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

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3
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2answers
103 views

Integral $\int_0^\infty e^{imx^2}dx$

In evaluating an integral in path integrals in QFT, I am stuck with this integral (that came up from evaluating a functional integral), $$I = \bigg( \frac{m}{2\pi i\tau}\bigg) \int ...
3
votes
2answers
108 views

Improper Integral of $x^2/\cosh(x)$

I need to compute the improper integral $$ \int_{-\infty}^{\infty}{\frac{x^{2}}{\cosh\left(x\right)}\,{\rm d}x} $$ using contour integration and possibly principal values. Trying to approach this as ...
3
votes
1answer
101 views

Would like help with a contour integral.

Disclaimer: the knowledge I have about contour integration is solely from the book "Mathematical Methods in the Physical Sciences" by Mary L. Boas. I am trying to understand how the following ...
3
votes
2answers
96 views

How to find this integral using Cauchy integral formula

How to obtain that $$\int\limits_{|z|=r} (\bar{z})^{-m} z^{-n-1}\, dz = \begin{cases} 2\pi ir^{-2m} &\text{if}\,\,n=m, \\ 0 &\text{if}\,\,n \neq m, \end{cases}$$ for $r>0$. I suppose I ...
3
votes
1answer
144 views

Analytic continuation of the Riemann zeta function using contour integration

To find the analytic continuation of the Riemann zeta function using contour integration one can integrate $\displaystyle f(z) = \frac{z^{s-1}}{e^{-z}-1}$ around a contour that consists of rays just ...
3
votes
1answer
39 views

Integration of exponential with square

It is known that $\int_\mathbb{R}e^{-tx^2}dx=\sqrt{\pi/t}$. What about $\int_\mathbb{R}e^{-t(x+ai)^2}dx$ for $a\in\mathbb{R}$? Is it still also $\sqrt{\pi/t}$? I can't simply change the variable ...
3
votes
3answers
201 views

Evaluating an improper integral that involves $\exp(-|x|)$

I am trying to prove that the function $f:\mathbb C\setminus\mathbb R\rightarrow\mathbb C$ defined by $$ f(z) := \frac{1}{2\pi i}\int_{-\infty}^\infty\frac{\exp(-|x|)}{x-z}dx $$ is holomorphic. I ...
3
votes
2answers
315 views

Complex analysis: contour integration

Evaluate by contour integral: $$\int_0^1{ dx\over (x^2-x^3)^\frac 13}$$ Should I go for some kind of substitution so that the limit changes to $0$ to $\pi/2$?
3
votes
1answer
209 views

Cauchy principal value of $\int_{-\infty}^{\infty} \frac{1}{x^{3}} \ dx $

By definition $\displaystyle\text{PV} \int_{-\infty}^{\infty} \frac{1}{x^{3}} \ dx = \lim_{\epsilon \to 0 ,\ a \to \infty} \Big(\int_{-a}^{0 - \epsilon} \frac{1}{x^{3}} \ dx + \int_{o+ \epsilon}^{a} ...
3
votes
2answers
48 views

find $\int _\gamma \frac{1}{z+\frac {1}{2}}dz$

I'm asked to find $$\int _\gamma \frac{1}{z+\frac {1}{2}}$$ where $\gamma (t)=e^{it}, 0\leq t\leq 2\pi$. To do this I deal with two different logarithms, one without the negative imaginary axis ...
3
votes
1answer
206 views

contour integration of logarithm function

I'm new to contour integral involving branch point and stuck on this particular integration. Here is the problem: $$\int_{\mathcal{C}}\log z\,\mathrm{d}z,$$ where $\mathcal{C}$ is a closed square ...
3
votes
1answer
253 views

Contour Integration: $\int_0^\infty\frac{1}{x^a(1-x)}\,dx$ for $0<a<1$.

I've been trying to calculate $$\int_0^\infty\frac{1}{x^a(1-x)}\,dx\quad\text{with }0<a<1.$$I haven't had much luck. I tried taking the branch cut with of the positive reals and estimating that ...
3
votes
1answer
561 views

branch cuts and contour integration

An exercise in a textbook says to evaluate $\displaystyle \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \cos (ax) \cos^{b} (x) \ d x \ (a > b > -1)$ by letting $\displaystyle f(z) = z^{a-1} ...
3
votes
3answers
237 views

Find the residues of singularities of the following function:

Let $$f(z)=\frac{e^{\pi iz}}{z^2-2z+2}$$ and $\gamma_R$ is the closed contour made up by the semi-circular contour $\sigma_1$ given by, $\sigma_1(t)=Re^{it}$, and the straight line $\gamma_2$ from ...
3
votes
2answers
288 views

Evaluate the following contour integral…

Let $\gamma(z_0,R)$ denote the circular contour $z_0+Re^{it}$ for $0\leq t\leq 2\pi$. Evaluate: $$\int_{\gamma(0,1)} \frac{\cos(z)}{z^2}dz$$
3
votes
3answers
137 views

Calculate $\int_{-\infty}^{+\infty}e^{-\frac{(x-it)^2}{2}}dx$ using contour integration

When one wants to calculate the characteristic function of a random variable which is of normal distribution, things boil down to calculate: $$\int_{-\infty}^{+\infty}e^{-\frac{(x-it)^2}{2}}dx$$ There ...
3
votes
2answers
329 views

Contour Integral Question

I'm working through a contour integral question, which is rounded off by finding the integral: $$\int^{\infty}_{0} \frac{x-\sin(x)}{x^3} dx$$ I have already shown that the residue at $0$ of the ...
3
votes
2answers
279 views

Finding $\frac{1}{2\pi}\int_{0}^{2\pi} \cos^{2n} x dx$

I have a question that asks me to find the value of $\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi} \cos^{2n} x dx$ $\ $ by considering the integral $$ \displaystyle \oint_{\gamma} \frac{1}{z}\left ( ...
3
votes
2answers
572 views

Proper application of Cauchy integral formula

I'm being asked to integrate $f(z) = \frac{e^z}{z^2 + 2z + 1}$ around a $5 \times 5$ square, centered at $i$, in the counter clockwise direction. It seems to me that applying Cauchy's integral formula ...
3
votes
1answer
59 views

Integrating around the upper half of $|z|=R$

In a textbook it says that you can show that $ \displaystyle\int_{-\infty}^{\infty} \frac{\cos(x^{2})+\sin(x^{2})-1}{x^{2}} \ dx = 0$ by considering $ \displaystyle f(z) = \frac{e^{iz^{2}}-1}{z^{2}}$ ...
3
votes
2answers
109 views

Trouble with $\int_0^\infty e^{-ix^2}\mathrm{d}x$

I'm trying to evaluate $$ \int_0^\infty \mathrm{d}x\ e^{-ix^2}. $$ I tried to integrate on the following contour $\Gamma_R$: the frontier of a circular sector, centered at the origin, of angle $\pi / ...
3
votes
1answer
131 views

Laplace transform of and impulse sampled function using “frequency” convolution

This is a long question, but assume we have this: The book uses the frequency convolution theorem to solve this problem. To solve the integral, it uses a contour + residue theorem to solve it. The ...
3
votes
1answer
53 views

Find the analytic continuation of the $ f(z) = \int_{0}^{\infty} \frac{exp(-zt)}{1+t^2} dt$

Find the analytic continuation of the function $f(z)$ defined by $ f(z) = \int_{0}^{\infty} \frac{\exp(-zt)}{1+t^2} dt$ , $ |\arg(z)| < \pi/2$ to the domain $ -\pi/2 < \arg(z) < \pi$ I ...
3
votes
1answer
130 views

An integral representation of the gamma function

An integral representation of the gamma function that would appear to be valid for all complex values of $z$ excluding the integers is $$ \Gamma(z) = \frac{2e}{\sin \pi z}\int_{0}^{\infty} ...
3
votes
1answer
81 views

How to prove $\operatorname{Log}(z) = \log(|z|)+i\arg(z)$.

The value of the principal branch of the logarithm can be evaluated by the formula \begin{align*} \operatorname{Log}(z) = \log(|z|)+i\arg(z), \end{align*} where $\arg(z) \in (-\pi,\pi)$ and ...
3
votes
1answer
123 views

Compute $\int_{|z|=1}\frac{\log z}{z}dz$.

Here is a question about contour integration in complex analysis: Compute $$\int_{|z|=1}\frac{\log z}{z}dz$$ I am not sure if I understand the question since the logarithm must be defined in a ...
3
votes
1answer
71 views

Sanity check for a contour integral, without using Cauchy's Integral Formula

Problem: Evaluate $\int_{\Gamma} \dfrac{z}{(z+2)(z-1)}dz$ , where $\Gamma$ traverses the circle $|z| = 4$ twice in the clockwise direction. As this is supposed to precede my knowledge of ...
3
votes
1answer
142 views

A Cauchy principal value integral, using contour integration and Plemjel.

I came across the following integral $$lim_{\epsilon->0+}\int_\mathbb{R}\frac{e^{-ax^2+ibx}}{x+i\epsilon}dx$$ with a,b>0. Using Plemjel's formula led me to evaluating ...
3
votes
1answer
84 views

Integrating around a pole

I brought this up in another thread. Is my observation correct? Let $f(z)$ have a Laurent expansion at $z=z_{0}$ of the form $$ f(z) ...
3
votes
1answer
161 views

Integral of involving Airy function without using its antiderivative

Inspired by my answer to this question, I am interested in evaluating the following definite integral $$ \frac{1}{2\pi i} \int_{c-i \infty}^{c+i\infty} \frac{dz}{\mathop{\rm Ai}^2(z)} =1 $$ without ...
3
votes
1answer
161 views

On lower bounds for contour integrals and their divergence

I want to find a lower bound for $\left|\int_\gamma f(z)dz\right|$. I know of the estimation lemma and Jordan's lemma for an upper bound, but I don't know of any for a lower bound. The motivation is ...
3
votes
2answers
465 views

Another residue theory integral

this is the last from me I need to evaluate the following real convergent improper integral using residue theory (vital that i use residue theory so other methods are not needed here) I also need to ...
3
votes
0answers
55 views

Branch-point order and Cauchy representation

This question is about the nature of branch points which arise in certain Cauchy-integral representations of functions of a single complex argument, $z$. Suppose we have the following representation: ...
3
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0answers
66 views

Contour Integration - Quantum field theory

I am a physics student. In calculating transition amplitude for Klein-Gordon real-scalar field, I encountered the integral, $$ \frac{-i}{2(2\pi)^2\Delta x} \int^{\infty}_{-\infty} \,dk ...
3
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0answers
77 views

Integral $\int_{0}^\infty\frac {(1-{{e}^{-i (q-p)t}})ln(|p^2-p_0^2|)}{(q-p)({{ p}}^{2}-{{p_1}}^{2})({{p}}^{2}-{{p_2} }^{2})}dp$

I am trying to get a closed form analytic result for the integral $$\int _{0}^{\infty }\!{\frac {\left(1-{{\rm e}^{-i \left( {q}-{p} \right) t}}\right){\rm ln}(|p^2-p_0^2|)}{ ( {q}-{p} ) \left( {{ ...
3
votes
1answer
56 views

Inverse FT of $Z(\omega) = a [- \frac{1}{i\omega}+\pi \delta(\omega)]$ (Contour integration)

Given is the Fourier transform of some function $z(t)$: $Z(\omega) = a [- \frac{1}{i\omega}+\pi \delta(\omega)]$ I now want to invert the tranform using contour integrals. How can I do that? I ...
3
votes
0answers
502 views

contour integration of a function with two branch points .

Many of us have seen the evaluation of the integral $$\int^{\infty}_0 \frac{dx}{x^p(1+x)}\, dx \,\,\, 0<\Re(p)<1$$ It can be solved using contour integration or beta function . I thought of ...
3
votes
0answers
161 views

Where's my mistake applying Perron's Formula?

I applied Perron's Formula to Riemann Zeta Function and got a weird result. First, I started with a simple definition of Riemann Zeta Function, $$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$$ where ...
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0answers
77 views

Show that $\int_{\alpha}\frac{1}{z}\, dz=\int_{\beta}\frac{1}{z}\, dz$.

Let $a$ and $b$ be positive real numbers. Define ways $\alpha,\beta\colon [0,1]\to\mathbb{C}$ via $$ \alpha(t):=a\cos(2\pi t)+ia\sin(2\pi t),~~~~~\beta(t):=a\cos(2\pi t)+ib\sin(2\pi t). $$ Show ...
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votes
0answers
133 views

Satisfying a Differential Equation and complex Laguerre

I have the following problem Show that $$L_n(x)=\frac{e^x}{2 \pi i}\oint \frac{t^n e^{-t}}{(t-x)^{n+1}}dt$$ satisfies $$x\, L_n^{\prime\prime}+(1-x)L_n^\prime+n\, L_n=0$$ where the contour is ...
2
votes
3answers
65 views

Help with a contour integration

I've been trying to derive the following formula $$\int_\mathbb{R} \! \frac{y \, dt}{|1 + (x + iy)t|^2} = \pi$$ for all $x \in \mathbb{R}, y > 0$. I was thinking that the residue formula is the ...
2
votes
2answers
122 views

Cool Integral $\int_0^{\pi/2}dx\ln \sinh x$

$$ I_1=\int_0^{\pi/2}dx\ln \sinh x,\quad I_2=\int_0^{\pi/2}dx\ln \cosh x, \quad I_1\neq I_2. $$ I am trying to calculate these integrals. We know the similar looking integrals $$ \int_0^{\pi/2}dx\ln ...
2
votes
1answer
75 views

Complex Analysis: Are these properties equivalent?

I am somehow thinking that these properties must be equivalent, unfortunately I do not know a theorem that says it: $f$ has a global antiderivative iff the line integral $ \int_{\gamma}f$ over every ...
2
votes
2answers
289 views

Evaluation of the contour integral $\int_\beta \frac{e^z}{e^z-\pi} dz$

Suppose $\beta$ is a loop in the annulus $\{z:10<\left|z\right|<12\}$ that winds $N$ times about the origin in the counterclockwise direction, where $N$ is an integer. Determine the value of ...
2
votes
2answers
353 views

Summation of series using residues

Let $P(n)$ and $Q(n)$ be polynomials such that $\displaystyle \sum_{n=-\infty}^{\infty} (-1)^{n} \frac{P(n)}{Q(n)}$ converges conditionally, that is, the degree of $Q(n)$ is exactly 1 degree more than ...
2
votes
2answers
217 views

integrating $\oint_C \dfrac{3z^3 + 2}{(z-1)(z^2 + 9)}dz$ on $|z|=4$

I am doing $\oint_C \dfrac{3z^3 + 2}{(z-1)(z^2 + 9)}dz$ on $|z|=4$ and I find that there are poles within the contour at $z = 1$ and at $z = 3i$, both simple poles. I find that the integral $I = 2\pi ...
2
votes
2answers
156 views

Evaluate the following contour integral:

Let $\gamma(z_0,R)$ denote the circular contour $z_0+Re^{it}$ for $0\leq t\leq 2\pi$. Evaluate $$ \int_{\gamma(0,1)}\dfrac{z^2+1}{z(z^2+4)}dz $$ I've tried to use the binomial expansion with ...
2
votes
2answers
337 views

Have I calculated this integral correctly?

I have this integral to calculate: $$I=\int_{|z|=2}(e^{\sin z}+\bar z)dz.$$ I do it this way: $$I=\int_{|z|=2}e^{\sin z}dz+\int_{|z|=2}\bar zdz.$$ The first integral is $0$ because the function is ...
2
votes
2answers
519 views

Evaluation of $\int_0^\infty \frac{x^2}{1+x^5} \mathrm{d} x$ by contour integration

Consider the following integral: $$\int_{0}^{\infty} \frac{x^{2}}{1+x^{5}} \mathrm{d} x \>.$$ I did the following: Since $-1$ is a pole on the real axis, I took $z_{1}=e^{3\pi/5}$ then ...
2
votes
2answers
104 views

Residues to solve an improper integral

I'm asked to solve the following improper integral: $$\int_0^\infty \frac{\rm {Log}^2(t)}{1+t^2}dt. $$ Do I consider the function $f(z) = \frac{\rm{Log}^2(z)}{1+z^2}$ or some variant? Is the ...