1
vote
0answers
30 views

Integral equation solution

I have an integral equations of the form $ \int s R(s) =s f(s)-\int f(s)ds \tag 1$ Can we solve this integral equation for $f(s)$ interms of $s,R(s)$ ? Means $R(s)=\psi(s,R(s))$ (with out integral ...
11
votes
2answers
134 views

Evaluating $\int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$

Good evening! I want to compute the integral $\displaystyle \int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$. However I find it extremely difficult. What I've tried is rewritting it as: ...
0
votes
1answer
24 views

An integral of Wolstenholme:$\int_0^{+\infty}\frac{\sum_1^n A_k\cos{a_k x}}{x}\mathrm {d} x$ where $\sum A_k=0$ and $a_k>0$

The book by Whittaker and Watson says it's equal to $-\sum_{k=1}^n A_k \log {a_k}$, and attibutes it to Wolstenholme. I believe this readily reduces to the simpler case of evaluating $\displaystyle ...
0
votes
0answers
33 views

What is the Riemann surface of the exponential integral?

I have recently encountered a differential equation whose solution has a term \begin{equation} \frac{1}{2}e^{-\frac{1}{2 \varepsilon} e^{i \tau}} \int_{\tau_0}^\tau e^{\frac{1}{2 \varepsilon} e^{i ...
5
votes
4answers
218 views

Wolfram alpha says that $\int_{-\infty}^\infty e^{-ix^2}dx = \sqrt{\frac{\pi}{i}}$

Wolfram alpha says that $$ \int_{-\infty}^\infty e^{-ix^2}dx = \sqrt{\frac{\pi}{i}}$$ holds. But it has two different values ($\sqrt{i}$). How should I understand this?
0
votes
0answers
45 views

Complex Gaussian Integral - $\int_{-\infty}^\infty e^{-p(t+c)^2}dt = \sqrt{\frac{\pi}{p}}$?

I found some formulas on books, especially the complex gaussian integral formula: $$ \int_{-\infty}^\infty e^{-p(t+c)^2}dt = \sqrt{\frac{\pi}{p}} $$ for $p,c\in\mathbb C$. Then if $p=i=\sqrt{-1}$, the ...
0
votes
0answers
24 views

The integral of a monomial over the complex sphere

Let $\alpha=(\alpha_{1},...,\alpha_{q})\in\mathbb{N}^{q}$ a multi-index. What is the expression for $$\int_{S}z^{\alpha}\,d\sigma(z),$$ where $S$ is the unit sphere of $\mathbb{C}^{q}$ and $\sigma$ ...
0
votes
0answers
46 views

Applying Green's Theorem to a Closed Complex Contour Integral

How would one apply Green's Theorem to the following complex contour integral: $\oint_\gamma $ $\frac{u^{s-1}}{e^{-u}-1)}du$. Where $\gamma$ is the Hankel Contour (counterclockwise) and R is the ...
4
votes
2answers
90 views

$\int_{0}^{\infty} \frac{\cos(x)}{1+x^2} dx$ and $\int_{0}^{\infty} \frac {\ln(x)}{x^2+b^2} dx$

Prove that $$\int_{0}^{\infty} \frac{\cos(x)}{1+x^2} dx = \frac {\pi}{2e}$$ My approach would be $$\lim_{n \to \infty} \int_{0}^{n} \frac{\cos(x)}{1+x^2} dx$$ and evaluate the limits of the sine and ...
10
votes
1answer
221 views

Contour integration with branch points inside the contour.

In my scientific research I ran into an unpleasant situation with specific type of contour integrals. Being more specific I have problems not with integrals themselves (I can use various numeric ...
0
votes
1answer
29 views

The value of the integral of $f(\zeta)/(\zeta-z)$ for a function holomorphic in exterior domain

Suppose that $f$ is a bounded analytic function on the domain $\{z ∈ C : |z| > 1\}$. (a) Prove that $\lim_{z→∞} f(z)$ exists. (b) Let $L$ denote the limit in (a), and let $Γ_R$ denote a circle $|ζ| ...
2
votes
1answer
33 views

integral calculate by complex analysis methods

Calculate using methods from comples analysis. $$ \int_0^{2\pi} \,\sin ^{2n} \phi\, d\phi$$ So this is how I started: $$\sin^{2n} \phi = \left[\frac{e^{i \phi}-e^{-i \phi}}{2i}\right]^{2n} = ...
3
votes
2answers
74 views

And another real integral to be solved by contour integration

I want to solve $$\int_0^\infty\frac{1}{x^3+x^2+x+1}dx$$ and i have really learned a lot already by failing to solve it. I want to solve it using a clever contour. It is possible to do it using ...
2
votes
0answers
108 views

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

I need help evaluating this with contour integration $$ \int_{0}^{1}{\ln\left(\,x\,\right)\over \,\sqrt{\vphantom{\large A}\,1 - x^{2}\,}}\,{\rm d}x $$ I am not sure as to how to work with the branch ...
4
votes
2answers
116 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
63 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
58 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
31 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
112 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
30 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
149 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
39 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
49 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
55 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
94 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
44 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
34 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
78 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
28 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
386 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
133 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
146 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
107 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
75 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
41 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
140 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
52 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 ...
11
votes
0answers
213 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
122 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
166 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
75 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
163 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
27 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
142 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 ...
2
votes
0answers
52 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 ...