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  1. Repetit

    Continuity of piecewise function undefined for 1<x<=2

    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...
  2. Repetit

    Conditional convergence of J1(kr)k

    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...
  3. Repetit

    Product of Legendre Polynomials

    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...
  4. Repetit

    Integral of exponential involving sines and cosines

    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...
  5. Repetit

    Sum of an infinitesimal variable

    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?
  6. Repetit

    Sum of an infinitesimal variable

    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...
  7. Repetit

    Sum of an infinitesimal variable

    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...