Instead of,
\(\cfrac{df(z)}{dz}\) or \(\int{f(z)}dz\)
we do,
\(\cfrac{df(z)}{d(e^{-i\theta})}\) or \(\int{f(z)}d(e^{-i\theta})\)
or equivalently,
\(\cfrac{df(z)}{d(-\hat{z})}\) or \(\int{f(z)}d(-\hat{z})\)
where \(-\hat{z}\) denote the unit vector opposite to the \(z\) direction.
This way \(-d\hat{z}\) or \(d(e^{-i\theta})\) is opposite to the direction of \(z\); at an angle \(-\theta\) to the \(x\) axis on the \(i-x\) plane. Furthermore,
\(\int { f(z)d(e^{ -i\theta }) }=- i\int { f(z)e^{ -i\theta } } d\theta \)
and
\( \cfrac { df(z) }{ de^{- i\theta } } =-\cfrac { 1 }{ ie^{ -i\theta } } \cfrac { df(z) }{ d\theta } \)
In the famous counter example to holomorphic function, \(f(z)=\bar z\)
\( -\cfrac { 1 }{ ie^{ -i\theta } } \cfrac { d\bar { z } }{ d\theta } =-\cfrac { 1 }{ ie^{ -i\theta } }\left\{R\cfrac { de^{ -i\theta } }{ d\theta }+e^{ -i\theta }\cfrac { d\,R}{ d\theta } \right\} =-\cfrac { 1 }{ ie^{- i\theta } } \left\{Re^{ -i\theta }.(-i)+e^{ -i\theta }\cfrac { d\,R}{ d\theta }\right\}=R+i\cfrac{d\,R}{d\theta}\)
this result is independent of \(\theta\) when \(R\) is a constant (a circle) or has a linear dependence on \(\theta\) (ie \(\small \cfrac{d\,R}{d\theta}=constant\), a linear spiral).
Furthermore,
\(\int { f(z)d(e^{- i\theta }) }= f(z)e^{-i\theta}-\int{e^{-i\theta}}df(z)=- i\int { f(z)e^{ -i\theta } } d\theta\)
when we have,
\(\theta=\omega t\) and \(z=R(\omega t)e^{-i\omega t}\)
\(f(z)=g(\omega t)\)
\(\int { f(z)d(e^{ -i\theta }) }= -i\int { g(\omega t)e^{- i\theta } } d\theta=-i\int{g(\omega t)e^{-i\omega t}}.td\omega\)
In fact,
\(\int ^{\infty}_{-\infty}{ f(z)d(e^{- i\theta }) }=-it\int^{\infty}_{-\infty}{g(\omega t)e^{-i\omega t}}d\omega=-it.G(t)\)
we have Fourier Transform on the RHS of the expression,
\(-it.G(t)\)
\(t\) being independent of \(\omega\), \(G(t)\) is the Fourier transform of \(g(\omega t)\) or \(f(z)\) with parameter \(t\), and the \(-i\) factor returns us to the positive \(\theta=0\) direction.
On the LHS, we have \(f(z)\) integrated around a unit circle \(e^{-i\theta}\), for all values of \(\theta\) from \(-\infty\) to \(\infty\).
As we go around in circles with \(f(z)\), we travel linearly through time \(t\) by \(G(t)\).
\(f(z)\) should be called the clock function, and \(G(t)\) our life function.
\(g(\omega t)\) is our habits, the sum of which in all magnitudes of frequencies is our life, \(G(t)\). \(g(\omega t)\) is the habit function.
Note: After the correction \(e^{i\theta}\to e^{-i\theta}\), life is no longer negative.
