Previous result was:
h(Omega^(pi/2)*Omega^ (-I/Omega)) = h((Omega^((1-I)/Omega))*(I^(1/I))=(-Omega+(I*(pi/2))/(pi/(2*Omega)+I)
h(((e^(pi/2))^(1/Omega))*(e^I)) =Omega*I=-I*0,56714329
Previosly here was damped harmonic oscillator. This time it has dissappeared, but will reappear later as undamped if W is real:
Let us take the formula:
W(-x*(pi/2) - I*x*ln(x) ) = ln(x) - I*(pi/2) for x>1
And substitute (1/W(y)) = x>1 ,so y<e for the time being. Than:
W(-(1/W(y))*(pi/2)- I* (1/W(y))*ln(1/W(y))) = ln (1/W(y))-I*pi/2
further, I only will write W instead of W (y):
W( -(1/W)*(pi/2) + I *(lnW)/W)=-lnW - I*pi/2
this means ln (arg h ) = (1/W)*pi/2-I*(lnW)/W and
h( e^((pi/(2*W))*e^(-I*lnW/W)) = (lnW+I*pi/2)/ ((1/W)*(pi/2) - I *(lnW)/W) which leads to :
h( e^((pi/(2*W))*e^(-I*lnW/W)) = I*(W+ W*pi^2/(4* (lnW)^2))/ ((pi^2/(4*(lnW)^2))+1)
by denoting
m = 1/((pi^2/4*(lnW)^2+1)
k= -(W^2*pi^2)/(4*(lnW)^2)*(1/((pi^2/4*(lnW)^2+1))=(W^2*pi^2)/(4*(lnW)^2)*m
k/m =-(W^2*pi^2)/(4*(lnW)^2)
The result above represents linear response function of undamped harmonic oscillator with equation:
y''+k/m*y=0
h(e^((pi/(2*W))*e^(-lnW/W) = Z= I*(W*m-k/W) which has imaginary resonance frequency (for W real and >1):
w rez =sqrt(k/m) = I*W*pi/(2*lnW) = I*pi/2*(log base W of (z)-1)
because from definition of W function, lnz=lnW+W; W=lnz-lnW; W/lnW = lnz/lnW-1 = log base W of z-1
f rez = (I/4)*(W/lnW)
So when y=1, W(1) = omega, w rez = (I *Omega*pi)/(2 *ln Omega) = - I*pi/2, f=I/4 - consistent with results obtained earlier. For other y, W(y), ln W(y) , k, m, w rez, Zm etc. will be different.
If W= I , w rez =sqrt(-(I^2*pi^2)/(4*(I*pi/2)^2)) = sqrt( -1) = +- I, f= (+-I/2*pi)
As we can see, both m and K are functions of "frequency" W, so this seems to be rather strange system.
Impedance module Zm =((W+ W*pi^2/(4* (lnW)^2))/ ((pi^2/(4*(lnW)^2))+1))
Phase arctan((W+ W*pi^2/(4* (lnW)^2))/ ((pi^2/(4*(lnW)^2))+1)).
Output of this system if driven by Fo*cos(Wt):
y=Fo/[W*((-W- W*pi^2/(4* (lnW)^2))/ ((pi^2/(4*(lnW)^2))+1))]*sin(Wt-arctan(Z/I))
I hope this is more correct than previous attempt.
Interesting? What would be oscillating in number world?
Little more play with rezonance frequency:
w rez =sqrt(k/m) = I*W*pi/(2*lnW) = I/2*pi*W/ln(W)
Since I/2= sinh (ln(phi)) where phi is golden ratio:
w rez = sinh(ln(phi) * pi* (W/lnW)
If y= 1, W(1) = Omega, ln(W(1))= - Omega and W/lnW=-1
w rez= -sinh (ln(phi))* pi = - I*pi/2
Ivars
h(Omega^(pi/2)*Omega^ (-I/Omega)) = h((Omega^((1-I)/Omega))*(I^(1/I))=(-Omega+(I*(pi/2))/(pi/(2*Omega)+I)
h(((e^(pi/2))^(1/Omega))*(e^I)) =Omega*I=-I*0,56714329
Previosly here was damped harmonic oscillator. This time it has dissappeared, but will reappear later as undamped if W is real:
Let us take the formula:
W(-x*(pi/2) - I*x*ln(x) ) = ln(x) - I*(pi/2) for x>1
And substitute (1/W(y)) = x>1 ,so y<e for the time being. Than:
W(-(1/W(y))*(pi/2)- I* (1/W(y))*ln(1/W(y))) = ln (1/W(y))-I*pi/2
further, I only will write W instead of W (y):
W( -(1/W)*(pi/2) + I *(lnW)/W)=-lnW - I*pi/2
this means ln (arg h ) = (1/W)*pi/2-I*(lnW)/W and
h( e^((pi/(2*W))*e^(-I*lnW/W)) = (lnW+I*pi/2)/ ((1/W)*(pi/2) - I *(lnW)/W) which leads to :
h( e^((pi/(2*W))*e^(-I*lnW/W)) = I*(W+ W*pi^2/(4* (lnW)^2))/ ((pi^2/(4*(lnW)^2))+1)
by denoting
m = 1/((pi^2/4*(lnW)^2+1)
k= -(W^2*pi^2)/(4*(lnW)^2)*(1/((pi^2/4*(lnW)^2+1))=(W^2*pi^2)/(4*(lnW)^2)*m
k/m =-(W^2*pi^2)/(4*(lnW)^2)
The result above represents linear response function of undamped harmonic oscillator with equation:
y''+k/m*y=0
h(e^((pi/(2*W))*e^(-lnW/W) = Z= I*(W*m-k/W) which has imaginary resonance frequency (for W real and >1):
w rez =sqrt(k/m) = I*W*pi/(2*lnW) = I*pi/2*(log base W of (z)-1)
because from definition of W function, lnz=lnW+W; W=lnz-lnW; W/lnW = lnz/lnW-1 = log base W of z-1
f rez = (I/4)*(W/lnW)
So when y=1, W(1) = omega, w rez = (I *Omega*pi)/(2 *ln Omega) = - I*pi/2, f=I/4 - consistent with results obtained earlier. For other y, W(y), ln W(y) , k, m, w rez, Zm etc. will be different.
If W= I , w rez =sqrt(-(I^2*pi^2)/(4*(I*pi/2)^2)) = sqrt( -1) = +- I, f= (+-I/2*pi)
As we can see, both m and K are functions of "frequency" W, so this seems to be rather strange system.
Impedance module Zm =((W+ W*pi^2/(4* (lnW)^2))/ ((pi^2/(4*(lnW)^2))+1))
Phase arctan((W+ W*pi^2/(4* (lnW)^2))/ ((pi^2/(4*(lnW)^2))+1)).
Output of this system if driven by Fo*cos(Wt):
y=Fo/[W*((-W- W*pi^2/(4* (lnW)^2))/ ((pi^2/(4*(lnW)^2))+1))]*sin(Wt-arctan(Z/I))
I hope this is more correct than previous attempt.
Interesting? What would be oscillating in number world?
Little more play with rezonance frequency:
w rez =sqrt(k/m) = I*W*pi/(2*lnW) = I/2*pi*W/ln(W)
Since I/2= sinh (ln(phi)) where phi is golden ratio:
w rez = sinh(ln(phi) * pi* (W/lnW)
If y= 1, W(1) = Omega, ln(W(1))= - Omega and W/lnW=-1
w rez= -sinh (ln(phi))* pi = - I*pi/2
Ivars
