In this video I used the gamma function to evaluate a definite integral that would otherwise, be very hard to evaluate.
Пікірлер: 35
@bridgeon75024 ай бұрын
Interesting how pi always shows up in integrals 🤔
@user-ky9kv5je9s4 ай бұрын
I think you make a mistake when you put Gamma(-1/2+1) at the end of video. You might say Gamma(1/2+1)
@PrimeNewtons4 ай бұрын
Oops 🙊 you're right.
@Heemashti4 ай бұрын
Please include your logarithms tutorial, for example log(log(logx))
@hosseinmortazavi790315 күн бұрын
Nice teacher
@marcolima894 ай бұрын
Your handwriting is seriously one of the prettiest I have ever seen. just one minor detail, in the end result, the 27 in the denominator should be +/- 27. Thanks a lot for these videos, amazing quality. As a mechanical engineer, I miss sometimes these math lessons.
@datboy038Ай бұрын
Should it? He wasnt solving for x and the square root always gives out the absolute value
@curtpiazza16884 ай бұрын
WOW! Great STUFF! 😊
@williammartin44164 ай бұрын
Thanks!
@PrimeNewtons4 ай бұрын
Thanks a lot
@emmanuelonah45964 ай бұрын
Very interesting
@jennymissen35234 ай бұрын
How groovy is that?!
@xinpingdonohoe39784 ай бұрын
In my opinion, I prefer to just use the actual factorial notation. Some people say "that's only defined for nonnegative integers" but I disagree. The Riemann zeta function, through analytic continuation, can be defined almost everywhere. Not all of it will satisfy the initial definition of ζ(x)=sum n≥1 of 1/n^x, but we still call it ζ. ζ(-1) is well defined and equal to -1/12, even if it's not the actual sum n≥1 of n. It's the same here. The factorial function has been analytically continued to all but countably many numbers. And whilst most don't satisfy the original definition of k!, the product n=1 to k of n, it's still the factorial function. (1/2)! is well defined and equal to √π/2, even if it's not the product from n=1 to 1/2 of n.
@adw1z4 ай бұрын
The factorial function n! is only defined on the non-negative integers n (0,1,2,...), and technically speaking the gamma/pi function is not an analytic continuation of the factorial, but rather an interpolation of the factorial function, as such an extension to the complex plane C is not unique (analytic continuation (AC) applies only to connected domains D, which is not the case here with the non-negative integers - as such, any AC must be unique). The gamma function is what we use by convention in place of this non-unique extension via the integral definition (satisfies gamma(z+1) = zgamma(z), gamma(1) = 1) which indeed only converges for Re(z) > 0. But it is not unique: for example, the so-called pseudogamma function also successfully interpolates the factorial. Hence, denoting something such as the equivalent of gamma(3/2) as (1/2)! can be ambiguous. The AC only applies to the gamma function itself, taking the domain to Re(z) < 0 (on which it is meromorphic, poles at non-positive integers); this is unique by the identity theorem, and this time our domains Di (from Re(z)>0, continued to slices -1 < Re(z) < 0, -2 < Re(z) < 0 etc... until ultimately onto Re(z)
@parthasarathy49902 ай бұрын
Errata - it is gamma(3/2) = 1/2 * gamma(1/2)= root pi/2 .... Not gamma (-half+1)
@sovietwizard1620Ай бұрын
Even from the answer we can tell that the error function is involved in the indefinite integral answer to this.
@Mutlauch4 ай бұрын
Hi, would't the Pi-function have a z-1 in the exponent of t, because you substituted z -> z-1 with respect of the Gamma-funciotn? Greetings from Germany :)
@lawrencejelsma81184 ай бұрын
He did something strange like evaluating π(Z) = √π/2 results instead. I was just as confused but just accepting that the "Indefinite integral" from 0 to infinity of √te^(-t)dt evaluates to √π/2 after substitution of those limits. Hopefully the ending part is correct! 😂
@dark7mc3 ай бұрын
Γ(z)= ∫₀ ᪲ xᶻ⁻¹ e⁻ˣ dx
@flavioc.bannwart12162 ай бұрын
Excelente!
@AminKhodadadi-h7s23 күн бұрын
😊
@OwusuOseiEmmanuel3 ай бұрын
Please Sir, you forgot to factor out the negative in your multiplcation
@PrimeNewtons3 ай бұрын
I used it to flip the boundaries
@sergeygaevoy64224 ай бұрын
ln(1/x) = -ln(x) so it could be simplier ... Plus -exp(-u)*du = dx 'cause exp(-u) = x so we can avoid it at all.
@serae40604 ай бұрын
Hi, I think I have a fun task: Find all solutions for sin(1/x)=0 in the interval ]0;1]
@surendrakverma5554 ай бұрын
Excellent explanation Sir. Thanks 🙏🙏🙏🙏🙏
@alpborakirte8014 ай бұрын
Can we use Gaussian Integral
@ADN0Infinity4 ай бұрын
Smart change in variable
@estanley01314 ай бұрын
This was a fun one!
@saarike4 ай бұрын
Simply Great!!!!!!!!!!!!!!!!
@courbe4534 ай бұрын
I like so much.
@auztenz4 ай бұрын
1000th view!
@greggwendorf22234 ай бұрын
So nice to see an intelligent black man talking high-level math as opposed to hip-hop. My hat is off to you. sir..