11b148fa-2f3e-44df-8a9b-42995677bfd4 29 4 5 false false false 0 3 decimal

&[DATE] &[TIME] - &[FILENAME]

Constructing Mathematics from a Single Function

x y μ e x^y ln - ≡

The exponential minus logarithm function, described by Andrzej Odrzywołek in https://arxiv.org/html/2603.21852v2

Functions

x E x 1 μ :

Exponential, Logarithm, identity, oposite, inverse, power ...

x L 11 x μ 1 μ μ :

x id x E L : 11 x 1 μ μ 1 μ μ

x op 110 μ μ x 1 μ μ :

x inv 110 μ μ x μ 1 μ :

x y pow y x L P E :

x y P 11 x μ L y 1 μ L μ 1 μ μ 1 μ :

Product, Sum, cos, sin, acos, asin

x y S x E y E P L :

x c 2 inv i x P E i x P op E S P :

x s 2 i P inv i x P E i x P op E op S P :

x ac i op x i 1 x x P op S 2 inv pow P S L P :

x as i op i x P 1 x x P op S 2 inv pow S L P :

Constants

0111 μ 1 μ μ :

i 1 op 2 inv pow op :

211 S :

π i 1 op L P :

11 :

SMath numerical implementation

x y μ e x^∞ - y 0 ≡ y ln cases - :

This fails for some zero values, but gives good results in almost cases

00

22

i 2.2204 1016 -^* i +

π 3.1416 3.1416 1015 -^* i * -

usual round-off errors

Basic functions

a b 12 mat 2312 mat :

c a b i * + :

c ln 1.2825 0.9828 i * +

c L 1.2825 0.9828 i * +

c exp 7.3151 - 1.0427 i * +

c E 7.3151 - 1.0427 i * +

c 23 i * +

c id 23 i * +

c - 2 - 3 i * -

c op 2 - 3 i * -

1 a / 0.5

a inv 0.5

a b^8

a b pow 8

a b + 5

a b S 5

a b * 6

a b P 6

Trigonometric functions

α a 12 mat 30 deg * 0.2 12 mat :

α cos 0.866

α c 0.866

α sin 0.5

α s 0.5 11015 -^* i * -

a acos 1.3694

a ac 1.3694 1.3694 1015 -^* i * +

a asin 0.2014

a as 0.2014 8.0543 1016 -^* i * -

x y μ x y ⨁ (:

showing eml as binary operator

x E x 1 ⨁

y L 11 y ⨁ (1 ⨁ (⨁

x id 11 x 1 ⨁ (⨁ (1 ⨁ (⨁

x op 11111 ⨁ (1 ⨁ (⨁ (⨁ (⨁ (x 1 ⨁ (⨁

x inv 11111 ⨁ (1 ⨁ (⨁ (⨁ (⨁ (x ⨁ (1 ⨁

x y pow 1111 y ⨁ (⨁ (1 ⨁ (⨁ (1111 x ⨁ (1 ⨁ (⨁ (1 ⨁ (⨁ (1 ⨁ (⨁ (⨁ (1 ⨁ (⨁ (1 ⨁ (1 ⨁

x y P 1111 x ⨁ (⨁ (1 ⨁ (⨁ (11 y 1 ⨁ (⨁ (1 ⨁ (⨁ (⨁ (1 ⨁ (⨁ (1 ⨁

x y S 111111 x 1 ⨁ (⨁ (⨁ (1 ⨁ (⨁ (11 y 1 ⨁ (1 ⨁ (⨁ (1 ⨁ (⨁ (⨁ (1 ⨁ (⨁ (1 ⨁ (⨁ (1 ⨁ (⨁

0 111 ⨁ (1 ⨁ (⨁

2 11111111 ⨁ (⨁ (⨁ (1 ⨁ (⨁ (1111 ⨁ (1 ⨁ (⨁ (1 ⨁ (⨁ (⨁ (1 ⨁ (⨁ (1 ⨁ (⨁ (1 ⨁ (⨁

etc ...

Alvaro

4 appVersion 1.73.9126.0