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    Raising to half power = PRINCIPAL square root?

    This may seem like a very elementary question...but here goes anyway. When a positive number is raised to the power 1/2, I have always assumed that this is defined as the PRINCIPAL (positive) square root, e.g. 7^{1/2} = \sqrt{7},. That is, it does not include both the positive and negative...
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    Differentiation of inverse trig

    Say we want to differentiate \arcsin x. To do this we put y=\arcsin x. Then x=\sin y \implies \frac{dx}{dy}= \cos y. Then we use the relation \sin^2 y + \cos^2 y = 1 \implies \cos y = \sqrt{1 - \sin^2 y} = \sqrt{1 - x^2}. Therefore \frac{dy}{dx} = \frac{1}{\sqrt{1 - x^2}}. My question is that...
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    Absolute value in separable ODEs?

    Suppose I have a variable separable ODE, e.g., \frac{dy}{dx} = 3y. We all know that the solution is y=Ae^{3x} where A is a constant. My question is as follows. To actually find this solution we rearrange the equation and integrate to get \int \frac{dy}{y} = 3 \int dx, which gives \ln...
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    Modified Bessel function with imaginary index is purely real?

    I'm trying to decide if the modified Bessel function K_{i \beta}(x) is purely real when \beta and x are purely real. I think that is ought to be. My reasoning is the following: \left (K_{i \beta}(x)\right)^* = K_{-i \beta}(x) = \frac{\pi}{2} \frac{I_{i \beta}(x) - I_{-i \beta}(x)}{\sin(-i...
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    2nd order PDE using integration by parts

    Homework Statement Find the general solution of the equation (\zeta - \eta)^2 \frac{\partial^2 u(\zeta,\eta)}{\partial\zeta \, \partial\eta}=0, where ##\zeta## and ##\eta## are independent variables. Homework Equations The Attempt at a Solution I set ##X = \partial u/\partial\eta## so that...
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    Difficult integral

    Mod note: Moved from the math technical sections. I need to show that \int_{-\infty}^\infty \frac{\sin^2 (pa/\hbar)}{p^2} \, dp = \frac{\pi a}{\hbar}. I haven't got a clue how to integrate this function! Any help would be much appreciated thanks.
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    Finding momentum distribution for particle in square well

    Homework Statement A particle of mass ##m## is trapped between two walls in an infinite square well with potential energy V(x) = \left\{ \begin{array}{cc} +\infty & (x < -a), \\ 0 & (-a \leq x \leq a), \\ +\infty & (x > a).\end{array} \right. Suppose the wavefuntion of the particle at time...
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    Parameterization for method of characteristics

    Consider the PDE xu_x + yu_y = 4u. Suppose that we want to find the solution that satisfies u=1 on the circle x^2 + y^2 = 1 using the method of characteristics. I have read that the boundary condition can be parameterized as x=\sigma, \qquad y=(1-\sigma^2)^{1/2}, \qquad u=1. My...
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    Show that inner product is zero.

    Let A be a Hermitian operator with n eigenkets: A|u_i\rangle = a_i |u_i\rangle for i=1,2,...,n. Suppose B is an operator that commutes with A. How could I show that \langle u_i | B | u_j \rangle = 0 \qquad (a_i \neq a_j)? I have tried the following but not sure how to proceed: AB -...
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    Show system of PDEs has no solution

    Homework Statement Show that there is no solution for the system u_x - 2.999999x^2 y + y = 0, u_y - x^3 + x = 0. Homework Equations The Attempt at a Solution I took the first equation and integrated w.r.t x, then differentiated w.r.t y. But I'm not sure if it helps: u_y -...
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    PDE change of variables Black-Scholes equation

    Homework Statement By changing variables from (S,t,V) to (x,\tau,u) where \tau = T - t, x = \ln\left(\frac{S}{K}\right) + \left(r - \frac{\sigma^2}{2}\right)(T-t), u=e^{r\tau}V, where r, \sigma, \tau, K are constants, show that the Black-Scholes equation \frac{\partial V}{\partial t} +...
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    Reducing angular Schrodinger equation to eigenvalue problem

    Homework Statement The angular part of the Schrodinger equation for a positron in the field of an electric dipole moment {\bf d}=d{\bf \hat{k}} is, in spherical polar coordinates (r,\vartheta,\varphi), \frac{1}{\sin\vartheta}\frac{\partial}{\partial\vartheta} \left( \sin\vartheta\frac{\partial...
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    Ambiguity with partial derivative notation

    Suppose I have some function f that depends on three variables, namely x, y, and t; i.e., f=f(x,y,t). Now suppose that y depends on x, i.e., y=y(x). Taking this into account, we see that f is really just a function of two independent variables, x and t. So my question is this: if I write down...