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...
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...
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...
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...
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...
I've found through the transformation u=pa/\hbar that it is equivalent to showing
\int_{-\infty}^{\infty} \frac{\sin^2 u}{u^2}\,du = \pi, if that helps anyone.
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.
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...
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...
Thanks for that - I think we both posted at the same time! My question is related to quantum mechanics. I think physicists use the words Hermitian and self-adjoint interchangeably. I know that pure mathematicians will distinguish them but it's not important for my area. Thanks!
Actually I think I got it:
a_i \langle u_i |B|u_j \rangle - \langle u_i |B| u_j \rangle a_j = 0 using the Hermiticity of A for the first term, and then since a_i \neq a_j we get
\langle u_i |B| u_j \rangle = 0
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 -...
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 -...
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} +...
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...
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...