[AntonioLao]

A holomorphic function of another holomorphic function gives yet again a holomorphic function. This is a property of 2-dimensional surface needed for creating a complex loop-manifold. Moreover, a necessary and sufficient condition for all functions having continuous partial derivatives is given by the Cauchy-Riemann partial differential equations (see http://cellular.ci.ulsa.mx/comun/complex/node16.html and http://ndp.jct.ac.il/tutorials/complex/node19.html and http://www.rit.edu/~pnveme/pigf/Complex/complex_afun_3.html and http://www.astro.cf.ac.uk/undergrad/module/PX3211/com/node3.html and http://mathworld.wolfram.com/Cauchy-RiemannEquations.html and http://en.wikipedia.org/wiki/Cauchy-Riemann_equations).

Suppose a scalar function F is holomorphic in the complex plane. This is in terms of the real and imaginary part of F and the complex number z = x + iy. Its conjugate is z* = x - iy and the product z*z = x² + y². If x² + y² = 1 then a single loop is formed which defined a unit volume or unit cube if and only if the inscribed sphere is of radius ½ with volume p/6 and the circumscribed sphere is of radius Ö3/2 with volume pÖ3/2.

[mkirkpatrick]

Originally Posted by AntonioLao A holomorphic function of another holomorphic function gives yet again a holomorphic function. This is a property of 2-dimensional surface needed for creating a complex loop-manifold. Moreover, a necessary and sufficient condition for all functions having continuous partial derivatives is given by the Cauchy-Riemann partial differential equations (see http://cellular.ci.ulsa.mx/comun/complex/node16.html and http://ndp.jct.ac.il/tutorials/complex/node19.html and http://www.rit.edu/~pnveme/pigf/Complex/complex_afun_3.html and http://www.astro.cf.ac.uk/undergrad/module/PX3211/com/node3.html and http://mathworld.wolfram.com/Cauchy-RiemannEquations.html and http://en.wikipedia.org/wiki/Cauchy-Riemann_equations). Suppose a scalar function F is holomorphic in the complex plane. This is in terms of the real and imaginary part of F and the complex number z = x + iy. Its conjugate is z* = x - iy and the product z*z = x² + y². If x² + y² = 1 then a single loop is formed which defined a unit volume or unit cube if and only if the inscribed sphere is of radius ½ with volume p/6 and the circumscribed sphere is of radius Ö3/2 with volume pÖ3/2.

I like the title to the thread Antonio,mine is ,die now,and live later?It looks like these
numbers are about to add up!

kind regards michael.

[AntonioLao]

Originally Posted by mkirkpatrick die now,and live later?

I always like the idea of enjoy now, pay later or laugh now, cry later. These processes imply periodicity not reversibility.