10 January 2026

St. Nicanor (76 AD); St. John Camillus Bonus (660 AD); St. William of Bourges (1209 AD)

 

Universal Transcendental Function and Universal Transcendental Constants derived from "π" and "e" (‘Universe's Numbers’ or ‘God's Numbers’).

 

Abstract.

This paper introduces Universal Transcendental Function and Constants similar to ‘π’ and ‘e’ and derived from them.

The following article addresses the properties of the Transcendental Function, such as index and subscript mathematics.

 The applications of this can be applied in Mathematics, Quantum Mechanics, Cosmology, and in Theology or Philosophy.

These numbers complete the number system, i.e., in addition to the numbers invented by humans, we also have the Transcendental Numbers, not invented but ‘created’. With these numbers, the Universe ‘talks to us.’ I also like the term ‘God’s Numbers’ which are part of Creation.

 

Introduction.

Derivation of Universal Transcendental Function.

How to derive the equation of the Universal Transcendental Function?

The first thing to note is that ‘π’ is at position "8" on the x-axis and ‘e’ is at position "7" on the x-axis. Only then can the formula be derived for the whole family of Transcendental Constants. (There may exist other placements of the constants ‘π’ and ‘e’, but I believe I chose the most precise, simple and elegant option).

 

How is it done?

1. We use two points in the X-Y plane:

P₁ (7, e)   Eqn. 1

 

And

 

P₂ (8, π)   Eqn. 2

 

Comment: This selection gives the most straightforward relation between Transcendental Constants on the Y-axis and Integers on the X-axis.

 

2. Now we have to use the general equation of the exponential function

 

y = P₀  * a^x   Eqn. 3

 

Now we can calculate parameter ‘a’ as follows:

 

a = ( y₂ / y₁ )^{(1 / (x}₂ ^{-  x}₁⁾⁾   Eqn. 4

 

Substituting numerical values:

 

a = (π / e) ^{( 1 / ( 8 – 7))} = (π / e)   Eqn. 5

 

3. Third step is to obtain P₀  - this is done by substituting P₂ for P₀ in Eqn. 3

 

where

 

P₂ (8, π)   Eqn. 6

And we have

 

π = ( P₀ ) * ( π / e )⁸    Eqn. 7

 

Rearranging for P₀

 

P₀ = π / ( π / e )⁸     Eqn. 8

 

Calculating the numerical value of that expression

 

P₀ = 0.986976350384… (Eqn. 8a)

 

4. Obtaining the general formula

 

We have to use Eqn. 7, where instead of π we use any constant in the equation of the universal transcendental function (UTF) with x as a value of the Universal Transcendental Constant, i.e., UTF(x) and for exponent determined on the x-axis equal to x as well. In other words, x is an integer from the x-axis and UTF(x) is a value of Universal Transcendental Constant for that particular integer x.

 

Continuing for the final formula using Eqn. 7

 

UTF(x) = ( P₀ = 0.986976350384… ) * ( π / e )^x     Eqn. 9

 

Now, point P0 has a parameter x equal to 0, and equals the first Transcendental Constant which we can name as C₀ and therefore the formula becomes

 

UTF(x) = ( C₀ ) * ( π / e )^x   Eqn. 10

 

* For a detailed procedure for finding the equation of the exponential function , visit the page of Mr. William Cherry http://wcherry.math.unt.edu/math1650/exponential.pdf

 

Graph of the Universal Transcendental Function UTF(x)

 

This is very simple – just substitute for x the integers 0, 1, 2, 3, … and the negative integers -1, -2, -3, … and calculate the numerical values.

Here are three values:

            UTF (-1) = 0.853987189…

            UTF (0) = 0.986976350384…

            UTF (1) = 1.140675562…

            Etc.

 

Figure 1. Graph of Universal Transcendental Function UTF (x) /Real Numbers/

 

Finding the equation of the straight line of ln(y) versus x (if Eqn. (9 and 10 are exponential, then the graph of ln(y) vs. x will give a straight line, and it does).

 

y = a_n * x + b    Eqn. 11

 

Calculating slope a₁

a₁ = ( y₂ – y₁) / ( x₂ – x₁ ) = ( ln ( π ) – ln ( e ) ) / ( 8 – 7 ) = 0.144 729 886…   Eqn. 12

 

Calculating intercept b

If x = 0 then y₀ = = 0.986976350384…   Eqn. 13

And

ln (y₀ ) = ln ( 0.986976350384… ) = -0.013109202…    Eqn. 14

 

and the linear equation equals

 

y = ( 0.144729886… ) * x – ( 0.013109202… ) Eqn. 14

 

Important.

 

X-axis values can be any number, complex, real, and so on, it does not have to be integer.

 

Here is an example.

UTF ( 7.5) = ( C₀ ) * ( π / e )^7.5 = 2.922282868…   Eqn. 15

A Special case is a Unitary Universal Transcendental Function (UUTF) where instead of Constant C₀ = 0.986976… we have the value set to 1.000000…, i.e., y-intercept is equal to 1.000…

This variation of the UTF (UUTF) will be very important in the next chapters – Universal Units of Measurement of the Universal, Physical, Nuclear, Chemical, Astronomical, etc., Constants, showing the Infinite Mind of the Holy Trinity (and Supreme Order) behind seemingly random numbers among ‘classical’ elements (‘visible’), such as mass (matter), temperature (heat or kinetic energy), space (length) and time.

In the future chapters there will be also other versions of Universal Transcendental Function (UTF) with different values of the constants in place of the constant C₀ . These will deal with possibly invisible elements, such as Dark Matter, Dark Energy, etc., and Charge.

 

Some of the other properties of Universal Transcendental Function

 

Derivate

d / dx (UTF ( x )) = d / dx (( C₀ ) * ( π / e ))^x = ( ln ( π ) – 1 ) * ( C₀ ) * ( π / e )^x  =

≈ ( 1 / 7) * ( π / e )^x    Eqn. 16

 

Where the reciprocal of the value of the coefficient in the derivative equals

Derivative coefficient = 1/ [ ( ln ( π ) – 1 ) * ( C₀ ) ] = 7.000596302…   Eqn. 17

 

Integral

∫ (UTF ( X )) = ( C₀ ) /  ( ( ln ( π ) – 1 ) * ( π / e )^x ) + C      Eqn. 18

 

Examples of complex exponential function:

 

https://commons.wikimedia.org/wiki/File:Complex_exponential_function_graph_spiral_shape_yvw_dimensions.jpg

 

https://www.geogebra.org/m/aEUGrASs

 

Image: Complex exponential function graph, spiral shape, yvw dimensions. Courtesy of Wikimedia Commons, link here  

>>>   https://commons.wikimedia.org/wiki/File:Complex_exponential_function_graph_spiral_shape_yvw_dimensions.jpg#:~:text=Summary.%20English:%20Graph%20of%20the%20complex%20exponential,green%20are%20for%20positive%20values%20of%20x.

 

We use only the Real Part of the Complex exponential Function for now.