#### 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)

_{m}

_{n}

#### So, in a concrete example, let’s say

*C*_{m} = *C*_{8}= π = 3.141 592 654...

_{m}

#### And

*C*_{n}= *C*_{7}= e = 2.718 281 828...

_{n}

#### Then

*C*_{8} × *C*_{7} = 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) = (*C*_{0}) × (π/e)^{x} (Eqn. 2)

^{x}

#### 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)

_{m}

^{p}

_{n}

^{q}

#### Let us use the previous example with some added powers:

#### (*C*_{8})^{1⁄4} × (*C*_{7})^{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)

_{m}

^{p}

_{n}

^{q}

_{o}

^{r}

_{x}

^{z}

#### = [*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)}) × (*C*_{0})^{(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).

#### (*C*_{8})^{1⁄4} × (*C*_{7})^{ − 3} × (*C*_{0} = 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*_{25.3333}we 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)}) × (*C*_{0})^{(0.25 − 3 + 2 − 1)}=

#### = (*C*_{ − 19}) × (*C*_{0})^{ − 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)

_{m}

^{p}

_{n}

^{q}

_{o}

^{r}

_{x}

^{z}

#### = p × ln(*C*_{m})+ q × ln(*C*_{n})+ r × ln(*C*_{o})+… + z × ln(*C*_{x}) = (Eqn. 5a)

_{m}

_{n}

_{o}

_{x}

#### = (*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(*C*_{0})(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}⁄(*C*_{0})^{ − 1}(Eqn. 6)

_{M}

_{n}

_{(-M+n)})

#### eg., *C*_{8}⁄*C*_{7}= *π*⁄*e* = 1.155 727 350...

#### Now, using (Eqn. 6)

*C*_{8}⁄*C*_{7}= *π*⁄*e* = (*C*_{( − 8 + 7)})^{ − 1}⁄(*C*_{0})^{ − 1}=

#### = (*C*_{ − 1})^{ − 1}⁄(*C*_{0})^{ − 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)

_{M}

^{P}

_{n}

^{q}

#### = (*C*_{(( − P × M + q × n)⁄ − P + q)})^{(P − q)} = (Eqn. 7a)

#### = (*C*_{( − P × M + q × n)})^{ − 1}⁄(*C*_{0})^{( − 1 − (P − q))}(Eqn. 7b)

#### e.g., from (Eqn. 7): (*C*_{8})^{2.5}⁄(*C*_{7})^{ − 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) = (*C*_{0}) × (*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}⁄(*C*_{0})^{( − 1 − (2.5 + 0.5))}=

#### = (*C*_{ − 23.5})^{ − 1}⁄(*C*_{0})^{ − 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}⁄(*C*_{0})^{ − 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)

_{M}

^{P}

_{N}

^{Q}

_{O}

^{R}

_{X}

^{Z}

_{m}

^{p}

_{n}

^{q}

_{o}

^{r}

_{x}

^{z}

#### = (*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}⁄(*C*_{0})^{[ − 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 *C*_{0} = 0.986 976 350... seems to be of utmost importance, as if all the other constants can be calculated with this particular Constant *C*_{0} plus *C*_{8} = π and *C*_{7} = e.

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

#### Return to the Syllabus >>> 132. Syllabus of the course – “Voyage through God’s Universe (Cosmos and beyond) according to St. Hildegard von Bingen and others, with the help of Mathematics, Cosmology and Quantum Mechanics."

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