258 reputation
19
bio website ondrejcertik.com
location Los Alamos, NM
age 31
visits member for 2 years, 7 months
seen Dec 16 at 17:06

Jan
16
comment Is odd continuous function differentiable at $x=0$?
N.S., you are right! $\frac{f(x)}{x} = \sin\frac{1}{x^2}$ which oscillates between -1 and 1 and so the limit does not exist. So this function is not differentiable at $x=0$. Thanks!
Jan
16
asked Is odd continuous function differentiable at $x=0$?
Oct
25
awarded  Tumbleweed
Oct
10
asked How to prove Gegenbauer's addition theorem?
Sep
21
awarded  Scholar
Sep
21
accepted How to prove asymptotic limit of an incomplete Gamma function
Sep
21
comment How to prove asymptotic limit of an incomplete Gamma function
I verified your steps, I think it's all correct. I am accepting your answer as it gives a simple proof.
Sep
21
comment How to prove asymptotic limit of an incomplete Gamma function
Thanks! As a matter of fact, I actually wanted to prove that $\gamma(z, x)\over\Gamma(z)$ goes to zero, but thought it'd be easier to do it with the upper incomplete gamma function.
Sep
21
comment How to prove asymptotic limit of an incomplete Gamma function
So you proved that $\gamma(z, x) < {x\over z-1-x}\Gamma(z)$ for $z-1 > x > 0$ and from that the limit follows immediately. Nice!
Sep
21
asked How to prove asymptotic limit of an incomplete Gamma function
May
9
awarded  Student
May
9
comment How to prove that Legendre polynomials form a complete basis using functional analysis
Ah -- I forgot about the boundary condition -- so requiring the solutions to satisfy some conditions (for example being non-singular, or I can even require something stronger, like being $\pm 1$ at the endpoints), then the only allowed $\lambda$ are of the form $n(n+1)$, thus proving that the spectrum is discrete. If we allow Legendre functions, then the spectrum is continuous --- because I think that the Legendre functions can be normalized to (Dirac) delta function (using physics terminology). What about 2?
May
9
awarded  Supporter
May
9
comment How to prove that Legendre polynomials form a complete basis using functional analysis
@J.M., for non-integer $n$ (i.e. a general $\lambda$), the solutions are Legendre functions (en.wikipedia.org/wiki/Legendre_function). So it would seem that all $\lambda$ are actually allowed, thus the spectrum being continuous...
May
9
asked How to prove that Legendre polynomials form a complete basis using functional analysis