<p> Golden ratio min/max </p>

2f239e41-f252-4640-9d7f-9edf42a16ab1 64 4 5 decimal

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Hypersinecosine . . . two styles

1. Exponential form

2. Phasor style

Latest version: 20221104 . . . from original jmG

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b k x Ω b k x *^:

q a b x f 12 / a - (b π / x q + (*^* 12 / a - (b π / x q + (*^Conjugate * + :

b 1.25 : k 0.9 : 12 mat

b k x Ω b k x *^: b - k 1.25 Ω 21 sys

x plot b k x Ω b k x Ω - b - k x Ω Re b - k x Ω Im axes 51 sys :

x y axes x y / - :

b - k 2.5 Ω b - k 1.25 - Ω 21 sys 1.1682 1.1682 i * + 0.7188 - 0.2977 i * + 21 sys

x plot

Hypersine ... thread wisely data export

reduced form Hypersine b k x Ω b k x *^:

X_Y defecates 
those ' ± k' 
Tip: set points
10*2range [180] k 0.9 0.8 0.7 0.6 41 sys ≡

b 1.25 : k 0.9 : 12 mat

# f Q i 1 U rows range vx i el U i el : vy i el U i el f eval : 21 line for vx vy augment 21 line :

x f b - k x Ω Re :

x Φ b - k x Ω Im :

Sanity 'N' ► plot lines.

L H N xd U 0 : dx H L - N / : 12 mat i 1 N range U i el L dx - (dx i * + eval : for U 31 line :

L 7 - : H 5 : N 997 : 13 mat

U L H N xd :

Redata x f Q : Imdata x Φ Q : 21 sys

Redata Imdata 2 1 sys

Solving the point(s) of tangency ...

<p> Golden ratio min/max </p>

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / 71 line :

a b + 2 / iter 21 mat

reduced form Hypersine b k x Ω b k x *^:

b 1.5 : k 0.75 0.75 - 21 sys : 12 mat

j 2 :

x plot b k j el x Ω b k j el x Ω - b - k j el x Ω Re b - k j el x Ω Im 41 sys :

y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ 21 sys

x f b - k j el x Ω Re :

ε 1015 -^:

X f 23 y1 y2 ≥ ε Extrema f 56 y1 y2 ≥ ε Extrema 21 mat :

x f

X 1 el f X 2 el f 21 mat 0.4482 0.1992 21 mat

X 2.612191512 5.27885818 21 mat

εopt 0.0544752 :

X 2 el X 2 el εopt + :

X 1 el X 1 el εopt + :

b k j el X 1 el Ω b - k j el X 1 el Ω Re - 2.04 1015 -^*

b k j el X 2 el Ω b - k j el X 2 el Ω Re - 9.9 1016 -^*

εopt can likely be automated ... RemToDo

x plot X 1 el X 1 el f + 30 black 1 5 mat X 2 el X 2 el f + 30 black 1 5 mat 3 1 sys

Hypersine phasor style

12 / a - (b π / x q + (*^* 12 / a - (b π / x q + (*^Conjugate * + a - (b x q + (* π /^≡

The very original complex  Hypersine jmG ... q a b x f 12 / a - (b π / x q + (*^* 12 / a - (b π / x q + (*^Conjugate * + :

x λ 10 x ≤ (x 3 π * ≤ (& cases :

a 0.75 : b 2.5 : q π 1 / - : 13 mat

cplx q a b 1.5 f : 0.8323 - 1.1949 i * +

x Δ q a b x - f Im q a b x - f Re - :

solve range depends upon 'q' & region X x Δ x 56 solve : 5.3748

Y q a b X - f Re q a b X f Re + 2 / : 2.4571 -

Experiment the phase differential u 2.25 :

explorer tracking blue q a b u - f Re q a b u f Re + 2 / : 0.6758

explorer tracking red q a b u - f Im q a b u f Im + 2 / : 1.8442 -

q a b x - f Re x λ * q a b x - f Im x λ * X Y + 20 black 1 5 mat u blue . 14 black 1 5 mat u red . 14 black 1 5 mat u 3 u 3 - 2 2 mat 6 1 sys

Differential measure D red blue > red blue - blue red > red blue - if if :

D 2.5200 -

x Δ q a b x - f Re q a b x - f Im - :

Algebraic Diffeential ...

x Δ -

More maths ... for convenience

reduced form Hypersine b k x Ω b k x *^:

b 1.25 : k 0.9 : 12 mat

x f b - k x Ω Re :

typical THIS demo h 0.0001 :

x h f D1 x 2 h * + f - 8 x h + f * + 8 x h - f * - x 2 h * - f + 12 h * / :

x f 0.5 x h f D1 * 2 1 sys

x f x 01 int 0.079

x h f D1 x 01 int 2.163 -

0 h f D1 0.2008

x f x 3 - 3 solve 2.7778 - 1.6667 - 0.5556 - 0.5556 1.6667 2.7778 61 mat

x h f D1 x 3 - 3 solve 2.197143 - 1.086032 - 0.025079 1.13619 2.247301 51 mat

sol x h f D1 x 0.024 0.026 solve : 0.025079179618866

sol h f D1 8.48 1012 -^* -

The Δ(x) ► phase explorer ...

start 0 : end 9 : 12 mat

x f x Δ :

t start end start 0.1 + (range :

i 1 t rows range X i el t i el eval : for

i 1 t rows range Y i el t i el f eval : for

XY t Y augment :

N XY rows : 91

step 1 N range :

t x X t el : t y Y t el : 12 mat

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

xy XY . 9 red plot :

piecewise explorer J 41 :

Cumulative explorer ... over Δ(x) phase differential

t 1 ≡ path t x t y + 30 red 1 5 mat : path path t x t y . 9 black 1 5 mat stack : if xy X J el Y J el + 20 blue 1 5 mat 3 1 sys

piecewise explorer ... X J el Y J el 12 mat 4 1.3248 - 12 mat

More conjugate operator ...

12 / 1 x - ln * - 12 / 1 x - ln Conjugate * - 1 x - ln - ≡

Polylogarithm [Jonquière function] x polylog 12 / 1 x - ln * - 12 / 1 x - ln Conjugate * - :

x polylog 1 2 / 1 x - ln Conjugate * 1 x - ln -