Suppose that I am given ${\tilde U_1,\ldots,\tilde U_N}$ as a sequence of numbers, and in addition, $U_1,\ldots,U_N$ is unknown, and $q$ is unknown and constant for all ${\tilde U_1,\ldots,\tilde U_N}$.
$\tilde U_1 = U_1 \exp (f(q, 1))$
$\tilde U_2 = U_2 \exp (f(q,2))$
$\tilde U_3 = U_3 \exp (f(q,3))$
The sequence continues up to:
$\tilde U_N = U_N \exp (f(q,N))$
Is there any numerical method or way to check and see if the exponential function has "disappeared", without knowing $f(q,p)$, but knowing that q is constant?
Suppose that for $ p = 1,...,N$:
${{\tilde U}_p} = {U_p}\exp ({k_{A,p}} + {ik_{B,p}})\exp \left[ { - \frac{{{\omega _p}}}{{2q}}\left[ {{{\left( {\frac{{{\omega _p}}}{{{\omega _h}}}} \right)}^{ - 1/\pi q}}} \right] - \left[ {\left[ {{{\left( {\frac{{{\omega _p}}}{{{\omega _h}}}} \right)}^{1/\pi q}} - 1} \right]{\omega _p}} \right]i} \right]$
The goal is to find a $q$ that will make $f(q,p) = 0$ without knowing $k_{A,p}$ or $k_{B,p}$, but knowing that $k_{A,p}$ and $k_{B,p}$ are positive.
${{\tilde U}_1} = {U_1}\exp ({k_{A,1}} + {ik_{B,1}})\exp \left[ { - \frac{{{\omega _1}}}{{2q}}\left[ {{{\left( {\frac{{{\omega _1}}}{{{\omega _h}}}} \right)}^{ - 1/\pi q}}} \right] - \left[ {\left[ {{{\left( {\frac{{{\omega _1}}}{{{\omega _h}}}} \right)}^{1/\pi q}} - 1} \right]{\omega _1}} \right]i} \right]$
${{\tilde U}_2} = {U_2}\exp ({k_{A,2}} + {ik_{B,2}})\exp \left[ { - \frac{{{\omega _2}}}{{2q}}\left[ {{{\left( {\frac{{{\omega _2}}}{{{\omega _h}}}} \right)}^{ - 1/\pi q}}} \right] - \left[ {\left[ {{{\left( {\frac{{{\omega _2}}}{{{\omega _h}}}} \right)}^{1/\pi q}} - 1} \right]{\omega _2}} \right]i} \right]$
${{\tilde U}_3} = {U_3}\exp ({k_{A,3}} + {ik_{B,3}})\exp \left[ { - \frac{{{\omega _3}}}{{2q}}\left[ {{{\left( {\frac{{{\omega _3}}}{{{\omega _h}}}} \right)}^{ - 1/\pi q}}} \right] - \left[ {\left[ {{{\left( {\frac{{{\omega _3}}}{{{\omega _h}}}} \right)}^{1/\pi q}} - 1} \right]{\omega _3}} \right]i} \right]$
This means that I am searching for a q such that the sequence above becomes:
${{\tilde U}_1} = {U_1}$
${{\tilde U}_2} = {U_2}$
${{\tilde U}_3} = {U_3}$
All that I know is the LHS, and I know that $q$ is constant on the RHS. I don't know $k_{A,N}, k_{B,N}$, $q$ and $U_1,\ldots,U_N$, but I do know $\omega_h$ as a constant and $\omega_p$ that changes for each element of the sequence. In the above, $i$ represents a complex number. In addition, $k_{A,p}, k_{B,p}$ are positive numbers.
Is there a way to check for the presence or absence of the exponential function in the sequence, and in doing so, determine $q$, which is constant for the entire sequence? Is there anything that I can do or change to get an approximation of $q$? Why or why not?
I suppose that the exponential function is still present in the expression, but I would like to make $\exp(f(q,p)) = 1$ for $p = 1,...,N$.