Clock Face And Life, Two Pies And Habits

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.