User talk:Nimo8

Alpha
Alpha-constant

10:19, 7 January 2009 (UTC)

test 12:19, 5 January 2009 (UTC)


 * $$\alpha\ =\ \frac{e^2}{\hbar c \ 4 \pi \epsilon_0}\ =\ \frac{e^2}{\ 2 h \ c \epsilon_0}\

=\ \frac{e^2 c \mu_0}{2 h}\ =\frac{1}{\sim137} $$.

\-- (1)- in "natural" units : $$\ \alpha = \frac{e^2}{\hbar c} =\frac{e^2 2 \pi}{h c} $$


 * $$e = \sqrt{\frac{h c \alpha}{2 \pi}} = \sqrt{\hbar c \alpha} $$

\--- (2)- to ease the logik~flow pointing to the two field-epsilons \varepsilon

(μ0;ε0) ,their relation to C & The Identity
 * $${c_0}^2 \, \epsilon_0 \, \mu_0 = 1.$$

treating $$\left(\frac{e^2}{2h}\right)$$ as-an "Logic-Entity" ,one can see that

$$\alpha\ =\left(\frac{e^2}{2h}\right) ((c)\mu_0) =\left(\frac{e^2}{2h}\right)\left(\frac{1}{(c)\epsilon_0}\right)$$

↓↓

$$\left(\frac{e^2}{2h \alpha}\right) =\left(\frac{1}{c \mu_0}\right) =\left(c \epsilon_0\right) $$

$$\left(\frac{2h \alpha}{e^2}\right) =(c \mu_0) =\left(\frac{1}{c \epsilon_0}\right) $$

&↓↓

$$\left(\frac{2h \alpha}{(c) e^2}\right) =(\mu_0) =\left(\frac{1}{c^2 \epsilon_0}\right) $$

$$ \left(\frac{e^2}{(c) 2h \alpha}\right) =(\epsilon_0) =\left(\frac{1}{c^2 \mu_0}\right) $$

\---

$$(2h \alpha) =(e^2 c \mu_0) =\left(\frac{e^2}{c \epsilon_0}\right) $$

↓↓
 * $$e = \sqrt{\frac{2 h \alpha}{c_0 \mu_0}} = \sqrt{2 h \alpha c_0 \epsilon_0} $$

.. ~ \--- Bohr_radius $$a_0 = \frac{4 \pi \epsilon_0 \hbar^2}{m_e e^2} = \frac{\hbar}{m_e\,c\,\alpha} = \frac{ h^2\,\epsilon_0}{\pi \,m_e\, e^2} = \frac{h}{2 \pi \, m_e\,c\,\alpha} $$

where:
 * $$ \epsilon_0 \ $$ is the permittivity of free space
 * $$ \hbar \ $$ is the reduced Planck's constant
 * $$ m_e \ $$ is the electron rest mass
 * $$ e \ $$ is the elementary charge
 * $$ c \ $$ is the speed of light in vacuum
 * $$ \alpha \ $$ is the fine structure constant

test
tutu אחת שלוש

08:48, 8 November 2009 (UTC)

Beta-&-Gamma-Functions
& zeta Gamma-Function $$\ \Gamma(n)=(n-1)!= \frac{(n)!}{(n)}$$ & $$ \Gamma(nx)=(nx-1)!= \frac{(nx)!}{(nx)} $$ & $$ \Gamma(n+x)=(n+x-1)!= \frac{(n+x)!}{(n+x)} $$ Beta-Function by Gammas $$ \Beta(x,n)=\frac{\Gamma(x)\,\Gamma(n)}{\Gamma(x+n)} =\frac{(x)!\,(n)!}{(x+n)!}\left(\frac{(x+n)}{(xn)}\right) =\frac{(x)!\,(n)!}{(x+n)!}\left(\frac{(1)}{(n)}+\frac{(1)}{(x)}\right) $$ ~ for some reasn WiKI render the background of "formula"  as "transparent" and not white as it use to be --12:20, 14 April 2010 (UTC) .. solved with http://userstyles.org/styles/23389 ~ ~ ~ ~

Other important functional equations for the Gamma function are Euler's reflection formula



\Gamma(1-z) \; \Gamma(z) = {\pi \over \sin{(\pi z)}} \,\! $$

and the duplication formula



\Gamma(z) \; \Gamma\left(z + \frac{1}{2}\right) = 2^{\frac{1}{2}-2z} \; \sqrt{2\pi} \; \Gamma(2z). \,\! $$

The duplication formula is a special case of the multiplication theorem



\Gamma(z) \; \Gamma\left(z + \frac{1}{m}\right) \; \Gamma\left(z + \frac{2}{m}\right) \cdots \Gamma\left(z + \frac{m-1}{m}\right) = (2 \pi)^{(m-1)/2} \; m^{1/2 - mz} \; \Gamma(mz). \,\! $$

t77
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