2
votes
0answers
56 views

Contour Integral $ \int_{0}^1 \frac{\ln{x}}{\sqrt{1-x^2}} \mathrm dx$

I need help evaluating this with contour integration$$ \int_{0}^1 \frac{\ln{x}}{\sqrt{1-x^2}} \mathrm dx$$ I am not sure as to how to work with the branch cuts of both $\ln{x}$ and $\sqrt{1-x^2}$ ...
4
votes
2answers
101 views

How to prove $\int^{\pi/2}_0 \log{\cos{x}} \, \mathrm{d}x = \pi/2 \log{1/2}$

ALREADY ANSWERED I was trying to prove the result that the OP of this question is given as a hint. That is to say: imagine that you are not given the hint and you need to evaluate: $$I = ...
0
votes
0answers
55 views

Residue with half order pole?

I'm having issues evaluating the following integral using Cauchy's residue theorem. $$\int_{-\infty}^{\infty} \frac{e^{ix}}{\sqrt{x^2 - 1}} dx $$ Here's what I have tried. We have to make a ...
0
votes
0answers
33 views

Find a Harmonic conjugate $v(x,y)$ to $u(x,y)$.

Show that $u(x,y) = \frac{y^2}{x^3+y^3}$ in some domain and find the harmonic conjugate $v(x,y)$ to $u(x,y)$.
1
vote
1answer
18 views

Differentiation under the integral sign (one complex variable)

Let $u(z), u'(z)$ be complex-analytic functions on an open neighborhood $\Omega \subseteq \mathbb{C}$ of the origin. Also, let $f(X)$ be a complex-analytic function. For $s \in [0,1],$ define $$g(s,z) ...
5
votes
1answer
102 views

Prove $\int_0^1 \frac{\ln(1+t^{4+\sqrt{15}})}{1+t}\mathrm dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3})$

Prove that: \begin{equation} \int_0^1 \frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}\mathrm dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3}) ...
0
votes
0answers
29 views

Complex analysis, cutoff integration

The diff-invariant distance between $z'$ and $z$ is (for short distances) $e^{w(z)}|z'-z|$, so a diff-invaraint cutoff would be at $|z'-z|=\epsilon e^{-w(z)}$. Then $ \int ...
6
votes
2answers
145 views

A strange answer for $\int _{-1}^1 \log x\; dx$

I typed $\int _{-1}^1 \log x\; dx$ on Wolfram Alpha. It is giving the answer to be $-2+i\pi$. Can someone please explain what is happening?
1
vote
1answer
36 views

If a real polynomial of degree $n\gt 1$ has a root of modulus exceeding all others, is that one a real root?

Suppose $a_nx^n+\ldots+a_1x+a_0=0\; (a_n\in \mathbb{R})$ has $n$ distinct roots $r_1,r_2,\ldots, r_n$ (no multiple roots), and if $\exists r_k$ s.t. $\forall r_i\in\{r_1,r_2\cdots r_n\}-\{r_k\}$, ...
3
votes
1answer
48 views

Finding all the possible values of an Integral in the Complex Plane

I am studying Complex Analysis by Lars V Ahlfors. I am unable to solve one of his exercises. It is: Find all possible values of $$\int \frac{dz}{\sqrt{1-z^2}}$$ over a closed curve. I do not have ...
2
votes
0answers
52 views

Can this modified Gaussian integral be calculated analytically?

In my research, I encounter this modified Gaussian integral $$\int_{-\infty}^{\infty}dx\,\frac{x+\sqrt{x^2-bx}}{2\sqrt{x^2-bx}}\exp\left[-a^2(x-x_0)^2+i\left(cx-d\sqrt{x^2-bx}\right)\right],$$ where ...
3
votes
1answer
90 views

Integrating $e^{a/x^2-x^2}/(1-e^{b/x^2})$

I want to solve the following two integrals analytically \begin{aligned} I_1 = & \int\limits_0^{\infty}\frac{e^{a/x^2}}{1-e^{b/x^2}}e^{-x^2}dx \\ I_2 = & ...
4
votes
2answers
81 views

Integrating $z^{2n}\cos(1/z)/(1-z^n)$ over a circle of radius $2$ around the origin

I'm stuck on the following integral computation: $$\int_C \frac{z^{2n} \cos (1/z)}{1 - z^n} \, dz,$$ where $C$ is a circle of radius $2$ around the origin. I tried making the substitution $u = ...
0
votes
1answer
42 views

How to find the area where $\frac{1}{z^2-4}$, $z \in \mathbb{C}$ is holomorphic?

Suppose that you are given a problem of finding the following complex integral: $$\int_\tau \frac{1}{z^2-4} dz$$ where $\tau = \{z \in \mathbb{C}: |z|=4 \}$. My question is (in the context of this ...
0
votes
3answers
47 views

Calculate complex integral with pole at zero

Calculate for $\alpha >0$ and $n \in {\mathbb Z}$. $$ \oint_{\left\vert\,z\,\right\vert\ =\ \alpha} z^{n}\,{\rm d}z. $$
2
votes
1answer
32 views

Cauchy Integrals

This was given to me as a $2$ part question. I was able to answer the $1$st part but the $2$nd part has me confused. a. Let C be the unit circle $z=e^{i\theta}$ where $-\pi\le\theta\le\pi$. Use the ...
3
votes
0answers
73 views

Residue Integral: $\int_0^\infty \frac{x^n - 2x + 1}{x^{2n} - 1} \mathrm{d}x$

Inspired by some of the greats on this site, I've been trying to improve my residue theorem skills. I've come across the integral $$\int_0^\infty \frac{x^n - 2x + 1}{x^{2n} - 1} \mathrm{d}x,$$ where ...
0
votes
0answers
24 views

Proof of Cauchy integral formula limit exchange

In the proof of the Cauchy integral formula there is a limit that exchanges places with the integral (which is itself a limit), my question is why can we do this? If $f(z)$ is a complex function, ...
8
votes
2answers
363 views

Which holomorphic function is this the real part of?

In the paper "The Inverse Function Theorem of Nash and Moser" by Richard S. Hamilton it is claimed that there exists a function $\phi$ such that: $$\int_{0}^{\infty}t^{n}\phi(t)dt=(-1)^{n}$$ For ...
1
vote
1answer
45 views

How to compute the integrals in inverse formula?

I have following characteristic function for certain random variable X: $$\Phi (t) = \frac{\beta_1\beta_2}{\eta_1}\frac{\eta_1 - it}{(\beta_1 - it)(\beta_2 - it)}$$ where $\eta_1 > 0, \quad\beta_1 ...
2
votes
1answer
124 views

How do you integrate Gaussian integral with contour integration method?

How do you integrate $$\int^{\infty}_{-\infty} e^{-x^2} dx$$ with contour integration method? I do not even know how to setup the problem.
2
votes
1answer
135 views

Calculating Riemann zeta function of a complex number given the complex contour integral

Can you please demonstrate how one would calculate the Riemann Zeta function of any complex number, given that the Riemann Zeta function is equal to the following (shown in ...
2
votes
0answers
101 views

Contour integration with merged pole/branch-cut type behavior?

I have the expression $$f(z)=\frac{-i}{\sqrt{z^2-a^2}},$$ where $a$ is a purely real number and $z$ is a complex variable. Numerical plotting gives the following. This leads me to the following ...
1
vote
1answer
70 views

solving integral with complex analysis

I have problems with understanding of the evaluation of this integral below. It has been a long a time ago since I had complex analysis. where $a = (1-\sqrt y )^2$ and $b = (1+\sqrt y )^2$. Now my ...
2
votes
1answer
39 views

Computing a contour integral over curve not centered at origin

Consider the integral $$ \int_C \frac{1}{z} \, dz $$ where $C$ is the circle of radius $R$ centered at the point $z_0 \in \mathbb{C}$. We parametrize the curve by $z(\theta) = z_0 + Re^{i\theta}$ ...
5
votes
0answers
57 views

Clarification of Contour Integration [duplicate]

I apologise if this seems like an elementary and silly question, but I am confused about the integral $$I=\int^{\infty}_{-\infty}\frac{\cos{x}}{1+x^2}dx=\frac{\pi}{e}$$ If I consider a semicircular ...
7
votes
1answer
137 views

Proving that a function is analytic

I'm struggling with the following problem: Problem: Suppose that $h$ is a continuous function on a simple closed curve $\gamma$. Define $$ H(w) = \oint_{\gamma} \frac{h(z)}{z - w} \, dz. $$ Show ...
2
votes
3answers
51 views

Parametrizing curve for complex analysis integral

I'm trying to show that $$ \int_{|z-z_0| = R} (z-z_0)^m \, dz = \begin{cases}0, & m \neq -1 \\ 2\pi i, & m =- 1. \end{cases} $$ Here's my attempt at a solution: We parametrize the curve at ...
1
vote
3answers
51 views

integrating $\int_{\gamma}e^zdz$ with $\gamma$ is the arc on the unit circle that unites one with i

I am stuck integrating $$\int_{\gamma}e^zdz$$ with $\gamma$ is the arc on the unit circle that unites one with i. I tried this : The integrand $\mathrm{e}^z$ is holomorphic for $\vert z \vert \le ...
10
votes
0answers
192 views

The closed form of $\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$

What tools, ways would you propose for getting the closed form of this integral? $$\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$$
2
votes
3answers
119 views

Integration by Euler's formula

How do you integrate the following by using Euler's formula, without using integration by parts? $$I=\displaystyle\int \dfrac{3+4\cos {\theta}}{(3\cos {\theta}+4)^2}$$ I did integrate it by parts, by ...
2
votes
2answers
164 views

How to calculate $\int_{-\infty}^\infty\frac{x^2+2x}{x^4+x^2+1}dx$?

I want to calculate the following integral: $$I:=\displaystyle\int_{-\infty}^\infty\underbrace{\frac{x^2+2x}{x^4+x^2+1}}_{=:f(x)}dx$$ Of course, I could try to determine $\int f(x)\;dx$ in terms of ...
3
votes
2answers
73 views

How to calculate $\int_{\partial B_2(0)}\frac{2z^2+7z+11}{z^3+4z^2-z-4}\;dz$?

I want to calculate $$\displaystyle\int_{\partial B_2(0)}\underbrace{\frac{2z^2+7z+11}{z^3+4z^2-z-4}}_{=:f(z)}\;dz\tag{0}$$ Partial fraction decomposition yields ...
2
votes
2answers
157 views

Gaussian integral with offset, and other cases

Consider the Gaussian Integral $$ \int_{-\infty}^{\infty} e^{-x^2} \ dx = \sqrt{\pi}$$ Numerically, it seems that for any arbitrary imaginary offset, ki, $$\int_{ki-\infty}^{ki+\infty} e^{-x^2} \ dx ...
0
votes
0answers
26 views

How to calculate $\int_{-\infty}^\infty e^{-t^2/2}\cos2t\ dt$ using Cauchy's integral theorem? [duplicate]

I need a hint. Where do I start if I want to calculate $$\int_{-\infty}^\infty e^{-t^2/2}\cos2t\ dt$$ using Cauchy's integral theorem?
4
votes
0answers
140 views

${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$

I have found the following new result connecting two rational log-cosine integrals. Proposition. \begin{align} \displaystyle & {\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos ...
1
vote
0answers
51 views

Choose appropriate contour for a complex integral

I have a problem to solve integral $$ I = \int^{\infty}_0 \frac{\mathrm{d}x}{(x-z)(1+x^2)^{\kappa+2}} $$ I can solve the same integral with borders $-\infty$ to $\infty$ using residue theorem but ...
0
votes
1answer
23 views

The integral along a circle of the inverse linear function is zero

Assume ${\rm C}$ is a circle and $a,b$ are distinct points in the interior of ${\rm C}$. How can we see that the complex integral $$ \frac{1}{b - a} \int_{\rm C}\left(\frac{1}{z - a} - \frac{1}{z - ...
8
votes
1answer
260 views

Evaluation of $\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$

I need some hints, clues for getting the closed form of $$\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$$
0
votes
2answers
35 views

Vanishing moments and integrability

Is this correct? $\int_\mathbb{R}x^m f(x) dx=0 \iff \int_\mathbb{R}x^m \overline{f(x)}\,dx =0$. If yes then please tell the conditions under which this holds.
2
votes
3answers
69 views

value of an integral depending on a parameter in complex plane

For each $z\in\mathbb{C}$, evaluate the integral $$ \int_0^1\int_0^{2\pi}\frac{1}{re^{i\theta}+z}d\theta dr. $$ How to evaluate it? Thanks.
0
votes
0answers
92 views

On an application of the Abel-Plana formula

Referring to a previous question, i am having a hard time trying to do the integral: $$f(s)=-i\int_{0}^{\infty}\frac{\log \left[1+\frac{\left(s\log(1+ix) \right )^{2}}{4\pi ^{2}} \right ]-\log ...
2
votes
0answers
60 views

Evaluate $\int_0^\infty x^{\lambda-1} \exp\left(-ax-b\sqrt x-\frac{c}{\sqrt x} - \frac{d}{x}\right) \: dx$

Is there a closed form for the integral $$\int_0^\infty x^{\lambda-1} \exp\left(-ax-b\sqrt x-\frac{c}{\sqrt x} - \frac{d}{x}\right) \: dx?$$ where $\lambda>0$, $a>0$, $d>0$ and where $b$, ...
1
vote
0answers
35 views

Conditions for changing the order of integration for contour integral.

I assume an integral $$I=\int_0^\infty f(x)g(x)\mathrm dx \tag{1}$$ where the function $f(x)$ can be represented as a contour integral in complex plane: $$f(x)=\oint_\Delta ...
0
votes
0answers
18 views

Divergence of Euler integral for non-positive arguments

Why is it necessary that $\operatorname{Re}(x),\operatorname{Re}(y) > 0$ for the Beta-function $$B(x,y) = \int_0^1 t^{x-1} (1-t)^{y-1} dt$$ I suppose it is because the integral diverges when ...
2
votes
0answers
64 views

Integrating $xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2})$?

I want to solve any of the two integrals for the complex number $a$ \begin{aligned} I_1 & = \int\limits_{0}^{\infty} xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2}) dx\\ I_2 & = ...
0
votes
0answers
29 views

Gaussian integral involving $\cos\circ\sin$

I stumbled upon an integral of the form $$\int_{\mathbb R} e^{-x^2/2}\cos(a\sin (bx+ic))\,{\mathrm d}x$$ for some real constant $a,b,c$. Has anybody ever seen such an integral? Mathematica doesn't ...
2
votes
1answer
33 views

Lusin's area integral

I was reading "Steven G. Krantz - Handbook of Complex Variables" and came around a complex surface integral called "Lusin's area integral": If $\Omega \subseteq \mathbb{C}$ is a domain and $\varphi: ...
0
votes
0answers
60 views

Integral $\int^\infty_{-\infty}\int^\infty_{-\infty}(\frac{(x-x_1)^2+(y-y_1)^2}{s_1^2}+1)^{-a_1-1}(\frac{(x-x_2)^2+(y-y_2)^2}{s_2^2}+1)^{-a_2-1}dxdy$

Under $x_i,y_i\in\mathbb R$, $s_i>0$ and $a_i>0$ for $i=1,2$, is there any good function to express the following integral? $$\int^\infty_{-\infty}\int^\infty_{-\infty} ...
0
votes
2answers
81 views

Calculate: $F(x)=\int_{0}^{+\infty}\frac{e^{i xt}}{t^{\alpha}}dt\quad \text{avec}~x\in \mathbb{R}~\text{ et }~0<\alpha<1$

I would like to calculate this integral: $F(x)=\int_{0}^{+\infty}\frac{e^{i xt}}{t^{\alpha}}dt\quad \text{avec}~x\in \mathbb{R}~\text{ et }~0<\alpha<1$ I calculated : $\displaystyle ...