66\int_0^\infty x^n e^{-x}\ \mathrm dx=n!
is true ONLY for positive integers. Otherwise, you have the generalization to Euler's Gamma Function
\Gamma(z)=\int_0^\infty x^{z-1} e^{-x}\ \mathrm dx
so that \Gamma(n+1)=n!.
Does n! = the integral from 0 to infinity of (x^n)(e^-x)dx hold true for all real numbers?
If so, can we find the derivative of n!?
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10 Answers
6666You guys should go back and check the definitions well. Then you would be probably able to answer whether or not one can differentiate n!.
@Gallardo, looking at our posts in the thread on Series summation (where you talked about the various tests on convergence) I got a very positive impression about you, but the post #2 in the current thread forced me to think otherwise.
1But sir if I am not wrong again , we can differentiate a gamma funtion and express the result in terms of polygamma funtions , can't we ? What I mean to say is ,
d Γ( x + 1 )d x = Γ ( x ) φ ( x ) , where φ ( x ) is the polygamma fuction .
again sir , for +ve integer j , we can write
Γ" ( j + 1 ) = j ! [ - ξ + summation of 1i over j terms starting from 1 ] ,
where ξ is Euler's constant .
1In fact in general , can't we write
dn Γ ( x )d xn = ∫ z y - 1 e - z ( ln z ) n dz , limit being from 0 to infinity ?
62Dude.. no egoes.. the last time i was amazed by knowledge of someone in maths was prophet sir. I dont visit aops very often.. so tiit is my only source..
Seeing u and ur knowledge makes me feel dwarfed..
If you are real (I mean a student in class XII and below) kudos.
prophet sir also bluffed us all for a couple of months before i finally asked him if he was a student or not :D
neways keep up... I din understand a lot of ur last post and instead of wiki i prefer to skip this one ;)
Will read it sometime when i am free
62Gallardo, Whatever you say, I am not very convinced yet that you are a student.. It is really good if you are. because then you have already studied complex analysis.. real you already know a lot and here too you know a bit more than me ( I am not in a position to judge teh accuracy..) which i am sure anant sir or prophet sir will be iin a better position to explain...
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62with the knowledge that I have I would only back what anant sir has said..
n! is not differentiable under the definition that we use for n!
infact you are trying to use a function whose value is n! at different values of x.. and then trying to differentiate it..
that does not mean that it will be the derivaitive of n!
1Exactly sir , by the definition its impossible . But sir , one question again arises . If , indeed , we diiferentiate a integral and say that it is the derivative of the result of that integral , are we wrong or are we right ?
1hey plz gallardo alias SUaMYA SINHA BABU
wat u r trying to show....?huh ? that u know very much than anyone else here ?
plz we dont need and there is no need for us to study all this higher complex analysis at this level.....u r going out of jee sylllab which is of no need rit now........and this is jus hampering our MORALE.....making us feel that we dont know anything in maths.......so if u wanna better hone ur skills in real analysis ...mathlinks is the place for u......and further this wont help u in jee for sure.....THEY WILL ASK U SIMPLE THINGS AND U WILL TRY TO LOOK STARS. ;)
and that ""boring syllabus"" which u r talking abt....that only will help u crack jee [3] and not this hi fi stuff.
1he ya soumya
sometime b4 u also told something related to some lagrange euler
mechanics and all
which went above our heads
so plz dont discuss very high level topic which are not even in olypiads
