Solving an integral equation using special functions.

🌟Support the channel🌟

🌟my other channels🌟

🌟My Links🌟

🌟How I make Thumbnails🌟

🌟Suggest a problem🌟

[xizarrg]

I love how, on one hand he glosses over the intricacies of trig substitution in integrals, but at the same time he carefully explains how we know that pi/4 is less than one.

[thomashoffmann]

10:30 in the definition of the beta function, the exponents should be v-1 and w-1. The second definition with sin and cos is fine though.

[throwaway]

Dude... I just want to say this out loud. As a Physicist(and teacher myself) I cannot say how happy I am to have found your video series. As you might expect I'm more of a PDE kinda guy than a Number Theory kinda guy. But you help keep me sharp. Much obliged.

[maths_]

The general case of the integral parametrized by alpha is pretty cool too

[Happy_Abe]

@5:10 while that inequality looks obvious, one has to be careful for x values between 0 and 1 where a higher exponent actually leads to a smaller number

[erosravera]

Wow! It starts as "plain" calculus, then it take a long steep route through the Gamma function, and finally an elementary quadratic formula, that yields the Golden Ratio!

[jcantonelli]

Somehow, once he said that 1 < a < 2, I suspected that phi would be a solution. Nice problem.

[MothRay]

As soon as he said it’s between 1 and 2, I said I bet it’s going to be phi. And it was phi. Glorious!

[thomashoffmann]

5:30 the obvious inequality is not clear to me 🤔
If x < 1, i can't see the inequality at first glance because 1+x^a gets smaller.
E. G. For a = 3 and x = 0.1, the left integrand is bigger.

[adfriedman]

Great video! For those of us lucky enough to guess the u-substitution u = 1/(1+x^a) with differential dx = -1/a u^(-1/a-1) (1-u)^(1/a-1) du, this gives the normally recognizable form of the Beta function directly without the trig substitutions. I got stuck on this problem trying to find a satisfying x for the equation with the Gammas.

[aadfg]

Fun fact, phi is the only value in (1, 2) for which antiderivative has a nice form.

[ThAlEdison]

I'm pretty sure that evaluates to
for a=phi, phi-1/phi=1, 1+1/phi=phi, tGamma(t)=Gamma(1+t),


This is based on making a u-substitution x^a=u, which gets it to the form (1/a)Int_0^Inf(u^(1/a-1)du/(1+u)^a), which is (1/a)Beta(1/a, a-1/a). One Euler's reflection formula later, and we're at the point where I started.

[cicik]

ohh there is another video in internet of similar or exact same task where also it a power of golden ratio, using that a² = a+1, etc...

[DmitriStarostin]

You get alpha to be 1.618. The limit of the divisor would then be x power 2.61, and integrated x power 1.61. This is about 1 divide by x power 3/2 (three halves). It is Kepler's law! The square root of time power 3 is an axis of an ellipse. So we get 1 / axis of an ellipse. This may be the eccentricity or the 1 / eccentricity if it has a limit. In other words, if it has a limit, it is an elliptical (and not hyperbolic orbit). It is not exactly 3/2 because 1. the Moon is moving away from the Earth and the planets are moving away from the Sun 2. the universe is expanding 3. one does not count in the Earth's motion against the fixed stars.

[Alan-zftt]

That was beautiful! Compliments to approaching the problem tying it in with gammas and betas and bringing it down to earth with some simple quadratics. And all in < 17 minutes

[Jack_Callcott_AU]

Gee..that was fun❕ The general formula for the integral could come in handy too.

[JianJiaHe]

I have never used calculus for years, but I found watching a math problem being solved satisfying.

[wolfmanjacksaid]

It's been scientifically proven that watching Michael Penn videos every day makes you more smarter.

[Happy_Abe]

The inequality in the HW should be the other way

[karn]

Shouldn't the inequality at the end be flipped?