5
While discussing theta functions, I thought:
$\zeta(s)=\sum n^{-s}=1+2^{-s}+3^{-s}+ \cdot\cdot\cdot$
and
$\Phi(s)=\sum e^{-n^s}=e^{-1}+e^{-2^s}+e^{-3^s}+\cdot\cdot\cdot $
What is the analytic continuation of $\Phi(s)?$
User @reuns had an insightful point that maybe, $\sum_n (e^{-n^{-s}}-1)=\sum...
The number of all two electron integrals
$$
\langle \phi_1 \phi_2|\phi_3\phi_4 \rangle = \int d^3\mathbf r' \int d^3\mathbf r'' \, \phi_1(\mathbf r'') \, \phi_2(\mathbf r') \frac{1}{|\mathbf r' - \mathbf r''|} \, \phi_3(\mathbf r') \, \phi_4(\mathbf r'')
$$
for $N$ number of basis functions (I am...
Problem:
Find $\frac{dy}{dx}$ by implicit differentation for the following:
$$ y^2 = \sin^4{2x} + \cos^4{2x} $$
Answer:
\begin{align*}
2y \frac{dy}{dx} &= 4(2) \sin^3(2x) \cos(2x) - 4(2) \cos^3(2x) \sin(2x) \\
y \frac{dy}{dx} &= 4 ( \sin^3(2x) \cos(2x) - \cos^3(2x) \sin(2x) ) \\
y \frac{dy}{dx} &...
Suppose $S$ is an immersed submanifold of $M$. Let $\iota: S\hookrightarrow M$ be the inclusion map. Since it is an immersion, at each $p\in S$, $\iota_{*}:T_pS\rightarrow T_pM$ is injective. Hence we identify $T_pS$ with its image under $\iota_{*}$ . Under this identification, $\iota_{*}$ is the...
