1.3.7 Complex Curvature

The math­e­mat­i­cal re­la­tion­ship for cur­va­ture in the real-valued Eu­clid­ean plane is well es­tab­lished. Still, per­haps one might find it hard to locate a math­e­mat­i­cal frame­work for com­plex-valued func­tion­al cur­va­ture in the lit­er­a­ture. A com­plex func­tion de­not­ed $f(z)$, has for its ar­gu­ment $z = x + iy,$for $x,y\in \mathbb{R}$. To prop­er­ly con­sid­er com­plex cur­va­ture, it is im­por­tant first to prove for $z = (x + iy)$, that

We will use the Cauchy-Rie­mann equa­tions to help prove this, which rep­re­sent a set of two par­tial dif­fer­en­tial equa­tions. Taken to­geth­er, the Cauchy-Rie­mann equa­tions ensure that a com­plex func­tion is holo­mor­phic, mean­ing it is dif­fer­en­tiable in the domain of com­plex space. For a com­plex func­tion $f(z) = u(z) + iv(z)$, and $z = x + iy$, the Cauchy-Rie­mann equa­tions are the fol­low­ing

For the func­tion $f(z)$, $u$ rep­re­sents the real part, and $v$ rep­re­sents the imag­i­nary part, and the Cauchy-Rie­mann equa­tions take the com­plex form

Each de­riv­a­tive is de­fined as a limit in the form

and as­sum­ing the limit for $h \rightarrow 0$ exists, the Cauchy-Rie­mann equa­tions detail if the result ap­pears along the real or imag­i­nary-axis. To apply the equa­tions using an ex­am­ple, we define a func­tion $f(z)$, as

and cal­cu­late its dif­fer­en­tial func­tions

find­ing the Cauchy-Rie­mann con­di­tions are met, given that

With this back­ground per­tain­ing the equa­tions, we can now prove that $\frac{d}{dz}\left( e^{z} \right) = e^{z}$.

The­o­rem (1.3.199) – The func­tion $f(z) = e^{z}$, for $z = (x + iy),$ $\left\{ x,y \right\}\in \mathbb{R}$, is a holo­mor­phic func­tion, and as such, is com­plex-dif­fer­en­tiable in its whole domain $\in \mathbb{C}$, with a dif­fer­en­tial, $\frac{d}{dz}\left( e^{z} \right) = e^{z}.$

Proof – We use the Cauchy-Rie­mann equa­tions to prove $f(z) = e^{z}$ is com­plex-dif­fer­en­tiable at all points in its domain $\in \mathbb{C}$, and fur­ther, is con­tin­u­ous­ly dif­fer­en­tiable. Writ­ing $e^{z}$, as

and by using the com­plex form of the Cauchy-Rie­mann equa­tions

we find the par­tial de­riv­a­tives

Plug­ging these into the Cauchy-Rie­mann form, we find they are sat­is­fied

Since $e^{x}e^{iy}$ is the com­po­si­tion of con­tin­u­ous trigono­met­ric func­tions, for $e^{x}e^{iy} = \left( \cosh(x) + \sinh(x) \right)\left( \cos(y) + isin(y) \right)$, we have a com­plex func­tion that is in­fi­nite­ly dif­fer­en­tiable at every point for all values $z = (x + iy)$ of the open subset of. To de­ter­mine the ex­is­tence and value of the com­plex de­riv­a­tive $f(z) = e^{z}$, we define it as the limit

which can be taken along either the imag­i­nary or real-axis

Using the limit along the imag­i­nary-axis, we apply l'Hôpi­tal's rule to $e^{z} = e^{x + iy}$

Since $e^{(x + iy + ih)}$ is an ex­po­nen­tial func­tion and con­tin­u­ous, we set $h = 0$, to arrive at our com­plex de­riv­a­tive

To con­sid­er com­plex cur­va­ture, we start with the ex­am­ple of a com­plex func­tion $f(z) = e^{z}$, for $z = x + iy$, whose tan­gent vector $T$ is cal­cu­lat­ed, as

Thus, from the above, we can see that the ar­gu­ment of the tan­gent vector $T$ is the imag­i­nary por­tion of the ar­gu­ment ${(e}^{z})'$

When con­sid­er­ing the cur­va­ture of a com­plex func­tion, we are look­ing to find the rate of ro­ta­tion of the tan­gent vector con­cern­ing the dis­tance along the curve. To find the arc, we would take the imag­i­nary por­tion of the de­riv­a­tive for the ar­gu­ment $f'(z)$. Let

and the rate of change for $\phi$, would be

As the tan­gent vector $T$ moves with unit speed, the prod­uct of the tan­gent unit vector $T$ and $\phi^{'}$, gives us the ver­ti­cal arc through a point that is one-unit length from the center of cur­va­ture, re­sult­ing in

As $T$ trav­els the com­plex curve with unit speed, $f(z)$ also trav­els the image with a speed $\left| f^{'}(z) \right|$. This ab­solute value will yield the real-valued por­tion of the com­plex func­tion and should be con­sid­ered as the in­verse radius of the func­tion. Di­vid­ing Eq. 1.3.204 by $\left| f^{'}(z) \right|$, we arrive at the fol­low­ing equa­tion of cur­va­ture $(\kappa)$

Need­ham¹ goes fur­ther, defin­ing the com­plex cur­va­ture of $f(z)$, as

where $(f)^{*}$ de­notes the com­plex con­ju­gate of $f$, and this ex­pres­sion il­lu­mi­nates that the tan­gent vector $T$ is not in­trin­sic to the com­plex func­tion $f(z)$, and $T$ re­tains the el­e­ment of di­rec­tion from the orig­i­nal curve. Need­ham argues that com­plex cur­va­ture is a very nat­ur­al math­e­mat­i­cal quan­ti­ty when thought of as a point $p$, from which vec­tors em­anate and project onto a line. The pro­jec­tion of $\mathcal{K}(p)$ onto a line through $p$ is the cur­va­ture of the image of that line at $f(p).$Now, $\mathcal{K}(p)$ can be re­lat­ed to $\kappa(T)$ by taking the imag­i­nary por­tion of $i\mathcal{K}^{*}T$, for $\mathcal{K} \bullet\mathbf{T}$, as the scalar prod­uct in $\mathbb{R}^{2}$

With a pre­cise math­e­mat­i­cal de­f­i­n­i­tion for com­plex cur­va­ture, we now take the ex­am­ple $e^{z}$, for $z = X + i\theta$. As pre­vi­ous­ly proven, we know that if $f(z) = e^{z},$then $f^{'}(z) = e^{z}$, and cal­cu­lat­ing $\kappa(T)$ first, shows that

Since $f^{'}(z) = \left( e^{z} \right)^{'} = e^{z}$, we know that $\left( e^{X + i\theta} \right)^{'} = \left( e^{X + i\theta} \right)$ , and $\kappa(T)$ above, be­comes

Eval­u­at­ing the com­plex cur­va­ture of $e^{X + i\theta}$, we would find

and it is easy to show that $\mathcal{K} \bullet\mathbf{T} = \kappa(T)$

The pre­ferred ap­proach to con­tem­plat­ing com­plex cur­va­ture starts by taking the com­plex func­tion in its polar form

Where in­tu­itive­ly, it is ap­par­ent that the seg­ment length re­lates to the radial com­po­nent $r(t),$which is then ro­tat­ed by $e^{i\phi(t)}$, in re­la­tion to the real-axis of the com­plex plane. Since cur­va­ture is de­fined as the rate of change for a curve, com­pared to the change in the dis­tance at a point on that curve, the signed cur­va­ture of $f(z)$ is

which sim­pli­fies to

In the next sec­tion, we apply cur­va­ture to func­tions in the com­plex plane by using ex­am­ples from the as­ymp­tot­ic number func­tions re­lat­ed to the $D_{3\mathbb{c}}$ group in Chap­ter 1.

1. Need­ham, Tris­tan, "Visual Com­plex Analy­sis", Oxford Uni­ver­si­ty Press, 1997