8
votes
Accepted
How to evaluate the integral $\int_0^{\infty} \frac{\sin(k \ln(t))}{\sqrt{t}} \left\{\frac{1}{t}\right\} \, dt $
Firstly, let's replace the sine function by a complex exponential, so that your integral corresponds to the imaginary part of the following integral :
$$
\int_0^\infty \frac{\exp(k\ln t)}{\sqrt{t}} \...
- 10.4k
3
votes
how to find a good change of variables to solve these kinds of multi-integral questions?
You do not need to solve for $x$ and $y$. These problems are usually constructed very carefully (and artificially) to work out just right. Remember that the Jacobian you need is the reciprocal of the ...
- 118k
3
votes
Accepted
how to find a good change of variables to solve these kinds of multi-integral questions?
When it comes to changes of variables, one possible approach is to determine formulas for $x(u,v)$, $y(u,v)$, compute the Jacobian, and apply the change of variables theorem as you outline. In some ...
- 7,297
2
votes
Proving an inequality on a monotonic sequence of positive real numbers
Case 1. Let $q=e^{-z_M}.$ Then $$\sum_{i=1}^Me^{-iz_i}>\sum_{i=1}^Mq^i={q(1-q^M)\over 1-q}$$ Therefore $$ p={1-q^M\over \sum_{i=1}^Me^{-iz_i}}<{1-q\over q}=e^{z_M}-1$$ When $z_i\nearrow z_M$ we ...
- 34k
2
votes
Formally find the domain of convergence of $\sum_{p\in P}x^p$ where P is the set of Prime numbers
Can't you note that
$$\Big|\sum_{p \in P} x^p\Big| \quad \le \quad \sum_{p \in P} |x|^p$$
$$\le \quad \Big(\sum_{p \in P} |x|^p + \sum_{n \in \mathbb{N} \setminus P} |x|^n\Big) \quad = \quad \sum_{n \...
- 21.2k
1
vote
Derivative of an Infinite Fraction
$y=\frac{x}{x^2+y}$ can be written as $y^2+x^2y-x=0$.
Solving this for $y$ gives $$y=\frac{-x^2\color{red}+\sqrt{x^4+4x}}{2}$$
Since
$$\begin{align}&(x^2+y)(x+y^2)
\\\\&=\frac{x^2+\sqrt{x^4+4x}...
- 145k
1
vote
What is the degree of the following differential equation?
The ODE is not polynomial, not even if that criterion is only applied to the highest order derivative. Thus the idea of "degree" does not apply.
The main branch of the inverse or arcus ...
- 128k
1
vote
Accepted
Integrating "mixed" integrals such as $\int_{10}^\infty x^a\cos(t\log(x))\ \tanh(x/2) \,\mathrm dx$
As the comments pointed out we want a lower bound of $0$ for a nice solution (otherwise the integral most likely does ot have a closed formula). Furthermore then we requiere $a \in (-2,-1)$ for ...
- 431
1
vote
Accepted
Find the point $P$ on an ellipse such that $\overline{AP} + \overline{BP}$ is minimum for given points $A$ and $B$
Given $p = (x,y,z)'$ eliminating $t$ in the equation
$$
p = v_0+v_1\cos t+v_2\sin t
$$
with the given values
$$
\cases{
v_0 = (2,3,7)'\\ v_1 = (1,0.5,0.5)'\\ v_2 = (-2,6,2)\\
p_A = (5,10,2)'\\ p_B=(1,-...
- 34.3k
1
vote
Accepted
An estimate involving Poisson Kernel
We have $$ u(x,y)={4y\over \pi}\int\limits_{x_0-x}^{x_0-x+h}{1\over t^2+y^2}\,dt \\ ={4\over \pi}\int\limits_{(x_0-x)/y}^{(x_0-x+h)/y}{1\over t^2+1}\,dt $$ The interval of integration contains $0$ ...
- 34k
1
vote
Solve this integral:$\int_0^\infty\frac{\arctan x}{x(x^2+1)}\mathrm dx$
Let $x=\tan y$, then
$$
\begin{aligned}
\int_0^{\infty} \frac{\arctan x}{x\left(x^2+1\right)} d x & =\int_0^{\frac{\pi}{2}} \frac{y}{\tan y\left(\tan ^2 y+1\right)} \cdot \sec ^2 y d y \\
& =\...
- 22.4k
1
vote
Solving the Riccati equation with constant coefficients $y^\prime = a y^2 + b y + c$
This is a separable ODE.
$$\frac{dy}{dx}=ay^2+by+c\quad\implies\quad dx=\frac{dy}{ay^2+by+c}$$
$$x=\int \frac{dy}{ay^2+by+c}dx=\frac{2}{\sqrt{4ac-b^2}}\tan^{-1}\left(\frac{2ay+b}{\sqrt{4ac-b^2}} \...
- 66.8k
1
vote
Solving the Riccati equation with constant coefficients $y^\prime = a y^2 + b y + c$
Set $y=pt+q$ and bring the RHS to the form $r(1\pm t^2)$.
By identification $a(pt+q)^2+b(pt+q)+c=r(1\pm t^2)$ yields $q=-\dfrac b{2a}, r=c-\dfrac{b^2}{4a},p=\sqrt{\left|\dfrac ra\right|}$ and the sign ...
- 337
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