23 September 2017 AD, the feast of St Pio and St Linus

Andrew Yanthar-Wasilik

Finding General Formula for Equations from the previous book, “Book 5 Integer Formula for Dimensionless Coupling Constants of Fundamental Forces”.

This time general term of constant similar to the fine structure constant, alpha, *α*_{E}, let’s name it just alpha, *α*, will be derived.

It is quite possible, that there are also other constants, we do not know yet, so the general term alpha, *α*, seems appropriate.

**In Part I of the Book 6 the Exponent Main, ExpM was derived: **

**ExpM = (A/B) ^{C} **

**✠Now, continuing further derivation from Book 5, we get part D** (Equation D is the exponent needed to calculate the value of the function at (x=D), represented by partial sequence:

D_{16} = 16 + ExpM_{16}

D_{17} = 17 + ExpM_{17}

D_{1} + 1 + ExpM_{1}

General formula for exponent D_{x}is:

D_{x} = [ x + ExpM_{x} ]

or just:

**✠D _{x} = [ x + ExpM ] (Eqn. D)**

**Having exponent D we may get the value of transcendent function at point x: **

**✠FT( x = D ) = ( C _{0}) * ( π/ e )^{D } (Eqn. FT)**

**✠Next step is getting the Partial Exponent ExpP, we have three terms of the the sequence:**

ExpP_{16} = ( 16 + (24 / 24) ) / ExpM_{16}

ExpP_{17} = ( 17 + (27 / 24) ) / ExpM_{17}

ExpP_{1} = ( 1 + (-21 / 24) ) / ExpM_{1}

...,1,...,16, 17, ... are just x

...,(-21 / 24),...,(24 / 24),(27 / 24),... are equal to exponent

y = ((3)/(24))*x + (-(24)/(24)) = (3**x* − 24)/(24) = (*x* − 8)/(8)

of (Eqn. A)

**A _{x} = ( C_{0})^{((}^{x}^{ − 8)/(8)) }(Eqn. A)**

Adding these two terms

x + y = x + (*x* − 8)/(8) = (8*x* + *x* − 8)/(8) = (9*x* − 8)/(8)

**This sum has to be divided by Exponent Main, ExpM, to get value of Exponent Partial, ExpP: **

**✠ExpP _{x} = ((9x − 8))/(8ExpM) (Eqn. EP)**

**✠Last sequence will be the denominator of **

**( α_{E} )^{( − 1 ⁄ 2)} = ( FT(x) / ( 8 + 2 * (24/24))) ^{ExpP} (Eqn. α_{E}) **

which is

For constant C _{16}: ( 8 + 2 * (24/24))

For constant C _{17}: ( 9 + 2 * (24/27))

For constant C _{1}: ( -7 + 2 * (-24/21))

First part of this sum is gives partial sequence:

...,-7,...,8,9,...

and this is equal to ( x - 8 )

Second part of the sum gives partial sequence:

...,2 * (-24/21),...,2 * (24/24),2 * (24/27),...

We have already calculated this. It is equal to

E_{part}*B* = 2 * ((8)/(*x* − 8)) = ((16)/(*x* − 8))

**Adding both terms gives: **

**( x - 8 ) + ((16)/( x − 8)) = (x^{2} − 16x + 80)/(x − 8) **

Taking reciprocal of this term we can avoid quotient and use product instead in (Eqn *α*_{E})

**✠Finally, the General Formula for for ( α )^{( − (1)/(2))} is: **

**✠( α_{x} )^{( − (1)/(2))} = {[(C_{0})(π ⁄ e)^{(}^{x}^{ + }^{ExpM}^{)}]*[(x − 8)/(x^{2} − 16x + 80)]}^{[(9}^{x}^{ − 8)/(8}^{ExpM}^{)]} (Eqn. α)**

To get (*α*)^{ − }^{1}simply square the previous equation

To get ( *α*) simply take reciprocal of the previous equation (for those not much in the math).

Getting the shorter formula for ExpM and *α*is probably possible but very difficult. I am not sure if I can do that, maybe some professional mathematicians can derive it.

Now, this General Equation for any x allows calculating any value of alpha.

**Next book, Book 7 will show coupling constant of gravity force (two candidates) and, switching to Quantum Mechanics Neutrino Mixing Angles and Weinberg Angle for weak interactions. **

**Quarks will follow later. **

Andrew Yanthar-Wasilik

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