Akanksh

Proof of Determinant to Dot and Cross Product

$M={\begin{pmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\end{pmatrix}}={\begin{pmatrix}{\overrightarrow {A}}\\{\overrightarrow {B}}\\{\overrightarrow {C}}\end{pmatrix}}$
$\parallel M\parallel ={\begin{vmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\end{vmatrix}}$
$=a_{1}{\begin{vmatrix}b_{2}&b_{3}\\c_{2}&c_{3}\end{vmatrix}}-a_{2}{\begin{vmatrix}b_{1}&b_{3}\\c_{1}&c_{3}\end{vmatrix}}+a_{3}{\begin{vmatrix}b_{1}&b_{2}\\c_{1}&c_{2}\end{vmatrix}}$

$=a_{1}\times (b_{2}c_{3}-b_{3}c_{2})-a_{2}\times (b_{1}c_{3}-b_{3}c_{1})+a_{3}\times (b_{1}c_{2}-b_{2}c_{1})$

$=a_{1}\times (b_{2}c_{3}-b_{3}c_{2})+a_{2}\times (b_{3}c_{1}-b_{1}c_{3})+a_{3}\times (b_{1}c_{2}-b_{2}c_{1})$

$={\overrightarrow {A}}\cdot \langle b_{2}c_{3}-b_{3}c_{2},b_{3}c_{1}-b_{1}c_{3},b_{1}c_{2}-b_{2}c_{1}\rangle$


$\langle b_{2}c_{3}-b_{3}c_{2},b_{3}c_{1}-b_{1}c_{3},b_{1}c_{2}-b_{2}c_{1}\rangle =\langle M_{1}\rangle$


$\Rightarrow \parallel M\parallel ={\overrightarrow {A}}\cdot \langle M_{1}\rangle$


${\overrightarrow {B}}\times {\overrightarrow {C}}={\begin{vmatrix}{\widehat {i}}&{\widehat {j}}&{\widehat {k}}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\end{vmatrix}}=(b_{2}c_{3}-b_{3}c_{2})\times {\widehat {i}}+(b_{3}c_{1}-b_{1}c_{3})\times {\widehat {j}}+(b_{1}c_{2}-b_{2}c_{1})\times {\widehat {k}}=\langle b_{2}c_{3}-b_{3}c_{2},b_{3}c_{1}-b_{1}c_{3},b_{1}c_{2}-b_{2}c_{1}\rangle =\langle M_{2}\rangle$


$\langle M_{1}\rangle =\langle M_{2}\rangle \Rightarrow \parallel M\parallel ={\overrightarrow {A}}\cdot \langle M_{2}\rangle$


$\Rightarrow \parallel M\parallel ={\overrightarrow {A}}\cdot ({{\overrightarrow {B}}\times {\overrightarrow {C}}})$

Special Functions

Complete Fermi Dirac Integrals

$F_{j}(x):=(1/\Gamma (j+1))\int _{0}^{\infty }dt(t^{j}/(\exp(t-x)+1))$

Incomplete Fermi Dirac Integrals

$F_{j}(x,b):=(1/\Gamma (j+1))\int _{b}^{\infty }dt(t^{j}/(\exp(t-x)+1))$

Airy Functions and Derivatives

$Ai(x)=(1/\pi )\int _{0}^{\infty }\cos((1/3)t^{3}+xt)dt$
$Bi(x)=(1/\pi )\int _{0}^{\infty }(e^{(}-(1/3)t^{3})+\sin((1/3)t^{3}+xt))dt$

Clausen Functions

$Cl_{2}(x)=-\int _{0}^{x}dt\log(2\sin(t/2))$

Normalized Hydrogenic Bound States

$R_{n}:=2(Z^{3/2}/n^{2}){\sqrt {(n-l-1)!/(n+l)!}}\exp(-Zr/n)(2Zr/n)^{l}L_{n-l-1}^{2l+1}(2Zr/n)$
Where:$L_{n}^{k}(x)=(-1)^{k}(d^{k}/dx^{k})L_{(}n+k)(x)$

Legendre Forms

$F(\phi ,k)=\int _{0}^{\phi }dt1/{\sqrt {(}}(1-k^{2}\sin ^{2}(t)))$

$E(\phi ,k)=\int _{0}^{\phi }dt{\sqrt {(}}(1-k^{2}\sin ^{2}(t)))$

$\Pi (\phi ,k,n)=\int _{0}^{\phi }dt1/((1+n\sin ^{2}(t)){\sqrt {(}}1-k^{2}\sin ^{2}(t))$

Carlson Forms

$RC(x,y)=1/2\int _{0}^{\infty }dt(t+x)^{(}-1/2)(t+y)^{(}-1)$

$RD(x,y,z)=3/2\int _{0}^{\infty }dt(t+x)^{(}-1/2)(t+y)^{(}-1/2)(t+z)^{(}-3/2)$

$RF(x,y,z)=1/2\int _{0}^{\infty }dt(t+x)^{(}-1/2)(t+y)^{(}-1/2)(t+z)^{(}-1/2)$

$RJ(x,y,z,p)=3/2\int _{0}^{\infty }dt(t+x)^{(}-1/2)(t+y)^{(}-1/2)(t+z)^{(}-1/2)(t+p)^{(}-1)$

Gamma Functions

$\Gamma (x)=\int _{0}^{\infty }dtt^{x-1}\exp(-t)$

Psi (Digamma) Function

$\psi ^{(n)}(x)=(d/dx)^{n}\psi (x)=(d/dx)^{n+1}\log(\Gamma (x))\cdot$

Riemann Zeta Function

$\zeta (s)=\sum _{k=1}^{\infty }k^{-s}$

Hurwitz Zeta Function

$\zeta (s,q)=\sum _{0}^{\infty }(k+q)^{-s}$

Eta Function

$\eta (s)=(1-2^{1-s})\zeta (s)\cdot$

Transport Functions

$J(n,x):=\int _{0}^{x}dtt^{n}e^{t}/(e^{t}-1)^{2}$

-- $\alpha \kappa \alpha \eta \kappa \varsigma h\ \nu \alpha \varsigma h\iota \varsigma \tau h\,\!$