My math book claims that the piecewise function f : [0,1] U (2,3] --> R defined by
f(x)=
x for 0<=x<=1
x-1 for 2<x<=3
is continuous. But it's undefined for 1<x<=2 so how can it be continuous? According to the definition of continuity, a function is at a point x0 if for a sequence x_n...
Hey!
I need to perform the following integration:
\int\limits_0^{\infty} J_1(k r)k dk
where J_1(x) is the cylindrical Bessel function of the first kind. This is an oscillatory function with amplitude decreasing as 1/\sqrt{x}. Due to the multiplication of k the integrand however, becomes...
Hey!
Could someone please help me find out how to express the product of two Legendre polynomials in terms of a sum of Legendre polynomials. I believe I have to use the recursion formula
(l+1)P_{l+1}(x)-(2l+1) x P_l(x) + l P_{l-1}(x)=0
but I am not sure how to do this. What is basically...
Hey!
Can someone help me solve the following integral
\int_0^{2\pi} exp[-i(G_x \cos\theta + G_y \sin\theta] d\theta
I've tried splitting the exponential into a product of two exponetials and rewriting the exponentials in terms of sines and cosines. But I always end up getting stuck...
Could you elaborate on that? Isn't there still a finite number of terms in the sum, and is r not still the same index variable indexing the (finite) number of energy levels?
Thanks for the answers to both of you. I see now that the sum must be zero because p_r is a probability distribution. And lalbatros you were right, I am studying statistical mechanics.
HallsofIvy:
Yes it seems strange that summing differentials (what I called infinitesimals before, but isn't...
Hey!
Can it be concluded generally that:
\sum_r dx_r = 0
...because we are summing an infinitesimaly small variable a finite number of times, in contrast to an integral which is an infinite sum of infinitesimaly small variables? In one of my books a probability is given by:
p_r...