55 views

Calculate contour integral

Calculate $\displaystyle \int_\Gamma \frac 1{z^4 + 81}$ where $\Gamma: |z+i| = \frac 34$ Can somebody help me with this question please or give me a hint on how to get started, as I have never ...
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2answers
69 views

The value of $\sum_{n=-\infty}^{n=\infty}\frac{1}{n^2-z^2}$ on $\mathbb{C}\setminus\mathbb{Z}$

Show, for $z\not\in\mathbb{Z}$, that $$\sum_{n=-\infty}^{n=\infty}\frac{1}{n^2-z^2} = \frac{-\pi}{z\tan(\pi z)}$$ Hint: You may assume that there exists $C$ such that $|\pi\cot(\pi w)|\leq ...
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What would be line integral along path number (iv) [closed]

In the above image what should be the line integration along path iv. Thanks.
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2answers
81 views

Computing $\int_{0}^{+\infty}\frac{\log(x)}{\sqrt x(1+{x^2})}dx$.

I would like to compute the following integral : $$\int_{0}^{+\infty}\frac{\log(x)}{\sqrt x(1+{x^2})}dx$$ using Residue theorem. I took the contour corresponding to half of the "donuts" ...
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0answers
12 views

contour integral of complex function

How would one compute the contour integral of along the wedge shape contour for the function $f = z^{-3/2} = \dfrac{1}{r^{3/2}}\dfrac{1}{\cos(\dfrac{3\theta}{2})+i\sin(\dfrac{3\theta}{2})}$ or ...
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1answer
58 views

How to use a contour integral to evaluate $\frac{2}{\pi }\int\limits_0^\infty {x{e^{ - {x^2}t}}\sin ax} {\rm{ }}dx$?

So I faced this question in our textbook: Using a contour integral, evaluate the improper integral $$\frac{2}{\pi }\int\limits_0^\infty {x{e^{ - {x^2}t}}\sin ax} {\rm{ }}dx$$ I don't need the ...
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1answer
78 views

Evaluate integral over complex path numerically to show that $C_\infty$ is equivalent to $-I$

I would like to evaluate $$C_\infty = \int_{R = -a}^{R = a} H_0^{(1)}(z) e^{-izt} dz $$ where $H_0^{(1)}(z)$ is the Hankel function of the first kind, $a \rightarrow \infty$, and $$ z = R - i ...
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1answer
40 views

Application of Complex Variables

By considering the integral of: $$\left(\dfrac{\sin\alpha z}{\alpha z}\right)^2 \dfrac{\pi}{\sin{\pi} z},\quad \alpha \lt \dfrac{\pi}{2}$$ around a circle of large radius, prove that ...
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1answer
44 views

Improper Integral $e^{-x}/\sqrt x$ [closed]

How do I find $$\int^\infty _0 \frac{e^{-x}}{\sqrt x}$$
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3answers
45 views

Integration of complex functions with trig functions: $\int_0^{2 \pi} \frac{ d\theta}{5-\cos( \theta )}$

$\int_0^{2 \pi} \frac{ d\theta}{5-\cos( \theta )}$ How should I integrate this? Using the exponantial identities of trig? Any hints will be great... Thank you!
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2answers
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What does “let $f\left(z\right)=\bar{z}$” mean in this context

I'm reading a paper/handout on contour integrals and Cauchy's Theorem which says in an example Let $f\left(z\right)=\bar{z}$. $\cdots$ Then \begin{align} ...
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1answer
54 views

Weird (?): Use Cauchy's Integral formula to calculate $\int _{|z|=3} \frac{z}{z^2-\pi^2}dz$

Weird (?): Use Cauchy's Integral formula to calculate $\large \int _{|z|=3} \frac{z}{z^2-\pi^2}dz$. But the function $\frac{z}{z^2-\pi^2}$ is holomorphic on all of $|z|=3$. Am I missing something? ...
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1answer
48 views

How i can find the fourier transform of $\frac{\sinh(ax)}{\sinh(\pi x)}$ where,$ |a| < \pi$

Using a rectangular contour in the complex plane, bypassing the poles at $z=0$ and $z=i$, i got $$\int_{-\infty}^{+\infty}\frac{\sinh(ax)e^{ikx}}{\sinh(\pi x)}dx - ...
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60 views

Integration Around Part of a Branch Cut

I am studying the integral, given by a Laplace transform, $$\int_0^\infty\!e^{-\alpha x}\sinh^{-2/3}x\left(1+\frac 12\sinh^2x\right)^{-1/6}\left(1-\beta\sinh^{4/3}x\right)^{1/2}\,\mathrm dx$$ From ...
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39 views

Where am I using the condition that $\gamma$ does not contain the origin

I am trying to evaluate $\displaystyle \int_\gamma z^n dz$ where $\gamma$ is a circle not containing the origin with positive orientation ($n\in \mathbb{N}$). I first calculated $\displaystyle ...
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1answer
61 views

Symbolic contour integral evaluation

Can anyone help with the evaluation of the following contour integral : $$\oint\limits_C \phi(x,y)\,dx+\psi(x,y)\,dy.$$ Where the contour $C$ is given by: What I am looking for is how to split ...
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1answer
27 views

Limit as $r$ tends to zero of integral $\int_C \frac{e^{iz}-1}z \mathrm dz$

Let $\mathcal C$ be a semi-circle of center $O(0,0)$ and radius $R$, such that $y \ge 0$. Find the limit as $R$ tends to zero of: $$\int_{\mathcal C} \frac{e^{iz}-1}z \mathrm dz$$ How can I find ...
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2answers
44 views

Inverse Laplace transform of s/s-1

Finding the inverse laplace transform: $$L^{-1}\left\{\frac{s}{s-1}\right\}$$ I wrote: $$L^{-1}\left\{\frac{s}{s-1}\right\}=L^{-1}\left\{\frac{1}{s-1}\right\} + L^{-1}\{1\}=L^{-1}\{1\} + e^{t}$$ And ...
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0answers
29 views

Integral of function has different values depending on contour?

What can I say if the integral of my function has different values depending on contour? If my function were analytic on a domain it would evaluate to $0$ right? My contours give values like $\pi ...
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3answers
225 views

How to know if an integral is well defined regardless of path taken.

I can calculate \begin{equation*} \int_0^i ze^{z^2} dz=\frac{1}{2e}-\frac12, \end{equation*} but why can I calculate this irrelevant to the path taken? Is this since it is analytic everywhere - if ...
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2answers
44 views

Using Cauchy Integral Formula $\int_C \frac2{z^2 -1}dz$

I want to understand why I can't use Cauchy Integral Formula for the following problem: $$\int_C \frac2{z^2 -1}dz\text{ on the contour } |z-1|=\frac12$$ Now it says that I need $f$ to be analytic ...
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3answers
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Contour integration of: $\int_C \frac{2}{z^2-1}\,dz$

I want to calculate this (for a homework problem, so understanding is the goal) $$\int_C \frac{2}{z^2-1}\,dz$$ where $C$ is the circle of radius $\frac12$ centre $1$, positively oriented. My ...
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42 views

Inverse Gamma function for integers (Hankel)

So I want to prove that for all integers $n \in \mathbb{Z}$ it holds that $$F(n):= \frac{1}{2\pi i} \int\limits_{\gamma}^{} s^{-n}e^{s} ds = \frac{1}{\Gamma(n)},$$ with $\gamma$ the 'Hankel'-contour, ...
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1answer
41 views

Evaluating the integral $\int_C \text{Re }z\,dz$ from $-4$ to $4$ via lower half of the circle

I want to evaluate the integral $\int_C \text{Re }z\,dz$ from $-4$ to $4$ via the contour being the lower half of the circle of radius $4$ centered at the origin. So I can take: ...
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1answer
60 views

Evaluating an integral across contours: $\int_C\text{Re}\;z\,dz\,\text{ from }-4\text{ to } 4$

This is for an assignment, describing the procedure is most beneficial for me, rather that solely computing the result. I want to evaluate the following integral: $$\int_C\text{Re}\;z\,dz\,\text{ ...
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1answer
32 views

Residue theorem, double pole, sinh.

how can I use the residue theorem to calculate $$\int_{-\infty}^\infty dx\, \frac{e^{-i x}}{(\sinh x)^2}$$ Im confused about how to tackle the double pole at $x=in\pi$. Thanks!
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29 views

Evaluating the inverse Laplace transform of $1/(s^2-\sum_{n=1}^\infty{n!s^{3-n}x^n})$

I want to evaluate at $t=1$ the inverse Laplace Transform $\mathcal{L}^{-1}\{F(s)\}\vert_{t=1}$ of $$ F(s) = \frac{1}{s^2-\sum\limits_{n=1}^\infty{n!s^{3-n}x^n}} $$ and find out the $x^n$ ...
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1answer
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Piecewise continuity for contours - Definition for complex analysis

We want piecewise continuity for any contours in complex analysis. What does this refer to? I imagine it refers to the nature of referring to each line arc being continuous. E.g. We want continuity ...
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1answer
101 views

Is this derivation of the Dirichlet Integral using a derivative under the integral sign, incorrect?

To find the integral of the Sinc function: Start with, \begin{equation} I(a)=\int_{-\infty}^{\infty}\frac{\sin\ ax }{x}dx %\hspace{20.0} ; (a>0) \end{equation} \begin{equation} \Longrightarrow ...
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1answer
20 views

Choosing contour to evaluate integral

I am practicing for a preliminary exam in complex analysis. I have struggled with the following problem for a while. I was hoping someone would have suggestions for contours to integrate over. (Or ...
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1answer
40 views

Gaussian Integral using contour integration with a parallelogram contour

I'm having trouble figuring out how to use contour integration to compute the Gaussian integral. The contour I'm using is a parallelogram with function, $f(z) = \Large \frac{ e^{i \pi z^2}}{sin(\pi ...
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3answers
94 views

Computing $\int_{0}^{\infty} \frac{x^2 + a}{x^6 + a^3}dx$

I wish to compute $$\int_{0}^{\infty} \frac{x^2 + a}{x^6 + a^3}dx, \quad a>0$$ but have no contour to work with. Does anyone have ideas on how to compute this integral?
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1answer
66 views

Analytic inverse of $f(z) \neq 0, f(0) = 0, f'(z) \neq 0 $ within minimum modulus on boundary.

Suppose $f(z)$ is analytic on closed disk of radius $r$ and $f(0)=0$, $f'(z) \neq 0$. Show that $f$ has an analytic inverse on $\{|z| \leq m\}$ where $m$ is the minimum of $|f(z)|$ on $\{|z| = r\}$. ...
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2answers
68 views

Computing the complex integral?

I am dealing with the following: $$\int_{0}^{\infty}\frac{x\sin(x)}{(x^2+a^2)(x^2+b^2)}dx$$ Furthermore, I know $a,b>0$ and I know $a\neq b$. I believe this is using Jordan's Lemma? I see that the ...
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23 views

How to determine contours by looking at the exponential integrands?

I know that we determine the contours in contour integrals by looking at the exponential integrand (assuming there is indeed an exponential integrand in the given integral) but I don't know how. For ...
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1answer
47 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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2answers
146 views

Dog Bone Contour Integral

Would someone please help me understand how to integrate $$ \ \int_0^1 (x^2-1)^{-1/2}dx\, ? $$ This is a homework problem from Marsden Basic Complex Analysis. The text book suggested using a "dog ...
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1answer
25 views

Let $C=\partial D_1(\mathbf i/2)$, compute $\int_C\frac{dz}{z^2+1}$

Let $C=\partial D_1(\mathbf i/2)$, compute $\int_C\frac{dz}{z^2+1}$ $C=\partial D_1(\mathbf i/2)$ is the boundary of the disc with center $\mathbf i/2$ and radius $1$, then $\mathbf i$ is ...
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2answers
20 views

Finding the value of $I=\int_C \overline{z} dz$ along $|z|=2$ from $z=-2i$ to $z=2i$

I want to find the value of the integral $$I=\int_C \overline{z} dz$$ When $C$ is the right hand half of the circle $|z|=2$ from $z=-2i$ to $z=2i$ Refer to beautifully made picture: Now I am new to ...