I cannot say my expertise in calculus is in 505 level, but i am still wondering, why this channel is not way more popular! Rarely i can find channels in which some slightly more difficult problems are shown with almost every step for solving them! Thank you very much!!
@cytos
Ай бұрын
Solving the dirichlet integral in UNDER 2 MINUTES Step 1: Plug the integral into wolframalpha
@Ben-wv7ht
Жыл бұрын
If L(f)_s is the laplace transform of f in s , and L^-1(g)_x is the inverse laplace transform of g , You can use the identity Int [0,+oo] f(x)g(x) dx=int[0,+oo] L(f)_x L^-1(g)_x dx And you also get the result in the blink of an eye
@srinitishsrinivasan8089
Жыл бұрын
Nice, another possible way is using contour Integration by writing it as e^iz/z and then equating the imaginary part of the above form
@maths_505
Жыл бұрын
Already did a video on that I've solved this using lots of ways: Feynman's trick, contour integration, laplace transforms, the generalized Dirichlet integral and now Lobachevsky's formula
@MikeBlaskiewicz
Жыл бұрын
If you use (exp(ix)-1)/x the function is analytic and goes to zero in the upper half plane so it's just the integral of dz/z over a circular arc: i*pi. No residue theorem needed.
@guyzan
8 ай бұрын
It's the fourier transform evaluated at DC.
@rohitashwaKundu91
Жыл бұрын
A Pretty useful formula indeed!
@violintegral
Жыл бұрын
Lobachevsky 😤😤😤
@hectorjosedelarosagutierre8998
Жыл бұрын
You are so fantastic 💪
@manstuckinabox3679
Жыл бұрын
The function obviously representing pi/2* how quick you can solve this integral.
@user-fz9go8pj4t
Жыл бұрын
The only problem with this is that the proof of the formula itself is pretty sophisticated, so I don't consider this way of solving as "fast"
@dimastus
Жыл бұрын
Well, it is debatable. For me, this is like when deriving Euler's formula for exp(ix) using power series. It's fast but require the knowledge of sin and cos expansions. And deriving them is the whole another story. The Euler's identity can be proved from the limit definition of e^(x) using just properties of complex numbers, but it is definitely longer. And the same for integral of sinx/x. Lobachevsky formula require the knowledge of the sec(x) expanstion into partial fractions. Once you proved it, you can use it like the instrument to solve other problems faster.
@user-fz9go8pj4t
Жыл бұрын
@@dimastus Yes, but deriving Euler's formula from the sin and cos Maclaurin expansions are much more "mainstream" knowledge than the derivation of Lobachevsky's integral formula. As I know, the derivation of the Euler's complex exponential is worldwide well known through calculus/functions of one complex variable classes, but the Lobachevsky's formula is not
@dimastus
Жыл бұрын
Sure. I've not known this formula before youtube. I guess it is wide spread among people who deal with signal processing.
@holyshit922
Жыл бұрын
His name should be read Wobachevsky (Your W sounds like Russian Л)
@daddy_myers
Жыл бұрын
UwUbachevsky.
@maths_505
Жыл бұрын
😂😂😂
@user-fz9go8pj4t
Жыл бұрын
Folks, did you know that Lobachevsky also worked on modern geometry? Actually he has very famous non Euclidean geometry axioms..."Lobachevsky's geometry"
@8mice979
Жыл бұрын
you could use si(x) function
@vikraal6974
Жыл бұрын
I would have done it with Taylor series but this is far quicker
@MrWael1970
Жыл бұрын
great 😍
@nightmareintegral5593
Жыл бұрын
Nice integral trick. You can also use laplace transform indentiti because it is from 0 to infinity or use ramanujan master theorem. But pls HANKEL! 💔
@maths_505
Жыл бұрын
Yes we can but this is faster And this time we were here for a good time not a long time😂😂😂
@nightmareintegral5593
Жыл бұрын
@@maths_505 laplace transform is also not long way.
@nightmareintegral5593
Жыл бұрын
It will give you integral from 0 to infinity of 1/(t^2+1) dt which is arctan of infiniti - arctan of 0 which is pi/2
@maths_505
Жыл бұрын
Yeah but I've done that before
@nightmareintegral5593
Жыл бұрын
@@maths_505 Oh I didn’t know. XD
@illumexhisoka6181
Жыл бұрын
What about when f(x)=sin(x)
@maths_505
Жыл бұрын
Odd function over symmetric interval gives you zero If you want the integral over the half libe from zero to infty, use Feynman's trick
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