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 Determinants II: Determinants of Order $n$  MathPhys Archive on Determinants I: Determinants of Order 2
 Inverses  MathPhys Archive on The Matrix Associated with a Linear Map
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Category Archives: Mathematical Physics
SelfAdjoint Differential Equations III: Real Eigenvalues, GramSchmidt Orthogonalization
In here, I mentioned that the eigenvalues of a Hermitian operator are real and that the eigenfunctions of a Hermitian operator are orthogonal. Let $\mathcal{L}$ be a Hermitian operator and let $u_i$, $u_j$ be eigenfunctions of $\mathcal{L}$ with eigenvalues $\lambda_i$, … Continue reading
SelfAdjoint Differential Equations II: Hermitian Operators
Let $\mathcal{L}$ be a secondorder selfadjoint differential operator. Then $\mathcal{L}u(x)$ may be written as $$\mathcal{L}u(x)=\frac{d}{dx}\left[p(x)\frac{du(x)}{dx}\right]+q(x)u(x)\ \ \ \ \ (1)$$ as we discussed here. Multiply (1) by $v^\ast$ ($v^\ast$ is the complex conjugate of $v$) and integrate \begin{align*} \int_a^bv^\ast\mathcal{L}udx&=\int_a^bv^\ast\frac{d}{dx}\left[p(x)\frac{du(x)}{dx}\right]dx+\int_a^bv^\ast qudx\\ … Continue reading
Posted in Differential Equations, Mathematical Physics
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SelfAdjoint Differential Equations I
Let $\mathcal{L}$ be the secondorder linear differential operator $$\mathcal{L}=p_0(x)\frac{d^2}{dx^2}+p_1(x)\frac{d}{dx}+p_2(x)$$ which acts on a function $u(x)$ as $$\mathcal{L}u(x)=p_0(x)\frac{d^2u(x)}{dx^2}+p_1(x)\frac{du(x)}{dx}+p_2(x)u(x).\ \ \ \ \ (1)$$ Define an adjoint operator $\bar{\mathcal{L}}$ by \begin{align*} \bar{\mathcal{L}}&:=\frac{d^2}{dx^2}[p_0u]\frac{d}{dx}[p_1u]+p_2u\\ &=p_0\frac{d^2u}{dx^2}+(2p_0^\primep_1)\frac{du}{dx}+(p_0^{\prime\prime}p_1+p_2)u. \end{align*} If $\mathcal{L}=\bar{\mathcal{L}}$, $\mathcal{L}$ is said to be selfadjoint. … Continue reading
Posted in Differential Equations, Mathematical Physics
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Legendre Functions III: Special Values, Parity, Orthogonality
Special Values From the generating function $$g(x,t)=\frac{1}{(12xt+t^2)^{1/2}},$$ when $x=1$ we obtain \begin{align*} g(1,t)&=\frac{1}{(12t+t^2)^{1/2}}\\ &=\frac{1}{1t}\\ &=\sum_{n=0}^\infty t^n, \end{align*} since $t<1$. On the other hand, $$g(1,t)=\sum_{n=0}^\infty P_n(1)t^n.$$ So by comparison we get $$P_n(1)=1.$$ Similarly, if we let $x=1$, $$P_n(1)=(1)^n.$$ For $x=0$, the … Continue reading
Legendre Functions II: Recurrence Relations and Special Properties
In this lecture, we derive some important recurrence relations of Legendre functions and use them to show that Legendre functions are indeed solutions of a differential equation, called Legendre’s differential equation. Differentiating the generating function $$g(x,t)=(12xt+t^2)^{1/2}=\sum_{n=0}^\infty P_n(x)t^n,\ t<1\ \ \ … Continue reading
Tensors I
Tensors may be considered as a generalization of vectors and covectors. They are extremely important quantities for studying differential geometry and physics. Let $M^n$ be an $n$dimensional differentiable manifold. For each $x\in M^n$, let $E_x=T_xM^n$, i.e. the tangent space to … Continue reading
Helmholtz Equation
Helmholtz equation $$\nabla^2\psi+k^2\psi=0\ \ \ \ \ \mbox{(1)}$$ is extremely important in physics. Solving many physically important partial differential equations such as heat equation, wave equation (KleinGordon equation), Maxwell’s equations, and Schrödinger equation, etc. often require solving Helmholtz equation (1). … Continue reading
Legendre Functions I: A Physical Origin of Legendre Functions
Consider an electric charge $q$ placed on the $z$axis at $z=a$. The electrostatic potential of charge $q$ is $$\varphi=\frac{1}{4\pi\epsilon_0}\frac{q}{r_1}.\ \ \ \ \ \mbox{(1)}$$ Using the Laws of Cosine, one can write $r_1$ in terms of $r$ and $\theta$: $$r_1=\sqrt{r^2+a^22ar\cos\theta}$$ … Continue reading
Posted in Engineering Mathematics, Mathematical Physics
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Spherical Bessel Functions
When the Helmholtz equation is separated in spherical coordinates the radial equation has the form $$r^2\frac{d^2R}{dr^2}+2r\frac{dR}{dr}+[k^2r^2n(n+1)]R=0.\ \ \ \ \ \mbox{(1)}$$ The equation (1) looks similar to Bessel’s equation. If we use the transformation $R(kr)=\frac{Z(kr)}{(kr)^{1/2}}$, (1) turns into Bessel’s equation … Continue reading
Neumann Functions, Bessel Function of the Second Kind $N_\nu(X)$
In here, we have considered Bessel functions $J_\nu(x)$ for $\nu=\mbox{integer}$ case only. Note that $J_\nu$ and $J_{\nu}$ are linearly independent if $\nu$ is a noninteger. If $\nu$ is an integer, $J_\nu$ and $J_{\nu}$ satisfy the relation $J_{\nu}=(1)^\nu J_\nu$, i.e. they … Continue reading