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Theoretical physicist turned programmer.


Jun
10
comment $\int_0^{+\infty}{ \sin{(ax)} \sin{(bx)}}dx=?$
@YvesDaoust I think it is consistent for the complex analysis argument to be correct within the sphere of its logical framework, while simultaneously the integral is simply undefined in the framework of real analysis. I'm a physicist by the way, so maybe I'm just missing your point since we tend to look at these things differently from how a mathematician would.
Jun
10
comment $\int_0^{+\infty}{ \sin{(ax)} \sin{(bx)}}dx=?$
@YvesDaoust I didn't see any indication in the original problem that $a\ne \pm b$. At any rate, to prove the convergence to 0, I think you could do a contour integral along the real number line, closing in the upper half-plane for $p>0$ or lower half-plane for $p<0$. There are no poles so the entire integral vanishes. Furthermore, the closing arc in the upper (lower) half-plane is exponentially suppressed so it vanishes as well. Thus the integral over the real line must also vanish.
Jun
10
comment $\int_0^{+\infty}{ \sin{(ax)} \sin{(bx)}}dx=?$
Yes, that looks correct. But I would double check all the constants and signs since I did not do out the calculation in full myself. And please accept my answer if it was helpful :)
Jun
10
comment $\int_0^{+\infty}{ \sin{(ax)} \sin{(bx)}}dx=?$
No, your last equation is false. Look at the four integrals you have. With a change of variables, you can combine the first and the last into a single integral over $(-\infty, \infty)$. You can combine the second and third similarly.
Jun
10
comment $\int_0^{+\infty}{ \sin{(ax)} \sin{(bx)}}dx=?$
You can completely replace the sines with exponentials, no need for cosines. The conjugate of the Euler equation is an independent algebraic equation for these purposes. So you have two equations $e^{\pm i \phi} = \cos\phi \pm i \sin\phi$, i.e. with both the plus and the minus. You can solve those two equations to get $2i \sin\phi = e^{i\phi} -e^{-i\phi}$. Using this last equation, you can write the integrand just as a sum of exponentials and then use the Dirac delta function formula on each term to get the final result.
Jun
10
comment $\int_0^{+\infty}{ \sin{(ax)} \sin{(bx)}}dx=?$
Correct me if I'm wrong, but isn't the integrand only periodic if $a$ and $b$ are commensurate? The logic here is that the period of the product must contain an integer number of both the periods corresponding to $a$ and $b$. This is impossible unless $a/b$ is rational. Editted
Jun
10
answered $\int_0^{+\infty}{ \sin{(ax)} \sin{(bx)}}dx=?$
Apr
19
awarded  Yearling
Apr
18
answered Interesting calculus problem advice
Apr
18
answered Reference Book on Special Functions
Jan
23
comment implicit derivates incorperating laplace's equation
@Jarryd Yes, at that stage, you now use the fact that $f(x,y)$ is itself harmonic. This means $f^{(0,2)}(A,B)+f^{(2,0)}(A,B)=0$ for any $A$ and $B$. Therefore $F(x,y)$ is harmonic.
Jan
23
comment What is the difference between a point and a vector
Well, my answer is not very precise or technical, so I could see how some people wouldn't like it. On the other hand, from the way the question was posed, I was intuiting that the OP is probably in the middle of undergrad, perhaps not even a math major, so I thought my answer would be more useful then some of the very detailed, technical responses that I saw. Anyway, I appreciate the support.
Jan
22
comment Interchanging a limit and integral
Note that the bottom example doesn't really make any sense. The definite integral doesn't depend on $x$ at all (since $x$ is the variable of integration) so taking its limit as $x\to\infty$ is pointless.
Jan
22
comment implicit derivates incorperating laplace's equation
You're welcome. If you found this useful, could you please accept it as an answer?
Jan
22
comment Help with Adventures in Stochastic Processes
You'll have better luck soliciting useful responses if you just copy down the statement and any explanatory text here in the body of your question. I couldn't even see the text when I followed your link.
Jan
22
answered Solving the differential equation: $ x = 11 (y')^{10} \cdot y'' $
Jan
22
answered Probability of having a complete random graph
Jan
22
answered Complex numbers and Roots of unity
Jan
22
revised Find the value of $\delta> 0$ for….
cleaned up the notation and formatting
Jan
22
suggested suggested edit on Find the value of $\delta> 0$ for….