Integration using the gamma function

In this video I used the gamma function to evaluate a definite integral that would otherwise, be very hard to evaluate.

[bridgeon]

Interesting how pi always shows up in integrals 🤔

[kykvjes]

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)

[xinpingdonohoe]

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.

[Heemashti]

Please include your logarithms tutorial, for example log(log(logx))

[surendrakverma]

Excellent explanation Sir. Thanks 🙏🙏🙏🙏🙏

[Mutlauch]

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 :)

[marcolima]

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.

[sergeygaevoy]

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.

[sovietwizard]

Even from the answer we can tell that the error function is involved in the indefinite integral answer to this.

[OwusuOseiEmmanuel]

Please Sir, you forgot to factor out the negative in your multiplcation

[parthasarathy]

Errata - it is gamma(3/2) = 1/2 * gamma(1/2)= root pi/2 .... Not gamma (-half+1)

[serae]

Hi, I think I have a fun task: Find all solutions for sin(1/x)=0 in the interval ]0;1]

[greggwendorf]

So nice to see an intelligent black man talking high-level math as opposed to hip-hop. My hat is off to you. sir..