I agree. However it comes at great cost. It makes me prone to mistakes. Then I redo the video or edit it. It's the only downside. I wish I could use prepared scripts. I would be bored 🤣
@@PrimeNewtons I get that too my class has spotted mistakes in my notes even pretty often. I try to avoid the notes but if I follow it they say what’s this math??
@@PrimeNewtons it’s the same reason why another person needs to review literature you write, you cannot see the obvious mistakes when reviewing your own work.
It is not true that the function is continuous as originally defined. True, we can plot the factorial "function" on a graph, but it will have only discrete natural numbers. Then we can draw lines between the points to pretend that the function, magically, becomes "continuous." Let's be accurate about how we should describe this attempt to find factorial values that "lie" between integers: "IF we ALL AGREE TO PRETEND this function were to be continuous by some miracle, then we calculate that the factorial of, SAY 3.5 IS THIS OR THAT. But in reality this does not exist because the definition of factorial is the multiplication of adjacent INTEGERS, not partials. Of course, mathematics being what it is, we can always reinvent the definition to fit our unnatural desires-------also known as "mathematicians with too much time on their hands!"" !!!!!
That's not a very good idea, I suppose. Given that the gamma (or pi) function satisfies the property n! = n ·(n−1)! for natural numbers, it would keep doing the same for the negative integers, and this property fails for negative integers (due to division by zero). Direct calculations of the integral should support this claim.
This will not work as the integral is divergent, it’s the same as dividing by 0 (it’s actually a simple pole). An equivalent definition of the gamma function is through the Euler product formula, where gamma(z) has terms lim n-> infinity (z-1)(z-2)…(z-n) on the denominator. From this u can quickly see that gamma(negative integer) is undefined
@@adw1z You probably meant to write Γ(z) = lim (n!·nᶻ)/(z(z+1)(z+2)...(z+n)) with "plus" signs instead, and also it is not defined for z = 0 either (because of the z factor in the denominator).
I know that nowadays the gamma (or pi) function are considered as definition of the factorial, but it seems like circular reasoning to me. You basically created a function that outputs the factorial as a result, which is fine. But now you are using that function to calculate something, that the factorial wasn't designed for and just decide that it is now part of factorial. To make it short, you do this: A → B B → A But how do you know, that "→" can be reversed? How do you know, that B didn't add something to A that wasn't there before or removed something from A, that is now missing (in terms of functionality, not value)?
A good question indeed. There are other possible extensions of the factorial, of course, though Euler's gamma function is the most famous and most widely used.
Start by defining gamma function. Then show gamma(1) = 0, and gamma(z+1) = zgamma(z). Hence deduce by induction that for z ==n a natural number, gamma(n+1) == n! , with no circular reasoning as u wrote. To be pedantic, the factorial is only valid for non-negative integers and we shouldn’t really be saying things like (1/2)! , but it’s just abuse of notation that is widely accepted at this point (e.g. by Demos!). In any literature, x! really means gamma(x+1) for x any complex number, so the poor abuse of notation makes it seem circular when it is simply a use of the base definition
@@adw1z The thing is, that you only proved it for the natural numbers, as those are the numbers that you can calculate factorial for. You HAVEN'T proved, that it is correct for any other number. Assuming, that it also works for any non-natural number is then circular reasoning, because you assumed that the gamma function IS the factorial and you define the factorial (of the non-natural number) as the result of gamma function. I don't know how to state this any clearer than this: You can only calculate factorial for natural numbers, therefore you cannot prove, that the gamma function is always equal to the factorial. Claiming otherwise is just plain wrong.
@@m.h.6470 yes because the factorial for any other number doesn’t exist, it’s nonsensical. The gamma function is (NOT UNIQUELY) a continuous version satisfying gamma(z+1) = zgamma(z) for all z complex and gamma(1) = 0, but there are infinitely many functions that also satisfy this property. Why I said, (1/2)! makes no sense and is an abuse of notation. But the CONVENTION is, it means gamma(3/2). It’s not something to be proven I recommend watching “Lines that Connect”’s video on extending the factorial to the reals, it may answer your question
The only issue about the gamma function as an extension to the factorial is that : Acording to the analytic extension of a function which demands that the function is defined on a set that has a limit point (accumulation point), in order to have a unique analytique extension.
Great video! But, of course, this is not the only possible extension of the factorial - there are other gamma functions (or, rather, pseudogamma functions) and "factorials", defined for real and complex numbers. Say, Hadamard's gamma function or Luschny factorial, just to name a few. Of course, they may give different results for (1/2)! or even for 0! (just like Luschny factorial does).
It is a good time to question the purpose. How can we use this (1/2)! and other riches of Γ and Π functions? Originaly n! was a number of permutations in an n-tuple. Can we invent sets of fractional size? Another use, is the binomial coefficients, which are fractions involving 3 factorials. Now we can replace the factorials with П. Does it give us some new possibilities? Binomial coefficients are used in Probability and Statistics in defining Binomial and Negative Binomial random variables. Can we consider them with fractional parameters now?
Yes, the binomial coefficients are defined using the gamma function for complex-valued binomials with complex exponents. Also the gamma distribution in statistics. The definition of the double factorial (used in the generalized formula of the volume of an n-dimensional hypersphere, as an example).
gamma(1/2) = sqrt(pi) is a lovely result that everyone should remember! If u remember the reflection formula, this should be easy to remember: gamma(z)gamma(1-z) = pi/sin(pi*z) , and plug in z = 1/2 ==> gamma(3/2) = 1/2 gamma(1/2) = sqrt(pi)/2 Video Ideas: I think it would be cool to make a video on the gamma function product formulas, such as the reflection formula, the Euler product formula, the Gauss product formula, the Stirling series and the Weierstrass product formula (which can be used to deduce the digamma function psi(z) == d/dz ln(gamma(z)), which is related to the Harmonic numbers and can be used to explain the Euler-Mascheroni constant). Plenty to take in! Can also do the Taylor series of the gamma function (related to above), the beta and the Riemann-zeta functions (all these functions are linked!)
You may calculate it numerically, by evaluating the corresponding integral: Γ(z) = (2ᶻ⁺¹/z) ∫₀¹[x·(−ln(x))ᶻ]dx This is another integral form of the gamma function, suitable for numerical calculations. With the trapezoid rule *n = 100000* is enough to obtain 6 correct digits.
I'll just keep distributing the factorial operator to both the numerator and denominator. (3/4)! = 3!/4! = 1/4. Much easier. Incorrect, but much easier.
We studied this function and its properties in the Analysis 3 method in the 2nd year. But we used it other methods later including the first year of master.
