Based on what I have been writing all along I have a tiny idea that something like that expressions:
\( e^{iwt*e^{-iwt}} \) in case of e.g. \( I^{1/I} = e^{pi/2} \) must be a "time" oscillator. Obviously non-linear.
When You try to solve any differential equation for a linear component \( w \) of Fourier spectrum, You always get:
Advanced solution : \( e^{-Iwt} \)
Retarded solution: \( e^{Iwt} \)
We know, \( e^{iwt} = \cos wt + I*sin wt \) , is a wave solution of a single infinite Fourier component with frequency w. Replacing t with -t just changes direction of time, so usually if there is no damping, and system is symmetric in time, one of solutions is enough.
Usually the full solution of the oscillator differential equation is the linear combination of both particular solutions: \( A*e^{-Iwt} + B*e^{Iwt} \).
The combination of particular solutions \( A*e^{Iwt}*B*e^{-Iwt} =A*B \) is not very useful since it is independent of time.
However, we can proceed further: if each of \( A*e^{Iwt}, B*e^{-Iwt} \) are solutions of some differential equation, what is the physical process and what is its differential or functional equation whose solution is exponential combination of particular solutions:
\( A*e^{{Iwt}* B*e^{-Iwt}} \) or
\( B*e^{{-Iwt}* A*e^{Iwt}} \)
which is also at the same time log Poisson type expression?
Here we exponate advanced time solution in retarded, or vice versa:
\( a^{(b*a^-b} \) or \( a^{-b*a^b} \)
Without doubt, it is an nonlinear oscillator since if we write the full Euler formula:
\( e^{Iwt*e^{-Iwt}} = (\cos wt+I*\sin wt)^{(\cos wt-I*\sin wt)} \)
It is a power combination of 2 complex helical sinusoidal waves of same frequency. Question: Solution of WHAT physical process and corresponding differential or functional equation it is?
I have no doubt it is related to delay differential equations, but I have to learn about them more.
Of course, this plugs DIRECTLY into tetration since it is a small power tower which can be extended.
Also obviously we can use instead of \( iwt \) imaginary part of tree function \( T(z) \). It than connects this solution to infinite tetration. Also, as another approach, we can create new variable that is somehow normed vs. infinite tetration so that:
\( p(z, a) = (z[4]a)/{h(z)} \)
Also, we can create new function \( (W_k(-ln z)* T_k(ln z)) / -(ln_k ( z)) ^2 = (h_k(z))^2 \)
In this case, the conjugate values of h(z) then would be 2 values of square root, a geometric mean between 2 expressions for h:
\( h_k(z)= +- sqrt((W_k(-ln z)* T_k(ln z))) / I*ln_k ( z) \)
Also, using Wright \( \omega \) function one can similarly define \( \tau(z) = T_{K(z)}(e^z) \)
I wonder if all k are the same? Corless has written that \( W_k(z) \) is an analytic ( in most part of k-plane) function of k where k can be any complex number. Is then \( T_k(z) \) also analytic in the same k region? Logarithm? \( (h_k(z))^2 \)?
More later when I have understood what I have written
Best regards,
\( e^{iwt*e^{-iwt}} \) in case of e.g. \( I^{1/I} = e^{pi/2} \) must be a "time" oscillator. Obviously non-linear.
When You try to solve any differential equation for a linear component \( w \) of Fourier spectrum, You always get:
Advanced solution : \( e^{-Iwt} \)
Retarded solution: \( e^{Iwt} \)
We know, \( e^{iwt} = \cos wt + I*sin wt \) , is a wave solution of a single infinite Fourier component with frequency w. Replacing t with -t just changes direction of time, so usually if there is no damping, and system is symmetric in time, one of solutions is enough.
Usually the full solution of the oscillator differential equation is the linear combination of both particular solutions: \( A*e^{-Iwt} + B*e^{Iwt} \).
The combination of particular solutions \( A*e^{Iwt}*B*e^{-Iwt} =A*B \) is not very useful since it is independent of time.
However, we can proceed further: if each of \( A*e^{Iwt}, B*e^{-Iwt} \) are solutions of some differential equation, what is the physical process and what is its differential or functional equation whose solution is exponential combination of particular solutions:
\( A*e^{{Iwt}* B*e^{-Iwt}} \) or
\( B*e^{{-Iwt}* A*e^{Iwt}} \)
which is also at the same time log Poisson type expression?
Here we exponate advanced time solution in retarded, or vice versa:
\( a^{(b*a^-b} \) or \( a^{-b*a^b} \)
Without doubt, it is an nonlinear oscillator since if we write the full Euler formula:
\( e^{Iwt*e^{-Iwt}} = (\cos wt+I*\sin wt)^{(\cos wt-I*\sin wt)} \)
It is a power combination of 2 complex helical sinusoidal waves of same frequency. Question: Solution of WHAT physical process and corresponding differential or functional equation it is?
I have no doubt it is related to delay differential equations, but I have to learn about them more.
Of course, this plugs DIRECTLY into tetration since it is a small power tower which can be extended.
Also obviously we can use instead of \( iwt \) imaginary part of tree function \( T(z) \). It than connects this solution to infinite tetration. Also, as another approach, we can create new variable that is somehow normed vs. infinite tetration so that:
\( p(z, a) = (z[4]a)/{h(z)} \)
Also, we can create new function \( (W_k(-ln z)* T_k(ln z)) / -(ln_k ( z)) ^2 = (h_k(z))^2 \)
In this case, the conjugate values of h(z) then would be 2 values of square root, a geometric mean between 2 expressions for h:
\( h_k(z)= +- sqrt((W_k(-ln z)* T_k(ln z))) / I*ln_k ( z) \)
Also, using Wright \( \omega \) function one can similarly define \( \tau(z) = T_{K(z)}(e^z) \)
I wonder if all k are the same? Corless has written that \( W_k(z) \) is an analytic ( in most part of k-plane) function of k where k can be any complex number. Is then \( T_k(z) \) also analytic in the same k region? Logarithm? \( (h_k(z))^2 \)?
More later when I have understood what I have written
Best regards,