Can we have negative factorial?

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We will figure out if we can do the factorial of -1/2 and the factorial of -1. We will be using the extension of factorial via the Pi function and the Gamma function. I will also give a summary at the end on when we can have negative factorials.
Pi & Gamma functions: • extending the factoria...
0^0 convention: • 0^0=1 is "seriesly" us... ,
negative factorial, • Can we have negative f...
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[@heliocentric1756]

10:50 Why the inequality method doesn't work? Can't we solve it like this? For t between 0 to 1: e^t is smaller than or equal e, so 1/e^t is greater than or equal 1/e so 1/(te^t) is greater than or equal 1/(et) so the integral from 0 to 1 of 1/te^t diverge

[@blackpenredpen]

OMG! You are right! I forgot for that bound is from 0 to 1... This is what happens when I have to do two different types of improper integral back to back...

[@Prxwler]

Are you argentinian?

[@udayadityabhattacharyya7496]

@@Prxwler ttrrrryyyrfi

[@mujahidrao7278]

What is the Integration of {x+1/x}½ ? Please solve this problem.🙏🙏🙏

[@Jacob-uy8ox]

try to do a complex factorial

[@SEBithehiper945]

You can do it like this: When doing Π(z), plug in complex integration.

[@Jeehd]

💀

[@sgiri2012]

​@@SEBithehiper945 How to actually calculate the negative number factorial without the intervention of gamma function plot. I want to plot (-1/3)!,(-2/3)!,(-5/3)!,... etc. i tried to solve by gamma integral. But didn't ended up in answer

[@ramez2775]

"How many ways can you arrange negative 1 apples?" ...

[@alexwang982]

R. 1

[@yosefmacgruber1920]

@@alexwang982 Oh really? Then n! for n

[@alexwang982]

@@yosefmacgruber1920 gamma function, mate and we meet again!

[@Fokalopoka]

@@alexwang982 using gamma function (-1)! diverges, and you cant even say that you can arrange it infinitely many ways, limit of x! as x->-1 doesnt exist

[@Fokalopoka]

also gamma of -1 is 0!, remember your definitions

[@ScottMaday]

*someone:* how many ways can you arrange negative half of a quarter *me:* square root of pi ways.

[@arnavanand8037]

You are too uneducated mathematically for this channel

[@ham1533]

Robin Sailo I think he meant -1/2 of a quarter (coin)

[@chaoticstorm8145]

@@arnavanand8037 Ohh and you're too educated for a joke?

[@chaoticstorm8145]

@@arnavanand8037 get over yourself buddy

[@sakuhoa]

@@arnavanand8037 you too sit, have a nice day.

[@blackpenredpen]

Why Gamma function? check out this page math.stackexchange.com/questions/1537/why-is-eulers-gamma-function-the-best-extension-of-the-factorial-function-to

[@Neo-po2xw]

blackpenredpen so what's 0.5! ?

[@donotlaughagain5093]

factOREO!

[@sdsa007]

Thank you! This is very advanced for me , but I am so glad I can find answers to my math questions! Awesome!

[@MichaelGrantPhD]

As I said in your poll, this is a definition issue. There are of course well defined ways to extend the factorial function beyond the nonnegative integers. But the exclamation mark is reserved for that original, integral definition. It's unfortunate that the Gamma function is "off by one" or it would be easy to just use that and call it a day.

[@GRBtutorials]

The Π function (mentioned in the video) is what you’re looking for, it’s the Γ function, but displaced by 1 unit: Π(x) = Γ(x+1).

[@MichaelGrantPhD]

@@GRBtutorials thank you!

[@Filip6754]

But can you do dis?

[@PeterAuto1]

Yes, cause e^t in the interval [0,1] is always less than e, so if you replace e^t with e the value gets smaller. and cause e is a constant it can be ignored entirely.

[@azharhaque9]

Chvocht - thats how i found this video aswell, dont even know why i clicked on it

[@quarkonium3795]

Peter Auto r/wooosh

[@thenixaless7493]

Yes *Leans Chair Backwards*

[@greenpewdiepie4207]

@@PeterAuto1 I know this was 4 years ago but woooooosh dude

[@nicholasleclerc1583]

14:11 “That I want my students to show...” OMG !! YOU HAVE STUDENTS !!!

[@blackpenredpen]

Nicholas Leclerc Yes

[@jorgeeduardopereztasso6134]

I remember that in my Calculus exam my teacher put a question with that divergent integral... It took me like all the exam time to realize that cannot be solved. XD

[@hritikworld9944]

Jorge Eduardo Pérez Tasso in which exam bro

[@jorgeeduardopereztasso6134]

In one of my College's exam @Hritik Rastogi

[@themsk9923]

You are literally bringing those questions which i always thought about 👍 thanks 😊

[@lill_m8]

Me: Can we have (-1)! at home? Mom: We have (-1)! at home. (-1)! at home: Undefined

[@factsheet4930]

Great this reminded me of the old questions we got back in elementary school where they asked things like: 2_2_2_2 = and you had to put signs in to make it equal as many numbers as you could usually like from 0 to 10 but now knowing whats -0.5! there is a cool question you can ask your friends 2_2_2_2=π and see if they can solve it! my solution is (-2^(-2)*2)!^2=π

[@Jee2024IIT]

Amazing

[@atrumluminarium]

If I remember correctly, there's a neat trick where you can "extract" the divergences/poles (on the negative integers) by using the by-parts expansion of Γ. This gives Γ(x)=Γ(x-n)/(x(x-1)...(x-n)) or something along those lines (it has been a while since I did complex analysis so my memory is a bit hazy) where you end up with the first n poles along the negative integers in the denominator.

[@ebz4125]

Was wondering if you can make a video on the analytical continuation / poles of the gamma function? That'd be interesting.

[@johi5951]

Ok ok i can do neg factorials... BUT CAN YOU DO THIS?

[@tomatrix7525]

Extremely nice vid bprp

[@bitcoinbuy]

Kudos to those who understood this

[@AbouTaim-Lille]

f(t) = 1/e^t is absolutely continuous over any closed interval and it has a max and min in the [O, 1] so it is easy to compare the initial integral with that of 1/t multiplied by some certain constant which is Devergent.

[@michaelspence2508]

You could also look at (-1)! as a sequence. Every time you subtract 1 you multiply by a larger value (in terms of absolute value) and change the sign. Roughly speaking it looks (vaguely) like the graph of x*sin(x) in that it approaches both infinity AND negative infinity which is why saying (-1)! = infinity is incorrect.

[@pon1]

If we treat the number line from negative infinity to positive infinity as a infinite circle then the undefined part is when both ends meet at infinity, I think they showed that each undefined part has its own infinity, I think there's some mathematical theory that uses that.

[@VenThusiaist]

Not necessarily. Remember, Infinity is a concept, so if you want, you can treat infinity like a number, but not exactly like one. Like the idea of ∞+n=∞ and 1/0≠+∞ or -∞, but instead is 1/0=±∞

[@orngng]

@@VenThusiaist What do you mean "not necessarily", only to reply with something else that doesn't follow up on the original comment

[@VenThusiaist]

@@orngng did you even look at the last part :| Listen, (-1)! gets you a vertical asymtope as you can literally see in the graph of Π(x), and a vertical asymtope has the value of 1/0, which could possibly be ±∞. The original comment literally described a vertical asymtope is and why it's a problem to 1/0. Do YOU understand what the comment is even saying?

[@VenThusiaist]

@@pon1 That is called "Wheel Algebra", my friend.

[@homer4590]

I don't understand all of this but it's fun to watch him get going on math

[@Qoow8e1deDgikQ9m3ZG]

instead of saying (-1)! is undefined or infinity, I think there is a need to put a strict and new definition to something like 1=0*infinity .... maybe something new symbol that is very super and like complex number i that avoid explain what is sqrt(-1)

[@linknero1]

so which for negative numbers are indetermined the factorial function?

[@flamingpaper7751]

Can you take the factorial of complex numbers, like i or 1+i? Or even quaternions like 1+i+j+k?

[@bonkuto7679]

That would be cool just try and plug it in and see what happens

[@wraithlordkoto]

@@bonkuto7679 I don't think there is a meaningful or useful notion of what it means to raise a number to a quaternion exponent power

[@The-Devils-Advocate]

@@wraithlordkoto maybe not in today’s conditions of math and science

[@wraithlordkoto]

@@The-Devils-Advocate I dont remember what it means, but quaternion exponents are a thing actually

[@The-Devils-Advocate]

@@wraithlordkoto I meant that they might not be useful today, but later they could be, like imaginary numbers

[@igorzigmaker5785]

I don't know why but hat scream at the very end just scared me so freaking much.

[@blackpenredpen]

sorry.... I think I forgot to lower the volume on that..

[@MrRyanroberson1]

3:48 actually i made a comment similar to this in a peyam video. as a polynomial of degree 6 differentiated 7 times should get 0, if we differentiate 5.5 times to get a degree of 1/2, differentiate again for a power of -1/2, and finally half-differentiate then the 7th derivative of x^6 ~ 1/x

[@VSP4591]

Very ingenious. Congratulation.

[@joaomarcosdossantos7593]

Hi, I really enjoy your videos. Could you show something about the wau(or digamn) number. I saw it, and got curious. Thanks for your amazing videos here.

[@dlevi67]

It's all one. Check the date of the "wau" video.

[@General12th]

Yay! This makes so much sense!

[@Mephisto707]

Can you show us the graph of the Pi function?

[@MathNerdGamer]

Take the Gamma function's graph and shift to the left by 1, because Pi(x) = Gamma(x+1).

[@giladu.6551]

Keep doing what you're doing!

[@yosefmacgruber1920]

Do you live in ישראל (Israel)?

[@EHTom]

Can you use the squeeze theorem to find a value?

[@MultiJoan09]

i have a question. pi(x) is a good function for factorials. But pi(x)*cos(2*pi*x) it's also a correct function for factorials. Why do use one and not the other one

[@williamwong03]

you are fooking instant. recall and teach my math a lot~~~

[@trueriver1950]

I prefer the second method, from 16:00 onwards - it is much more intuitively appealing If you insist on using the PI function we can still do the same Having already shown the relationship PI (n) = n.PI (n-1) in an earlier video, we can simply apply this result instead of repeating the Laplace integral again.

[@yosefmacgruber1920]

Would ∏(n) = n • ∏(n-1) be an improvement upon your syntax, or did I do it wrong in some way?

[@madhavstalks3925]

Amazing stuff

[@madhavstalks3925]

Loved video

[@cicik57]

can you also view it as the series of sum ,and easy compare to known harmonic series and 1/n^2 series

[@meryemnour9996]

Thank you

[@chrisrybak4961]

Great! So, as 0! = 1! = 1 but n!

[@EAtheatreguy]

So when I look up the gamma function on wikipedia, why is it defined for negative non-integers? This integral diverges for all exponents of t less than negative 1, right? So this integral can't be used to find those values. Is it just analytic continuation using the identity x!=(x+1)!/(x+1)?

[@MysteryHendrik]

Yes, it is analytic continuation using that identity.

[@fanamatakecick97]

Is it only the antiderivative of e^-1 that is a logarithm?

[@user-ns4et7de4z]

Thank's very much

[@assiddiq7360]

Where do I find the graph?

[@protasov-by]

Hey! Maybe if factorial of (-1/2) i.e. -0.5! is sqrt(pi) then maybe if there is some rule as one factorial solution can be written as summ of other factorial solutions, then ! (-1) can be calculated and finally be defined from that?

[@kamoroso94]

So what is the domain of the factorial function?

[@elliottmanley5182]

Have you thought about bringing your accessible approach to explaining derangements, subfactorials and the partial gamma function? When I first researched this, I find the appearance of e unexpected and delightful and the appearance of the nearest integer function completely counter intuitive. Now I've started looking at the analytic continuation of subfactorials and I find it counter intuitive in two ways. First that, as far as I can tell, it's defined everywhere, including negative integers and second that it maps real numbers into the complex plain.

[@elliottmanley5182]

*plane. Oops.

[@mondherbouazizi4433]

In the second part of the check, why not just say that (1/(t*e(t))) is always bigger than (1/t*e) {which is here 1/(t*e(1))} in the interval (0,1] and since the integral of {(1/t) * Constant} diverges, the other integral diverge as well? Edit: Recall, the integral of the function between these two values is the area beneath it.

[@josearita6440]

Hello! Can you make a video explaining this optimization word problem? I would really appreciate it! Love your videos btw! A woman in a rowboat 3 miles from the nearest point on a straight shore line wishes to reach the dock which is 4 miles farther down the shore. If she can sail at a rate of 6 miles per hour and run at a rate of 4 miles per hour, how should she proceed in order to reach the dock in the shortest amount of time? I can't figure this out! Thanks

[@AhsimNreiziev]

Trick question. If she sails -- although _rowing_ would be more consistent with her stated mode of transportation -- faster than she runs, and the shortest path is only rowing, then obviously taking the shortest path is not only the path of least distance, but also the path of least time. Rowing the *sqrt((3^2) + (2 ^ 2)) = 5 miles* (by Pythagoras' Theorem) at *6 miles / hour* would take her 50 minutes. All other paths are slower than that. There is a much more interesting type of problem that's similar than this, but it only works if the speed in the water -- or whatever travel medium the starting point is in -- is actually _slower_ than the one on the sand (or whatever type of medium the end point is in). It also only works if the end point is _not_ on the line that is the transition from one medium to the other (the shore in this case). Instead, it must be at least marginally "land-inwards", so to speak If you're interested, watch this video by VSauce. kzbin.info/www/bejne/qZzZn51sbL56o9k The whole thing is brilliant and I definitely recommend watching the whole thing, but the type of problem I was talking is given an example at around the 6:25 mark (or maybe a few seconds after that -- it's the one with the mud and the road).

[@ARAVINDKUMAR-ug7gt]

Great sir

[@veetaha]

Hey blackpenredpen. May I clarify something about your students? According to me, I study at the top 2 (or 1) university of Ukraine, and our students are so lazy that about 60% of all of them at my specialty do not pass calculus exam ('cause we have a strict tutor=)). So the question is: how many students pass your exams in average?

[@zubiiiiii_]

I can barely understand anything but its so satysfying to watch lol

[@NekoChan_TV]

So can we conclude that f : x => x! is defined on R/Z-* ?

[@haydenkarkainen1167]

Rip negative integers, thanks for the video!

[@odysseus9672]

(-1)! is undefined, but 1/(-1)! = 0 just fine. You can show this without resorting to the gamma function by considering the number of ways to write n symbols in a list of length k. That is given by n! / (n-k)!. First: how many ways are there to write the list when it has length n? n!, obviously, but that requires 0! = 1. Similarly, if k = 0 the formula gives 1, but that is also 0!. Put another way, there is one way to write an empty list, just put the grouping symbols (that avoids the philosophical worry over how to arrange 0 things). Second: how many ways are there to write the list when k > n? Zero ways, because you can never successfully write such a list, but that requires n!/(n-k)! = 0 for k > n.

[@purim_sakamoto]

ふぇええ　こうやって拡張できるのがガンマ関数の面白いところですねえ そして(-1)!がこれまた面白い

[@snakespeak]

Brilliant!

[@thepipe6397]

That was a good clickbait tittle, i stoped immediately what i was doing.

[@blackpenredpen]

hehehe

[@tejing2001]

The Pi function has a singularity at each negative integer, but because those singularities are poles, not essential singularities, it is reasonable (so long as you take appropriate care) to say the value at those points is projective infinity in much the same way that other intuitive processes (like splitting up dy/dx and working with the dy and dx as individual values, or pretending the dirac delta is a function even though it isn't) are not only reasonable, but helpful, so long as you properly account for the caveats. That said, it is safer, if you're not confident of your ability to properly handle the caveats, to just say the value is undefined.

[@XESolar]

Complex factorials possible?

[@davidrheault7896]

Yes, GAMMA in complex analysis is a meromorphic function, it has poles with residues at negative integers, and you can compute the integral in all the positive complex domain (Re(z)>0 otherwise the integral in undefined). Beware the real part of z being negative though since you need the mirror to compute the analytic conitnuation, example , GAMMA (-3.15) = pi/sin(pi*(-3.15_)/GAMMA(4.15), same thing goes for any complex value with negative real part, you need to mirror into the original domain, GAMMA(z) = pi/sin(pi*z)/GAMMA(1-z)

[@PhasmidTutorials]

materiasacra

[@davidrheault7896]

Yes, since Re(z)=1 >0, the integral is convergent, GAMMA(1+i) = .4980156681-.1549498284*I

[@dlevi67]

Whatever the hell a "factorial" of a non-ordered field means. :-D

[@alexwang982]

reddit.com

[@__fahim.__123]

I used feynman's technique although its undefined its a pleasure to use that technique like its soo gud yk

[@rifaturrahman5779]

Wait can't we integrate e^(-t)/t by Feynman/Leibnitz rule?

[@pizzy1992]

Eventi with that technic it doesn't converge

[@VikasKumar-jz5mv]

Sir U r great,👍👍👍💥💥💥💥

[@palmtree6794]

How do we find i!

[@mathtips877]

DEAR SIR, I REQUEST YOU TO POST VIDEOS ON MULTIPLE INTEGRALS

[@andreamonteroso8586]

can you do Gamma(n+1/2) and Gamma(-n+1/2) formula? pls

[@atharvaverma5013]

Blackpenredpen when I tried it on Hiper scientific calculator shows that (-1)! is -1

[@aaronrashid2075]

So this means that the factorial is undefined for all negative integers right?

[@user-qb5gw7tc9e]

15:58 i definitely heard "Ладно я шучу"

[@randomviewer896]

It's worth noting that as you approch (-1)! from the negative side, then it diverges to negative infinity too.

[@VenThusiaist]

Like I have said many times before, It is *_undefined_* due to *_definition issues._* A common solution used by the math community is ±∞ as it's own value instead of +∞ or -∞.

[@factsheet4930]

I challenge you to make cool solutions to the indeterminate form 0^i with limits!

[@wojtek9395]

What about 1/n factorial f.e 1/3? I will try it on my own but I will probably fail. It may be e^t^3 but then I don't know what to do next, maybe it is related to higher dimensions.

[@davidrheault7896]

wo997 we don't usually use factorial but GAMMA function. And yes you can compute GAMMA(1/3) and by Euler reflection the other ones. This number is transcendental (PROVEN!) I guess what you mean is 1/3! = GAMMA 4/3

[@peanut12345]

Yet sqrt -1 is Real, no -1! is UNREAL, Euler had Whiskey on Weekends.

[@himanshumallick2269]

Riemann Zeta function's integral expression involves gamma function. We know that Riemann Zeta function is defined everywhere in the complex plane (by analytic continuation), except for the line where Re(z)=1. Thus Zeta(-n) is defined (where n a positive integer). But when (-n) is plugged into the integral expression,the zeta function (LHS) is defined whereas RHS is undefined because gamma (-n) is undefined. How to resolve this problem????

[@davidrheault7896]

himanshu mallick zeta integral is only convergent for Re(z)>1 the same thing for GAMMA function (Re(z)>0). For any negative complex number I mean the real part you need to use the mirror for the analytic continuation (functional equation)

[@sleekweasel]

Why don't you need to put ± in front of u for (-½)!, since you've taken the root of t?

[@seemajois3263]

Can you please make a video on i factorial? !'i!'¡ looks good with Spanish exclaimation mark.😆

[@newmathexperiments4702]

2iπ=what. plese tell me

[@Cardgames4children]

Gamma upsets me. The pi function is much more logical! What gives?!

[@brandonklein1]

Also, putting in any n+1 to the gamma function... You'll find that it gives the same integral as the pi function by subtracting 1! The gamma function has a very useful property that gamma (x+1)=x gamma (x) which flirts very closely with the Reimann Zeta Function and a ton of other series in higher math

[@atrumluminarium]

Gamma function is usually much more convenient when studying the Riemann Zeta function

[@tracyh5751]

gamma function also arises in statistics very naturally.

[@CharlesPanigeo]

@@tracyh5751 yep! Gamma distributions are useful for modeling continuous random variable distributions that are positively skewed. Also, other distributions like the chi squared distribution, and the exponinetial distribution are really special cases of the gamma distribution.

[@MarcoMa210]

The gamma function is indeed very convenient for the riemann zeta function, but what i really don't get is people using it for calculating simple factorials, WHY??? You are doing more work when you could be using the capital pi function which is simpler.

[@ffggddss]

Behavior of factorial in the vicinity of a negative integer: When n is a positive integer, and ε is an infinitesimal quantity, (-n + ε)! ~ (-1)¹⁻ⁿ·n!/ε An interesting plot to show this, is y = 1/x! It oscillates for x < 0, crossing the x-axis for each negative integer; the amplitude increases "factorially" as x becomes more negative. For x > 0, y > 0, and goes asymptotically to 0 as x increases toward ∞. y has a local maximum for x between 0 and 1. Fred

[@Xnoob545]

James Grime (on numberphile) extended the function and it didn't work... i mean it kinda worked... i guess... -1! = 1÷0

[@themanofiron785]

Well the result is right...you still end up with infinity :)

[@MarioPlinplin]

What if you just expand your number set? Maybe you can call (-1)! something like "t" and just use it to divide by 0 (since 1=0xt then n/0=nt), kinda like what you do with i and complex numbers. Anyone knows if this is consistent wiht the rest of the math?

[@koskovictor5095]

Yes only if no 0/0 develope

[@MysteryHendrik]

Problems arise if we still want multiplication to be associative: 2 = 2×1 = 2×(0×t) = (2×0)×t = 0×t = 1

[@thesardaunatv]

Thank you Mentor

[@NLproductions-xo2gq]

This method can work because 0!=0*(0-1)! which gives you 1=0*(-1)! which if you do in another way 0-1=-1 then you can say 1=1/-1*(-1)! then you get (-1)!=1/1*(-1)! which you tern the 1/1=1 then you put the factorial in front of the positive one and multiply it by (-1) which would look something like (-1)!=1!*(-1) which then 1!=1 so then you get (-1)!=1*(-1) which then equals to (-1)!=-1 because 1 multiplied by -1 = -1 so this means this works and (-1)! does exist and equals -1

[@rakshithgowda1606]

Let g be a continuous function which is not differentiable at 0 and let g(0) = 8. If f(x) = x.g(x), then f(0)? A) 0 B) 4 C) 2 D) 8.

[@iloveevermore13]

4! 24 We divide 4 and 3! 6 We divide 3 and 2! 2 We divide 2 and 1! 1 We divide 1 and 0! 1 We divide "0" and -1! 1/0 nondefined We divide -1 and -2! -1/0 nondefined And for other negatife numbers x/0

[@alexmeanin8049]

Cooool!

[@charliebaker1427]

why did you multiple by 2

[@Sans-bd3bn]

Me: hmmm lets open youtupe because i am tired of studying This man:

[@vietnambarca233gmail]

Please do a video on complex numbers factorials

[@blackpenredpen]

I will try!

[@yosefmacgruber1920]

@@blackpenredpen And what about quaternion factorials? Is there any such thing? Are quaternions the ultimate numbers?

[@juliankemmink]

Is it allowed to say (-1)!=1/0

[@jowillll]

Love the accent ❤️

[@adamkangoroo8475]

Maybe make the next video about the derivative of factorial?

[@davidrheault7896]

Adam Kangoroo the derivative of the GAMMA function is called the digamma function and it is also meromorphic with the same poles

[@cosimobaldi03]

Well if we know half! = root pi /2, then by a property of the gamma function (half - 1)! = half! /half = 2*half! = root pi.

[@pervezdar7340]

In general sir.... Tell me that ... Can we find ( R )! Where R is any real number...

[@ronitmandal7301]

But can you do this?!

[@replicaacliper]

Its a meme you dip

[@RafaxDRufus]

Steve, could you integrate sqrt(1+4x²)? I'd appreciate it so much

[@kaszimidaczi]

Set tan(u)=2x, then sec^2(u)du=2dx. After that you end up with half of the integral of sec^3(u). You can find the solution to that on his channel. Then use the fact that u=arctan(2x).

[@RafaxDRufus]

kaszimidaczi Oh thank you! I haven't though it could be possible with tangent. Thanks :D

[@igorzigmaker5785]

Simply set 2x = sh(u), then 2dx = ch(u) du, dx = ch(u)/2 du. Putting everything in your integral and knowing that 1 + sh^2(u) = ch^2(u) and ch(u) is always possible, you get ch(u) * ch(u)/2 du = 1/2 (ch(u))^2 du. After that use fact ch(u) = 1/2 (e^u + e^(-u)) and you'll get pretty simple integral with exponents.

[@kaszimidaczi]

Rafa xD No problem :)

[@obinnanwakwue5735]

His name is Steve?!

[@alexhancu969]

Can you try to do "(1/3)!"?

[@VSP4591]

Excelent

[@billprovince8759]

While (-1)! is undefined, it seems that you should be able to show that lim(x --> -1+, x!) approaches +inf.

[@dlevi67]

Yes, but the same limit approached from the left approaches -inf, hence the "undefined".

[@billprovince8759]

Agreed: There's a nice plot of the Gamma function (not the Pi function) at Wikipedia: en.wikipedia.org/wiki/Gamma_function

[@billprovince8759]

@Gerben van Straaten Agreed: The value is undefined, because you reach different limits if you approach from left vs right. In fact, my comment shows that approaching -1 from above (i.e., x --> -1+), it approaches infinity.