9. Book 2-b - Some of the properties of Transcendental Function

 

Andrew Yanthar-Wasilik

26 December 2016 AD, Feast of St Stephen

 

I derived general formulas for multiplication, division, powers, and logarithms for the Transcendental Constants.

Addition and subtraction are more challenging to derive; it was done only partially.

Transcendental Constants have their unique way of calculations, i.e., they use what I call,

Index Mathematics.

It means that indexes (subscripts) of the given constants are used to calculate new values of multiplication, division, powers, and logarithms, possibly integrals and derivatives.

I will start with simple examples so it is easier to understand and then derive the general formulas.

1. For example, the multiplication of two constants can be described as follows:

C_m × C_n= (C_{(m + n)⁄2})² (Eqn. 1)

So, in a concrete example, let’s say

C_m = C₈= π = 3.141 592 654...

And

C_n= C₇= e = 2.718 281 828...

Then

C₈ × C₇ = π  ×  e = 3.141 592 654...  ×  2.718 281 828... = 8.539 734 223...

Now,

(C_{(m + n)⁄2})² = (C_{(8 + 7⁄2)})² = (C_(15⁄2))² = (C_7.5)²

Using formula Eqn. 11 from "Book 1 - Transcendental Constants - Introduction."

we can calculate any value of constant with real index, as follows:

FT(x) = (C₀) × (π/e)^x (Eqn. 2)

FT(7.5) = (0.986 976 350...)  ×  (1.155 727 350...)^7.5 = 2.922 282 364...

Squaring that we get

(2.922 282 364...)²= 8.539 734 216...

Relative error is

ε = -0.000 000 001

i.e., minimal error (if at all) - calculations are made on a hand-held calculator.

2. Adding powers to the formula for multiplication of two constants (Eqn. 1) gives:

(C_m)^p  ×  (C_n)^q = [C_{(p × m + q × n)⁄p + q)}]^(p + q) (Eqn. 3)

Let us use the previous example with some added powers:

(C₈)^1⁄4  ×  (C₇)³ = (C_{((0.25 × 8 + 3 × 7)⁄0.25 + 3)})^0.25 + 3

Left Hand Side equals to

= 26.740 585 61...

And Right Hand Side (again, using Eqn. 2) is

= (C_23⁄3.25)^3.25 = (2.748 713 730)^3.25 = 26.740 585 57...

Relative error

ε = 0.000 000 001

3. The general formula for multiplication for any powers and number of factors.

(C_m)^p × (C_n)^q × (C_o)^r × ... × (C_x)^z= (Eqn.4)

= [C_{(p × m + q × n + r × o + ... + z × x)⁄(p + q + r + ... + z)}]^{(p + q + r + ... + z)}= (Eqn. 4a)

= (C_{(p × m + q × n + r × o + ... + z × x)}) × (C₀)^{(p + q + r + ... + z − 1)}(Eqn.4b)

In (Eqn. 4a) p + q + r +...+ z  ≠ 0; so the (Eqn. 4b) is much more robust.

Example of the last formula for three factors and three powers with one index equal to "0"; (exception: to make it work, 2X0 must equal 2).

(C₈)^1⁄4  ×  (C₇)^− 3 × (C₀ = 0.986976350...)²= 0.064 568 027...

The second part of the general formula gives

(C_{((0.25 × 8 − 3 × 7 + 2 × 0)⁄(0.25 − 3 + 2)})^{(0.25 − 3 + 2)}= (C_25.3333)^− 0.75=

Using Eqn.2 to calculate C_25.3333we get

= (38.604 978 32...)^− 0.75= 0.064 568 027...

The third part of the general formula gives

(C_{(0.25 × 8 − 3 × 7 + 2 × 0)})  × (C₀)^{(0.25 − 3 + 2 − 1)}=

= (C_− 19) × (C₀)^− 1.75=

= (0.063 103 627...)  × (1.023 206 273) = 0.064 568 027...

So, all three results are the same.

4. The general formula for the logarithm of the products and power.

There is not much to it. But, taking logarithms of Equation 4, we get:

ln[(C_m)^p × (C_n)^q × (C_o)^r × … × (C_x)^z] = (Eqn. 5)

= p × ln(C_m)+ q × ln(C_n)+ r × ln(C_o)+… + z × ln(C_x) = (Eqn. 5a)

= (p + q + r + … + z) × ln(C_{(p × m + q × n + r × o + ... + z × x⁄p + q + r + ... + z)}) = (Eqn. 5b)

= ln(C_{(p × m + q × n + r × o + ... + z × x)})+ [(p + q + r + ... + z) − 1] × ln(C₀)(Eqn.5c)

Again, in the (Eqn. 5b) p + q + r +...+ z  ≠ 0

Multiplication of power and index of the constant with index equal to “0”; (exception: to make it work (“power” or “index”) X 0 must equal to power or index not equal to 0).

5. Division of two constants.

C_M⁄C_n= (C_(-M+n))^− 1⁄(C₀)^− 1(Eqn. 6)

eg., C₈⁄C₇= π⁄e = 1.155 727 350...

Now, using (Eqn. 6)

C₈⁄C₇= π⁄e = (C_{( − 8 + 7)})^− 1⁄(C₀)^− 1=

= (C_− 1)^− 1⁄(C₀)^− 1= (0.853 987 189...)^− 1÷ (0.986 976 350...)^− 1 =

= 1.155 727 350...

The values of constants are from the Blog section “Table of Transcendental Constants...”

Same results.

6. Division of two constants with powers.

(C_M)^P⁄(C_n)^q = (Eq. 7)

= (C_{(( − P × M + q × n)⁄ − P + q)})^{(P − q)} = (Eqn. 7a)

= (C_{( − P × M + q × n)})^− 1⁄(C₀)^{( − 1 − (P − q))}(Eqn. 7b)

e.g., from (Eqn. 7): (C₈)^2.5⁄(C₇)^− 0.5= (π)^2.5⁄(e)^− 0.5= 28.841 770 89...

From (Eqn. 7a): (C_{(( − 2.5 × 8 − 0.5 × 7)⁄ − 2.5 − 0.5)})^{(2.5 − ( − 0.5))}=

= (C_{( − 20 − 3.5⁄ − 3)})³ = (C_7.8333)³

Using (Eqn. 2) to calculate this result:

From Transcendental Function general formula:

TF(7.8333) = (C₀) × (π⁄e)^7.8333 = 3.066 718 931...

And from equation 7a:

(C_7.8333)³ = (3.066 718 931...)³ = 28.841 770 86...

From (Eqn. 7b):

(C_{( − 2.5 × 8 − 0.5 × 7)})^− 1⁄(C₀)^{( − 1 − (2.5 + 0.5))}=

= (C_− 23.5)^− 1⁄(C₀)^− 4=

Using (Eqn. 2) to calculate this result:

TF(-23.5) = (0.986976350...) × (π⁄e)^− 23.5 = 0.032 900 694...

Now:

(C_− 23.5)^− 1⁄(C₀)^− 4= (0.032 900 694...)^− 1÷ (0.986976350...)^− 4=

(30.394 495 37...) ÷ (1.053 835 963...) = 28.841 770 86...

i.e., the same result.

7. General formula for division with any number of factors and any powers.

((C_M)^P ×  (C_N)^Q ×  (C_O)^R ×  ...  ×  (C_X)^Z) ÷ ((C_m)^p × (C_n)^q × (C_o)^r × ...  × (C_x)^z) = (Eqn. 8)

= (C_{(( − P × M − Q × N − R × O − ... − Z × X) + (p × m + q × n + r × o + ... + z × x)⁄( − P − Q − R − ... − Z) + (p + q + r + ... + z))})^{[(P + Q + R + ... + Z) − (p + q + r + ... + z)]}= (Eqn.8a) = (C_{(( − P × M − Q × N − R × O − ... − Z × X) + (p × m + q × n + r × o + ... + z × x))})^− 1⁄(C₀)^{[ − 1 − ((P + Q + R + ... + Z) − (p + q + r + ... + z))]}(Eqn. 8b)

(Eqn. 8a) has a limitation, as before, with powers or indexes equal to "0."

( − P − Q − R − ... − Z)+ (p + q + r + ... + z) ≠ 0

8. Logarithms.

We get similar equations to (Eqn. 5a, 5b, and 5c), taking the logarithm on both sides.

It is too tedious to write it here.

Comments:

In all these Index Math formulas, Constant C₀ = 0.986 976 350... seems to be of utmost importance, as if all the other constants can be calculated with this particular Constant C₀ plus C₈ = π and C₇ = e.

 

* In light of further research, there are more constituents of the Universe once the complex numbers are used.

 

Here are the links related to these articles

 

1. Universal Transcendental Function and Universal Transcendental Constants derived from "π" and "e"   >>>   https://luxdeluce.com/15-new-version-book-1.html

 

 

2. Table of Transcendental Constants going Down   >>>   https://luxdeluce.com/37-book-4a-table-of-transcendental-constants-going-down.html

 

 

3. Updated Table of Transcendental Constants going Up   >>>   https://luxdeluce.com/38-book-4b-updated-table-of-transcendental-constants-going-up.html

 

 

4. Index Mathematics – a Property of Transcendental Constants   >>>   https://luxdeluce.com/27-book-2-b-some-of-the-properties-of-transcendental-function.html