#### 21. Book 7 - Graphs of General Form of Transcendental Function Alpha

#### 29 September 2017 AD; Feast of St Michael the Archangel

#### 30 September 2017 AD; Feast of St Jerome

#### Andrew Yanthar-Wasilik

#### A couple of thoughts on the parameters of the equation

**✠**** ExpM = ( A / B )**^{C}** (Eqn. M)**

^{C}

#### This is the equation for Main Exponent ExpM.

#### Parts “A”, “B”, and “C” contain some fractional exponents and terms which were a matter of choice.

#### After many tests, I have chosen denominator equal to 24, in Equation “C”:

**✠**** C**_{x}** = ( ( C**_{0}**) ( ***π***/ e )**^{x}** ) * (****(***x* − 8)/(24)**) (Eqn. C)**

_{x}_{0}

*π*

^{x}*x*− 8)/(24)

**Reasons:**

**The axis between Real and Complex parts falls on constant C**_{8}** = ***π*

_{8}

*π*

**In other cases, they would fall in between the whole numbers or some other whole constants, which would “undermine” the role of C**_{8}**= ***π*** as a first constant, the parent constant.**

_{8}

*π*

**Choosing other parameters would give the only exact value of fine structure constant alpha, ***α*_{E}**, not the other coupling constants such as weak, gravity, and strong nuclear force (checked).**

*α*

_{E}**Neutrino mixing angles would not show up as properties of the integer values for constants (probably).** I might come back to this topic later on and run some extra tests for other possibilities of the values of the denominator in Eqn. C.

#### The other problem is the choice of the numerator in the same equation, and parameters in other Equations, such as Equation 1 from Book 3:

**✠**** ****(***C*_{0}** ⁄ ((***C*_{16}** ⁄ 10)**^{(11 ⁄ 3)}**))**^{(C}_{16}^{ ⁄ 3) }**(Eqn. 1)**

*C*

_{0}

*C*

_{16}

^{(11 ⁄ 3)}

^{(C}

_{16}

^{ ⁄ 3) }

#### The exponent could be run differently, for example, using whole numbers instead of C_{x}. This was also checked, and I will not return to this issue.

_{x}

#### Nagoya University team, in my opinion, got the most precise value of fine structure constant, alpha, *α*_{E}:

_{E}

**✠**** ***α*^{( − 1)}** = 137.035 999 181 727 13 (Res. 1)**

*α*

^{( − 1)}

#### However, using the General Formulas gives a slightly different result starting on the 12th decimal position - it may be the calculator error; I have to run it on the FORTRAN later on.

#### Let us move to the main topic of this article - Graphs of coupling constants versus constant and Graphs of mixing angles for Neutrinos versus constant.

#### As can be seen from Figure 1 and Figure 3 - Graph of *α* and Graph of *θ* versus x (constant), there is a division between the Real and Complex section at constant C_{8}= *π*.

**In the Real part are three coupling constants (marked in red on graphs):**

#### alpha electromagnetic, *α*_{E}, (aka fine structure constant) at C_{16 }x-coordinates,

_{E}

#### alpha weak, *α*_{W}, weak force coupling constant at C_{17} x-coordinates,

_{W}

#### alpha gravity,*α*_{G}, gravity force coupling constant at C_{17 + 8 ⁄ 11} x-coordinates and, at C_{17 + 13 ⁄ 18} x- coordinates two candidates. (2 March 2019 - Gravity should fall at integer value 18, i.e., C_{18} gives weaker force than the present educated guess).

_{G}

**In the Complex part, there are:**

#### alpha strong, *α*_{S}, strong nuclear force coupling constant at C_{1 }x-coordinate.

_{S}

#### The approximate values of the constants are:

*α*_{E} = 7.3× 10^{ − }^{3}

_{E}

*α*_{W} = 4.4 × 10^{ − }^{7}

_{W}

*α*_{G} = 2.2 × 10^{ − }^{39} or 8.8 × 10^{ − }^{39}

_{G}

*α*_{S} = 1.07 × 10^{0}

_{S}

**Theses numbers agree quite well with official guesses:**

*α*_{E} = exact

_{E}

*α*_{W} = 10^{ − }^{6} - 10^{ − }^{7}

_{W}

*α*_{G} = 5.9 × 10^{ − }^{39}

_{G}

*α*_{S} approx = 1

_{S}

#### (See References in Narrative for Book 5.)

**As one can see (especially in Figure 3), the shape of the graph resembles the letter M, with two hills.**

#### As I wrote in the narrative for Book 7, the left hill may be an invisible universe (Heaven in religious terms), or it may be just dark matter and dark energy and other “dark” components of the Universe.

#### This blog is “the work in progress”, and it will be clear later on what the left hill is.

**✠**** Neutrino Oscillations Angles and Weinberg Angle (marked in green on graphs).**

#### (In Narrative for Book 7, in References, you can get the idea of what these angles are).

#### All Neutrino Angles (*θ*) are in the Complex part, which in polar coordinates gives the length of the Vector (Modulus) and Angle called Argument.

**Weinberg Angle** describes some properties of electro-weak force and equals to from best estimates from 2004:

*θ*_{W}**= 28.74 ****±****0.7 degrees**

*θ*

_{W}#### Now, constant C_{ − }_{7} gives the following results:

#### MODULUS (OF POLAR FORM) 4.8353102814099902; ANGLE THETA OF COMPLEX NUMBER (ARGUMENT) IN DEGREES 61.148536003448903

#### 90.0 degrees - ANGLE THETA OF COMPLEX NUMBER (ARGUMENT) IN DEGREES 61.148536003448903 = **28.85 degrees**

#### It agrees with an experimental result.

**Neutrino Mixing Angle Solar Oscillations angle is:**

*θ*_{23}** = 36.78 with 90% confidence**

*θ*

_{23}

#### Now, this angle can be found at constant C_{ − }_{4}:

#### MODULUS (OF POLAR FORM) 2.7003870767540996 ANGLE THETA OF COMPLEX NUMBER (ARGUMENT) IN DEGREES -36.078748676662492

*θ*_{23}** = -36.078748676662492 degrees**

*θ*

_{23}

**For Atmospheric Neutrinos with an angle of:**

*θ*_{23ATM}** = 45 ****±****7.1 degrees**

*θ*

_{23ATM}

#### Constant C_{5} has an angle of:

*θ*_{23ATM} = ANGLE THETA OF COMPLEX NUMBER (ARGUMENT) IN DEGREES **-45.073642716639654**

**For Neutrino ***θ*_{13}** from recent experiments :**

*θ*

_{13}

*θ*_{13}** ****approx = ****9.0 degrees**

*θ*

_{13}

#### Now, this angle can be found at constant C_{ − }_{2}:

*θ*_{13} = ANGLE THETA OF COMPLEX NUMBER (ARGUMENT) IN DEGREES **9.0863333097870669**

**Neutrino mixing angle ***θ*_{12}** equals:**

*θ*

_{12}

*θ*_{12}** = 34.22****±1****degrees**

*θ*

_{12}

#### Now, the angle of constant C_{6 }is:

*θ*_{12} = ANGLE THETA OF COMPLEX NUMBER (ARGUMENT) IN DEGREES 124.26162879014910 = 90 +** 34.26 degrees.**

#### (2 March 2019 - The angles are calculated here, not quite according to the complex number system, but they work nicely. Future articles will look into that, and they are just around the corner. Please remember that this is still a work in progress until the final formula is derived).

**How do you explain that?**

#### As to quarks, things are more complicated, and they will be presented later.

#### Andrew Yanthar-Wasilik

#### Graphs.

#### I have some difficulties posting the graphs. They will be posted after the issue is solved. - See "Book 7 - Graphs only."

#### Links with graph paper:

#### Print a Ruler

#### Dr. Philippe Marquis

#### Thanks to:

#### Hearst

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