Pi Function (Euler Factorial Function)

[@flowingafterglow629]

This is a lot of fun, with the back and forth and your quick responses to the comments. You posted a video yesterday in response to comments from the day before, and this video is now including a response to comments from literally yesterday. Very enjoyable!

[@PrimeNewtons]

Glad you enjoyed it!

[@souravgarain2835]

You are an amazing teacher

[@prakrit1280]

Sir's teaching methodology is awesome 😎👍Plus Sir is genious - math mentor, youtuber, singer 😎😇👍 Always looking forward to your upcoming videos😁

[@adibjauhari]

I like how you explain math concepts. With such a fun expression, it makes me watch this video till the end. Thanks 😄

[@ProactiveYellow]

0 as a member of the natural numbers depends on the field. In axiomatic set theory, specifically using Von Neumann's construction of the natural numbers, the natural numbers are constructed as follows: Let 0 be defined as the empty set ∅, and let the function s(n) on the natural numbers be defined as s(n)=n U {n} where U is the union of sets. The set of natural numbers is therefore the intersection of all sets that satisfy the axiom of infinity. That is, the natural numbers are precisely the set of 0 and all its successors. This has the benefit of any natural number n containing precisely n elements, and for all n and m, n

[@allozovsky]

As a compromise solution, we may use *ℕ* for {1, 2, 3, ...}and *ℕ₀* for {0, 1, 2, 3, ...}, though ISO 80000 defines the set of natural numbers as the set of positive integers and zero, and denotes it by simply *ℕ* (without a subscript).

[@patrickfrei9322]

Came here for this, thank you! It really puzzles me when people just claim that 0 isn't part of the naturals 😅

[@benshapiro8506]

the function under the integral sign has 0^0 as part of its expression evaluated at the left end pt of the interval. therefore special consideration must be given to evaluate the integral.

[@jlmassir]

You're a great singer too! May I suggest two more reasons for 0! = 1! = 1? First, it allows a uniform formula for the binomal coefficients. Second, n! is the number of permutations of n elements, and there is only one permutation of 1 or 0 elements (a permutation of 0 elements is somewhat spooky, but it is a real thing in combinatorics). Please do a video about this!

[@allozovsky]

Also 0! is an _empty product,_ which (by convention) is equal to the neutral element of multiplication (multiplicative identity), that is 1. Just like 0⁰ is also an empty product and is equal to 1 for the same reason (when treated as exponentiation with a _natural_ exponent).

[@jlmassir]

@@allozovsky So this argument resolves the controversy about 0^0?

[@allozovsky]

@@jlmassir Yeah, that is how exponentiolation with a _natural_ exponent is normally defined: *a⁰ = 1* (for any base *a,* as an empty product), and *aⁿ⁺¹ = aⁿ·a,* which gives us *0⁰ = 1, 0¹ = 0⁰·0 = 1·0 = 0,* and so on. No division by zero is needed to define 0⁰ with this approach, so it is well-defined under exponentiation with a natural exponent.

[@Gremriel]

"There's no mystery," he says. Me, watching this: "uh huh".

[@surendrakverma555]

Very good lecture Sir. Thanks 🙏

[@lucdutreiz5135]

Loved when you sang

[@kartikbhardwaj2604]

Bring more real analysis vids❤

[@EvilSandwich]

Man, when you set out to correct a faux pas, you don't mess around. lol

[@prabhatrexkira398]

Amazing....Never Stop Learning, Never Stop Roasting 😅😅😅😅

[@YixuanREN-q7q]

love your video so much！！！

[@anglaismoyen]

The factorial plot thickens... by the way, what watch do you wear?

[@valemontgomery9401]

Do we really need the symbols for pi and gamma in the first place? We already have the integral and the ‘!’ for the factorial function, what’s the point of using pi just to say the same thing?

[@mikefochtman7164]

Just a shorthand way of writing an often used function. Like the upper-case 'L' used for LaPlace function, or many others. Saves a little space for a well documented and often used idea.

[@wkmartins]

Integral notation might be a littlle too long. But i agree just using '!' would be better because we don't invent a new symbol when we extend multiplication to the reals

[@derzahlenmensch6089]

It's more of a formal thing. Usually, factorials using the "!"- notation are only defined for natural numbers (n! = n(n-1)!) while the Γ - and the Π - function extend that concept to any complex number z. So it's basically just used to make clear that you're trying to calculate the factorial of a complex number (not specifically a natural number). But yes, you're right you could use the "!"-notation for any factorial (as long as it's clear that the integral definition can be used which every person knowing that definition should think of at first though).

[@sphakamisozondi]

Euler, Gauss and the Bernoulli brothers are in the Mount Rushmore of mathematics. Modern mathematicians, I gotta give it to David Hilbert and Henri Poincaré

[@patrickfrei9322]

There were more than just 2 berboullis, whole family of geniuses 😄

[@haotj]

At 5:59 shouldn't it be t^(-1) since PI(x) = Gamma(x+1)?

[@bowlineobama]

Mr. Newtons, since you and I are big fans of Euler, I would like to send you a gift. Where can I send it? No kidding.

[@DragomirGąsieniec]

Never stop teaching, cause those who stop teaching stop developing the world 😁

[@mrvortex8885]

Anyone know how to solve 12.8(x+2^x)=x^x

[@allozovsky]

12.8 = 256/20 = 4⁴/20 4⁴·(x+2ˣ) = xˣ·20 x = 4 But we need to prove there are no other solutions (real or complex).

[@curtpiazza1688]

GREAT! 😂

[@Ahmad-yi6d]

❤

[@Heemashti]

why 0⁰=1?

[@МаксимАндреев-щ7б]

I usually see pi(x)=number of primes that

[@สุธีร์แซ่ฮุ้น]

F(s)=1/s F(1)=1

[@serae4060]

√-1 8 Σ π and it was delicious

[@축복이-x6u]

asnwer=1 isit